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My Favorite Theorem

Join us as we spend each episode talking with a mathematical professional about their favorite result.

94 avsnitt • Längd: 30 min • Månadsvis

Om podden

Join us as we spend each episode talking with a mathematical professional about their favorite result. And since the best things in life come in pairs, find out what our guest thinks pairs best with their theorem.

The podcast My Favorite Theorem is created by Kevin Knudson & Evelyn Lamb. The podcast and the artwork on this page are embedded on this page using the public podcast feed (RSS).

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Episode 93 - Robin Wilson

2 december 2024 | 22 min

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, and I am joined, as always, by our other host. Will you introduce yourself?

Kevin Knudson: Hi, I’m Kevin Knudson. Yeah, I will. I'm Kevin Knudson, professor of mathematics at the University of Florida. It's been a while.

EL: Yeah.

KK: You know, I've actually gotten a few emails from our listeners saying, hey, where, where the hell is My Favorite Theorem? And I always have to reply, you know, we're trying, but everybody's busy.

EL: Yeah, and we're busy.

KK: And we're busy. But here we are. We are still committed. We're still into this. This is — we're going to go to year eight soon enough, which is kind of mind-blowing.

EL: Yes.

KK: I had less gray and more hair in those days. So here we are.

EL: You’re as lovely as ever.

KK: As are you, Evelyn, as are you. Yeah, although I kind of missed the green hair. I wish you would start coloring your hair again.

EL: Yeah. Honestly, like many people, during COVID, I just lost my ability to put forth more than minimal effort into my appearance.

KK: Yeah.

EL: Sorry, that sounds like a smear on other people. I just, I've heard this from other people. I'm not judging based on what I see from other people. But yes, it's just, like, the bleaching, the dyeing, it just, it's more maintenance than I'm willing to put forth right now.

KK: It’s a whole thing. And as one gets older, you just go, who cares?

EL: Yes. But anyway, we are delighted today to have Robin Wilson on the show. Robin, will you introduce yourself? Tell us where you're joining us from, and a little bit about yourself?

Robin Wilson: Yeah, hi everyone. So I am joining you from Los Angeles, currently in my office at Loyola Marymount University, where I'm a professor in the math department. And so I have been a professor since about 2007 and work in topology and math education. It's great to be here today.

KK: What part of town is Loyola in? I don't think I actually know where that is.

RW: Loyola Marymount is right on the coast, near LAX. So if you've ever visited us here and flown into that airport, then you've flown right over campus, and depending on which runway you land, you can actually, like, see the fountain.

KK: Okay. I'll be flying through LAX in December. I will try to take a look.

RW: Come say hello, yeah.

EL: And I have to say, if we were broadcasting a video of this, you have one of the best backgrounds, the beautiful bookshelf, and then the chalkboard behind you has the appropriate level of mathiness. So our listeners can't enjoy that. But I will say that, you know, it looks very math professor.

KK: It does. It could actually go in that book of math chalkboards. Have you seen this? What's it called? I can't remember. Anyway.

RW: I’m so honored.

EL: I think it’s something like Please Do Not Erase, or something. [Editor’s note: Very close! It’s just Do Not Erase by Jessica Wynne.]

KK: Please Do Not Erase. That’s right, yeah.

EL: But yes, we are so happy to have you here. And Robin and I have actually worked together before a few years ago, on a publication for — at at the time, known as MSRI. It has a new name now, the SL Math Institute now, but we worked together putting together a booklet for the math and racial justice workshops that they did in summer 2021, was it? And it was such a delight to work with you, and I'm glad that lo these many years later, we can get to chat about your favorite theorem. So what have you chosen for your favorite theorem today?

RW: Okay, so the theorem that I've chosen for my favorite theorem today, this was a tough one, and so I chose one that was sort of important for me on my journey. So the Poincare-Hopf index theorem.

KK: Oh, I love this theorem.

EL: All right!

RW: So should I tell you more about theorem?

KK: Please.

EL: Please.

RW: Okay, so the theorem, so I'll state the two-dimensional version of the theorem, which is the one that I can kind of see. So you take a surface and you add a smooth vector field on the surface, and there's an extra condition that the vector field has only finitely many critical points. And so it turns out that the sum of the indices of the critical points of the vector field is equal to the Euler characteristic of the surface. And so the theorem connects these two different areas of math, sort of analytical and topology. And so it was one that I encountered kind of at the beginning of my journey as a topologist. And so it's the one that I picked to share today.

EL: I love that, and I've got to say, I love how many people on our podcast come in and say, it connects this and this. And like mathematicians, we just love these theorems that connect, you know, calculus and topology, or, you know, algebraic geometry and topology, or something like that. It's just something, mathematicians just all love these bridges here. And so I, yeah, can you talk a little bit? You said you encountered it first as a kind of starting out as a beginning graduate student. You know? Can you tell us, bring us back to that moment, tell us about its importance in your life?

RW: Sure. So I was an undergraduate student, and there was a graduate student that was a TA for one of my classes at the time, and I asked him about what type of math he studied, and he drew this picture of a trefoil knot on the board. So shout to Aaron Abrams if you're out there, Aaron. And so I asked him, you know, how could I learn more about this? And he pointed me to a class, an algebraic topology class, that was being offered the next semester. So I signed up for it, and the class was real challenging, but I really enjoyed it. And then the following semester, I got a chance to do a senior thesis, and so I ended up going back to the instructor for that class, and he said yes. So also want to shout out Charlie Pugh for saying yes. And he chose the project that I would work on, the proof of this theorem. And so I'd never, I'd never heard of Poincare. I didn't know much about the historical context of the theorem, but it was — looking back, it was pretty significant that I got a chance to try and think about how to come up with my own proof of something that Poincare had proven, who was right there at the beginning of the field.

KK: And it's a surprisingly tricky theorem to prove. I mean, I was reading this actually, last year sometime. I was trying to remember how this proof goes, because what's remarkable about it is, there's lots of things involved, right? Because no matter which vector field — and it could have, like you said, it can only have finitely many zeros or singularities, but it might have 10 million of them, or it might only have two, but that number, the Euler characteristic, of course, is an invariant of the surface. So no matter how you wiggle this vector field, if you create something, some weird swirl on one side, well, some sort of opposite swirl has to happen somewhere else, effectively, to cancel it out. It’s really remarkable.

RW: Yeah, yeah, that's right. So I was trying to remember what we did to prove the theorem. We, you know, we really were wrestling with a lot of the tools that we used in the course, like, a lot of the details about triangulations of surfaces and trying to find the right ones and paths of vector fields and things that build mature for a very young mathematician back then.

KK: Yeah, yep, yep. So. And I guess one of my favorite corollaries is what, it's the hairy ball theorem, right?

RW: Yeah, that’s right. I was thinking about this. It's got to have, like, one of the worst names in all of mathematics.

KK: Yeah.

RW: So yeah, you can't comb a hairy ball flat without a cowlick, also known as the you can't comb a coconut theorem.

KK: Okay, that’s a little better.

EL: Yeah.

KK: A little little less innuendo, right?

RW: Yeah. That was a close, a close second for my favorite theorem. It's nice that they're connected.

EL: Yeah. We were talking earlier a little bit that, you know, it is hard for people to pick favorite theorems, and, you know, what does it mean if a theorem doesn't immediately leap to mind as a favorite? I just want everyone to know that we are, we might be mathematicians, but we are very not pedantic and mathematician-esque about definition of your favorite. We are very loose, and you know, it can be your favorite of the moment or your favorite for what it meant for your mathematical development. And we’re very imprecise with the definition of favorite on this podcast. All favorites are welcome.

RW: I must say that I had not thought about this theorem for years, until I was asked to find a favorite theorem.

KK: Well, it's sort of like on Instagram, there's this account we rate dogs. Do you know this one? So anyway, basically the guy rates dogs, but the lowest possible rating is 10.

EL: 10 out of 10.

KK: 10 out of 10. Theorems are sort of that way too.

EL: I don't know. I would say, I'm not going to name any theorem names. But I would say there are some theorems that I would put below 10 out of 10. Cancel me if you must. But you know, I’m going to put that out there.

KK: This is it. So we have to start our new Instagram account, clearly.

EL: We Rate Theorems.

RW: 10 out of 10.

KK: That’s right.

EL: Yeah. So another thing we like to do on this podcast is we ask our mathematicians, as if it weren't hard enough to choose a theorem, to choose a pairing for their theorem. You know, be it art, music, food, wine, any delight in life. What have you chosen to pair with the Poincare-Hopf [index] theorem?

RW: So I think I might have actually started with the food and then went back to the theorem. But there was this example that also really like captivated me, captured my attention as a student, and that's the hot fudge flow. So it's a vector field over a surface. And so the idea is to imagine a ball of ice cream, and you do what you do with ice cream. You take the hot fudge and you drizzle it on top of the ice cream, and you try and hit the center. And then what happens to the fudge? It sort of, you want it to expand and wrap around and then come back as a source and drip out of the bottom, if this was, you know, suspended in the air. So that's the hot fudge flow. And you can compute the sum of the indices of the critical points of that vector field, and it'll match of the Euler characteristic of the sphere. So the pairing is a hot fudge sundae.

KK: Okay.

EL: Excellent.

KK: That’s exactly perfect. Yeah.

EL: Of course we have to ask. What is your number one ice cream flavor for a hot fudge sundae?

RW: I was actually hoping you wouldn't ask that I'm the most boring ice cream person. Vanilla is my favorite.

KK: Look, you can't go wrong.

RW: Yeah.

EL: I will say, it is very unfair to vanilla that it has become this word in in our our language, for something that's boring, or pedestrian, because, like, it is an incredibly complex flavor, like, if you get an actual vanilla bean, it's like, there's so much going on. And I don't, I don't know the the history of how vanilla became “boring,” but, you know it is, it is anything but boring. Justice for vanilla.

KK: And so complicated to grow, right? It only grows in very specific places.

EL: A few places. And it’s expensive. Isn’t it, like, the second or third most expensive spice after definitely saffron.

KK: Saffron, I think, is number one.

EL: Maybe something like cardamom. Cardamom is up there too, I think.

KK: It’s not cheap.

EL: No hate to vanilla.

KK: It’s not cheap, because one little pod of vanilla, one little pod at the store is like, $4 or something. You know, it's like, it's really, really absurd. But it's an orchid, right? I mean, so, I live in Florida. We can actually get orchids to grow here, but it's still not easy.

EL: Right. Do you know if the vanilla orchid can grow there?

KK: I doubt it. If it could, they would be cultivating it left and right. I actually think it's too hot here. It's not humid enough, somehow, yeah, so some orchids will work.

EL: Because I think, like Madagascar, Tahiti and maybe Mexican? Is it grown in Mexico also?

KK: I think there might be some spots in Mexico, yeah, like, maybe in southern Mexico, Oaxaca or something. But, yeah, anyway, okay, all right, this is not a vanilla podcast.

EL: Yeah, three mathematicians speak extemporaneously on vanilla cultivation. Tune in next week for the exciting conclusion.

KK: That’s right. Yeah, so Robin, we always like to give our guests a chance to plug anything they're doing. Where can we find you online, what sort of, any big projects you're working on that people might be interested in, or anything like that?

EL: Or have done recently?

RW: Yeah, so I have a really bad online presence right now. At the moment, the website could use some dusting off. But one of the projects that I'm working on that I'm excited about right now is in math education. So we've been making videos of Black mathematicians talking about their work, their educational experiences, and giving advice to young people. And so these are for K to 12 students, but also, I think they're going to be of interest to lots of folks. And so we do have a website, but the URL isn't in on my mind to pass on to you right now. Maybe I could share it with you afterwards.

EL: Yeah, we'll, we'll get that from you and put it in the show notes, so it’ll be easy for people to get.

RW: That’ll be fantastic, but thanks for letting me make that plug.

EL: Yeah, well, and I remember seeing recently, you did a talk at the Museum of mathematics, right with and was that a conversation with Ingrid Daubechies?

RW: It was so much fun. It was a conversation.

EL: Do you know if that is available in video form somewhere? I meant to look for that before we got on. But of course, I didn’t.

RW: You know, I had the same question cross my mind as I was approaching this as well. And I think it might be available, but it could be, like, for museum members.

EL: Okay.

RW: I need to check.

EL: Yeah, I remember seeing your saying your name in my inbox, and thought, well, that's cool. And you've also, do you mind talking a little bit about the Algebra Project and and Bob Moses?

RW: Sure.

EL: Because I know that's something that you've — I know I've talked with you about it before, and Bob Moses passed away around the time we were putting that book together.

KK: Yeah, it was a couple years ago.

EL: So, yeah, do you mind talking a little bit about it? I thought it was really interesting.

RW: Yeah, sure. That's something that I could talk about for a long time. So just just check me if I start going on too long. I met Bob Moses as a graduate student, and I think I was kind of wrestling with some identity issues about my interest in math, but also, you know, interest in social issues, and kind of wanted to make a difference in my community, and trying to figure out how these two things came together, and if I was doing one, did that mean that I couldn't do the other? And so I came across his book, Radical Equations. It was about math literacy and the civil rights movement, and he brought his work in the civil rights movement together with his work as a math teacher in in Boston, and it really kind of spoke to me. And so I got a chance to meet him, and ended up staying connected with him and Ben Moynihan at the Algebra Project, and so I worked with them in different ways, attending teacher professional development. We helped spearhead an effort in Los Angeles, where the Algebra Project curriculum was used in four different high schools supported by an NSF grant. We had a second effort here, where we've been running some summer programs for students through the Algebra Project. And recently I joined the board of directors, and so I’ve been involved with them since I was in my 20s, and so it was a real honor to be asked to kind of be a part of that, that part of the leadership for the project.

KK: Bob Moses really, really impressive man. And then, this idea that you know, that every you know, things are really important. You know, education is so important to advancing, you know, civil rights and things like that. I mean, Bob Moses was really spectacular. Our listeners, if they don't know much about him, should just look him up, because he was really impressive and influential, and by all accounts, a very kind man. Like I said, I've never met him, but just a really great human being.

RW: And I think what people, a lot of people don't know about him, is he was a math teacher first, he was teaching math and and the sit-in movements happened, and he got drawn into the sit-ins. And then when, when things kind of settled down, he went right back the math classroom. And so kind of think of him as one of us.

EL: Yeah. I think reading, reading radical equations a few years ago, I remember, you know, it's just like sometimes when you're a mathematician, especially if you're really involved in the academic math world you get so, you know, drawn into these very abstract questions that you feel like have nothing to do with, you know, anything resembling reality, or anything resembling social issues, and just the way that he writes about how access to good math education, like is so important for people to be prepared to, you know, have careers that they want, be able to have financial stability in their lives then, and just the, you know, the doors that it opens to have access to math at, you know, the middle school, high school level, really reminds you as a mathematician, like, oh, yeah, we are part of this society.

RW: Yeah, that's right, and we do have a really important role to play. That's one of my biggest takeaways from him that as mathematicians, we do have a really important role to play in how this whole thing turns out.

EL: Well, thank you so much for joining us. Really great to talk with you again.

RW: Thank you so much.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Robin Wilson, a mathematician at Loyola Marymount University, about the Poincare-Hopf index theorem and the importance of math education. Below are some links you may enjoy after the episode.
An interview with Wilson for Meet a Mathematician
More on the Poincare-Hopf index theorem
The 2021 SLMath Workshop on Mathematics and Racial Justice and its follow-up, to be held in May 2025
Storytelling for Mathematics
The Algebra Project
The 2025 Critical Issues in Mathematics Education workshop, to be held in April 2025, focusing on mathematical literacy for citizenship

Episode 92 - Kate Stange

10 juni 2024 | 30 min

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance writer in Salt Lake City, Utah, where it is gorgeous spring weather, perfect weather to be sitting in my basement talking to people on Zoom. This is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I don't know, Evelyn, I saw the pictures on Instagram over the weekend and it looked cold in Utah. You wrote that you rode a century, right?

EL: Metric century.

KK: Okay. Metric.

EL: Just in case — you know, I don’t want people to think I'm quite that hardcore. Yeah, at least at this point in the season. Yeah, I hadn't managed to ride more than about 25 miles since last fall because weather, travel, just things conspiring against me. The week before I was like, I really need to get 30 or 40 miles in on Monday. And then it was, like, 20 mile an hour winds and sleet and I was like, well, I guess I'm just going into this cold, but it was fine. It was actually gorgeous weather. It was a little chilly at the start, but better than being too hot.

KK: Well, you know, the muscle memory takes over, right? So you can do — I mean, 62 miles isn't that much more than 25, really, once you have the legs, so congratulations.

EL: Yes, thank you. Well, we are delighted today to be joined by Kate Stange, who is in the Mountain Time Zone, something that I always feel thrilled about because I'm constantly converting time zones when I'm talking to people, and finally, someone I didn't have to do that for! So Kate, other than being in the Mountain Time Zone, what would you like to tell us about yourself?

Kate Stange: Oh, geez. Um, well, I'm also a cyclist. And so I'm jealous hearing about your rides.

EL: Wonderful!

KS: Here in Colorado we have we have this ride called the The Buff Classic. And so it has a 100 mile option where they close Boulder Canyon so that you can bike up the canyon without any cars.

EL: Oh, wow.

KS: Then you bike along the peak to peak highway. It's just wonderful.

EL: Yeah.

KK: Yeah. That sounds great.

EL: So you're at CU Boulder?

KS: Yes. And it's run from the campus. It starts right outside the math department.

EL: Oh, perfect. Yeah, just drop your stuff in your office and hop on and ride it?

KS: Yeah.

EL: Yeah, great. Well, we are thrilled to have you here today. And I guess we can just dive right in. What you're on what do you like to tell us about?

KS: My favorite theorem, at least for today, is the bijection between quadratic forms and ideal classes.

KK: That’s a lot of words.

EL: Yeah, and I'm so excited to hear about this, because I am honestly a little nervous about both quadratic forms and ideal classes, and a little embarrassed about being nervous about quadratic forms, not so much with ideal classes. So yeah, can you tell us a little bit about what that all means?

KS: Yeah, sure. So quadratic forms is probably what sort of comes first in the story, at least sort of the way that the mathematics tells it, and also probably the historical way. And so a quadratic form is just a polynomial with an x2, a y2 and an xy. So it's like 3x2 + 7xy − y2. So that's quadratic form. And, as number theorists, one of the things that we're most interested in studying is what are the integer solutions to polynomial equations? And so first you start with linear equations. And there's a wonderful story to do with Euclidean algorithm and stuff there. And then you move on to quadratic. And really, these are sort of some of the first equations that you would start studying next, I guess. And so they go back to the classical days of number theory, Gauss and Euler and everybody. And, yeah, so they come in, what happens is that they come together in families. So different quadratic forms, you can actually just do a change of variables. And it'll look different, but it won't really be too different, particularly if you're interested in what numbers it can represent when you put in integers. So say I take x2 + y2, which is the simplest one, if I put in various different integers to that I get various different integers out. And then if I do a change of variables on that, just a little change of variables — like maybe I change x to x + y, but I leave y alone — the formula will look different after I do that change of variables, but as I put in all integers and look at all the stuff I get out, those two sets, the in and out, they're going to look the same. And so we kind of want to mod out, we want to ignore that difference. So I'm really thinking of equivalence classes of quadratic forms. So that's the first object.

EL: And that change of variables is kind of the only equivalence class thing that happens with them?

KS: Yeah. Yeah. Because they could really behave differently between the different classes.

KK: And you only allow a linear change of variables, right?

KS: Yes, exactly. Yes. Thank you.

EL: Yeah. Okay. So now, ideal classes.

KS: Now ideal classes. So this is an interesting one, to describe where it comes from, I think. So there's sort of — if you think about the history of math, I would say there are sort of two versions, there are sort of two histories of math at the same time. There's one, which is sort of the human history, which is fascinating and human and quirky. And then there's sort of the way that the math would like to unfold to human understanding the way that as any human coming to it, they might discover the pieces of the mathematics. And I don't know too much about the details of the human history. But in terms of how you might discover this, if you're just looking at the integers, you are interested in how they behave, you discover things like prime numbers, you've got addition, you've got multiplication, you've got powers, you might ask how these things interact. And at some point, maybe when complex numbers are discovered, you think about whether there are possibly other collections, or other number systems, other collections of numbers in which you could do the same kind of thing. And so one of the first examples of this would be what's called the Gaussian integers, which is where you take complex numbers, I don't know whether I should dive into complex numbers, but you take complex numbers with integer coefficients.

EL: So that means things like 1 + 2i or something. So the i and the 1 both have integers in front of them.

KK: Right.

KS: Yeah, exactly. And so this is a collection of integers, kind of, right? And we ask things like, okay, are there prime numbers? And so it turns out that there are in that system, there are Gaussian prime, so, like, 1 + i is a prime number. And so you kind of start to develop this whole theory that you have for the integers. But what you find is that in some of these systems, you lose unique factorization. So we love unique factorization in the integers, right? Every integer, up to reordering the factors and maybe putting on a minus sign, you have always a unique factorization into prime numbers. And in the Gaussian integers, that's true. But in some of these other systems, you lose that. And so what people tried to do is to try to fix it. And it turns out, the way to fix it is to add in what were I think, originally called ideal numbers. They were thought of as numbers that should be in the system that weren't in the system. And what they actually were were collections of numbers. They were sets of numbers instead of individual numbers. And the idea here is that, say, you were to take — like in the integers, if you took the number two, you could replace that idea, that idea of two-ness with just the collection of even integers. And so that's an ideal now. Instead of a number, it’s an ideal, and it's really carrying the same information. But now it's a subset. And so by moving from individual elements of the ring, of the collection of numbers, you move to subsets of them. Now you have more things, and so now you can recover unique factorization in that world. So those are ideals.

EL: Yeah. And so the Gaussian integers do have unique factorization.

KS: They do. Yeah.

EL: So this — actually, I kind of forgot, but recently, this came up in something I was writing where I wanted the example to be the Gaussian integers so bad because it doesn't have any square roots in it. But then it didn't work because it isn't true for that. I was trying to show how unique factorization could fail, but I didn't want to have to use square roots. But as far as I know, you can't do that. So then I fixed it by putting a square root of negative five in there and hoping that people would be patient with me about it. But yeah.

KK: So that's the example of one where you don't get unique factorization, right? So you take the integers and you join the square root of minus five. That's one example.

KS: That’s one of them. Yeah.

EL: And then it's like two and three are no longer primes.

KS: So if you multiply (1+ √ −5) × (1− √ −5)

KK: You get six. Yeah.

KS: You get six, which is also two times three. And those are two different prime factorizations of six.

KK: Right.

EL: Yeah, but it's so fun that you can do that, and I like your way of putting it where regardless of how these ideas actually formed, you can as a human, looking at some of these basic pieces like primes and then or primes and integers and square roots and things, you can kind of come up with this, like, what happens if I do this? And create this new thing where this this property that I know I always assumed — like unique factorization, when you're growing up, you know, when you take math classes in school and stuff, it just seems like so basic, like, how could you even prove that there's unique factorization? Because how else could you factor anything?

KS: Yeah, exactly.

EL: It feels so basic.

KS: Yeah. And this is what happened, I think, historically, too, is that people didn't expect it to fail. And so they were running into problems and it took a while to figure out that that's what was going wrong.

KK: Wasn’t this part of, was it Kummer who had a reported proof of Fermat's Last Theorem, and he just assumed unique factorization?

KS: That’s what I've heard, although I never trust my knowledge of history. Yeah.

KK: It’s probably true.

EL: Well, and there are a lot of good stories. And they may or may not be true sometimes. But yeah, okay. So we've got these, these two things.

KK: Yep.

EL: The quadratic forms and the ideal classes. So yeah, I guess either historically or mathematically, what is this connection? And how do you know that these two things are going to be related?

KS: Yeah, so they seem like different things. So I think quadratic forms were studied earlier. And at some point, people noticed that quadratic forms had an interesting property, which is that sometimes you could multiply them together and get another quadratic form, which is kind of hard to explain. But like, if you actually wrote out (x2 + y2) × (z2 + w2) and you multiplied that all out, you'd have a big jumble. But then you could factor it out. So it looked like, again, a square with some stuff inside, z’s and w's and whatever inside the brackets, plus a square. And so this meant that sometimes if you picked your forms correctly, and they had this sort of relationship, then if you looked at the values they represented, the numbers that can come out, when you're putting integers in, you would take that set of things the first one represents and the set of things the second one represents, and then you’d look at what the third one represents, and it would represent all of the products of those things. So there was this definite relationship, but the way I'm describing it to now is a little awkward, because it's a lot of algebra. But this is, I think, what was noticed first, somehow. And again, I might be mixing the human story with how math tends to want to unfold. I don't know exactly the history. But anyway, so you notice that there's this relationship. And that's kind of reminiscent of an operation, like a multiplication law. And what happens is that, in fact, that's coming from the fact that these ideal classes, each one of them — sorry, my mistake — so it's from the fact that each of these equivalence classes of binary quadratic forms, each one of them is associated to an ideal. And the ideals as the sort of generalization of the idea of number, they can be multiplied together to get new ones. And so on the ideal side, it makes sense that there's an operation because you're already living in a number ring where you've gotten an operation. But on the quadratic forms side, it's a surprise. And so that's one of things I like about this theorem is that you see some structure and you want to understand why. And the reason to understand why is just to change your perspective and realize these objects can be viewed as a different kind of object where that behavior is completely natural. Yeah, so that's one thing that I like about it.

EL: And does this theorem have a name or an attribution that you know?

KS: Oh, it's such a classical theorem that no, I don't know.

KK: Right. It's just the air you breathe, right? So what's the actual explicit bijection? So you've taken a quadratic form. What's the corresponding ideal?

KS: Well, actually, the other way is a little bit easier to figure it out.

KK: Yeah, let's go that way.

KS: So let's take the Gaussian integers, okay. And in the Gaussian integers, you've got — for your ideal, so think of it as a subset of the Gaussian integers. But because it's an ideal, it has the property that it has the same shape as the Gaussian integers. I actually usually like to draw a picture. So I'm going to try to draw a picture just out loud. So if you think of the Gaussian integers in the complex plane, they fill out a grid, right? It's all the integer coordinates in that plane. So that's a grid. And if you want to see what the ideals are, they’re subsets that are square grids as well, but fit inside that grid that we started with, maybe rotated or scaled out.

EL: Okay.

KS: But they're square again.

EL: Okay.

KS: And so, what you can do is with this example, specifically, you can take the norm of each of these elements in the Gaussian integers. So the norm of a complex number, usually I think of it as the length from the origin. But I don't want to do the square root part. So if I have a Gaussian integer x + iy, I'm going to take x2 + y2, and that's the norm.

KK: Okay.

KS: All right. And so if I take the whole Gaussian integers, which is itself an ideal, that's one of the subsets that is valid as an ideal, then if I take all of the values, all the norms of all those elements, that's all the values of x2 + y2. So from my collection of integers, I take all of the values and that's actually a quadratic form.

KK: Okay.

KS: Okay?

KK: Okay.

KS: And so you can do this with the other ideals as well. So for each one, you look at the norms of all of its elements, and that is a quadratic form and the values of that quadratic form?

KK: Right. So the Gaussian integers are Euclidean, right? So it's PID, right?

KS: It is. It’s a principal ideal domain.

KK: So everything's generated by one element, basically every ideal?

KS: That’s right.

KK: So that makes your life a little simpler, I suppose.

KS: Yeah. So the ideals, in that case, really, they're not so different than the numbers themselves. This is one of those ones where you don't have to go to ideals. But by going to it, you think about instead of just, say, 1+i the number, you think about all the multiples of 1+i and you take all of those, and you take their norms.

EL: Okay. And I told you, when we were emailing earlier, that you'd have to hold my hand a little bit on this. So yeah, sorry, if this is a too simple question or something. But like, what is the quadratic form like the x2 + whatever xy +whatever y2 that you get from the the Gaussian integers that you just said?

KS: Right. So if we take the Gaussian integers, if I take x+iy as a Gaussian integer, its norm is x2 + y2. That’s the form right there.

EL: Okay. Yeah. All right.

KS: And then if I were to take a subset, like all the multiples of 1+i, I'm not plugging in all x's and y's. I'm plugging in only multiples of 1+i, so you end up with a slightly different form popping out.

EL: Yeah, so I guess it's kind of like x+x then.

KK: 2x2 squared basically, right?

KS: Yeah. Yeah. You could have Yeah, various things in various different situations, but yeah.

EL: Okay, thank you. Yeah. And so, yeah, can you talk a little bit about how you encountered this theorem? If it was something that like you really loved to start out with? Or if your appreciation has grown as you have continued as a mathematician?

KS: Yeah, well, it's one of these things, so I think everybody has things that they're attracted to mathematically, they all have a mathematical personality. And there's some sort of particular kinds of things that attract you. And for me, one of the things is the sort of projection theorems that tell you that a particular structure, if you look at it a different way, has a whole different personality. And it's actually the same thing, but it has just become totally different. So I really love those things. And I've always loved number theory, because it has such simple questions. But then when you dig into them, you always run into such fascinating, complex structure hidden. And so this is one of those things that if you have that kind of personality thing you, just keep bumping into. And so for me, and all of the research that I've done and things I've been interested in, I keep coming back to this theorem and bumping into it in different places. It shows up when you study complex multiplication of elliptic curves, it shows up when you study continued fractions, it shows up all over the place. And it just seems so fundamental. And it's sort of like maybe one of the most fundamental examples of this special kind of theorem that I really enjoy.

EL: Okay.

KK: Cool.

EL: All right, well, then the next portion of the podcast is the pairing. So yeah, as you know, we like to ask our guests to pair their theorem with something that helps you appreciate the theorem even more. What have you chosen for that?

KS: So, when I think about this theorem, I just it's a treat. So I think the only thing that comes to mind really over and over again is just chocolate. I love chocolate. And that's what you should enjoy this theorem with because maybe you should just be happy enjoying it.

KK: I mean, chocolate pairs with everything.

KS: That’s true. It's a bit of a cop out.

KK: No, no, that’s okay. So our most recent favorite chocolate is Trader Joe's has this stuff. And it's got, I don’t even know what’s in it, pretzels and something else crunched up in these like bark of chocolate. And it’s a dark chocolate I really recommend it. So you must have a Trader Joe's in Boulder, right?

EL: Are you a dark, milk, or white chocolate person?

KS: Oh, definitely dark. Yeah, I have a dark chocolate problem, actually.

EL: Yeah, the Trader Joe's. For me the dark chocolate peanut butter cups are are always purchased when I go to Trader Joe’s.

KK: Dark chocolate feels healthier, right? It's got more antioxidants and a little less sugar. So you're like this is fine, less milk. Okay. All right. It's actually it's a fruit, right?

EL: It’s a bean. You’re having a black bean pate right there.

KK: That’s right.

EL: Yeah, well, Salt Lake is actually a hub of craft chocolate. We have some really wonderful chocolate makers here, like single origin, super fancy kind of stuff. So if either of you are here, we'll have to pick up some and enjoy together. And yeah, along with quadratic forms and ideal classes.

KS: Sounds wonderful.

EL: Yeah. So something I meant to talk about this earlier in the episode, but you mentioned that you'd like to illustrate things, and that is how we first met is, through mathematical illustration. So I don't know, maybe it's a failure of imagination on my part, but I always, I'm always fascinated by like, number theorists who are really into illustration as well, because I think of, like, geometry, as you know, it shapes it as the more naturally illustrate-y parts of math. But would you talk a little bit about it, you know, illustrating number theory? And if if you've done anything related to this particular theorem, or if there's something else you want to talk about with your mathematical illustration?

KS: Oh, yeah, that's a that's a great idea. Yeah. So there's actually building up gradually a wonderful community of people who are interested in illustrating mathematics. And so that's maybe one of the things that you could add a link for is the website for the community.

EL: Definitely.

KS: Yeah. And so I've always found that the way I think about mathematics is very visual. I mean, I think as human beings, we have access to this whole facility for visual thinking, because we're embedded in this three-dimensional world that we're living in. And another way that we think about mathematics, I think often, is we're using another one of our natural facilities, which is our sort of social understanding facility, where we imagine characters interacting with each other and having motivations and stuff like that. But for me, it was always a very visual thing. And so even though it wasn't taught in that way, in my mind, somehow these things were always very visual things. And so I've always been really attracted to situations where you can see some hidden geometry in number theory. And with this particular theorem, there is a little bit of nice hidden geometry. I mean, the first hint of this is that when I talked about ideals in the Gaussian integers, I visualized them as a lattice.

EL: Yeah.

KS: And in all of these number rings, you can do this, you can you can think about lattices. And you're really talking about lattices, and lattices have things like shape. And you know, there's lengths and angles and stuff like that to talk about. And so one of the really cool things that you can do is you can think about, for example, with the Gaussian integers or with some other ring of interest that you can put in the plane like this, into the complex plane, then you can ask this question, it's a natural question that people ask: how can I study the collection of objects instead of the individual objects themselves? So if you want to study the collection of lattices, say, two-dimensional lattices in the plane, then one way to do it would be okay, how do I decide on a lattice? Well, I have one vector that's generating it, and then another vector that's generating it. So let's put the first one, let's sort of ignore issues of scaling and rotation, let's put the first one down pointing from like zero to one. And then the other one is somewhere, but now you don't have any choice anymore. No more freedom. And so you can think of the plane itself as a sort of moduli space, as a parameter space for the collection of lattices. And this space has a lot of beautiful properties. So you might as well order your vectors so that we're just talking about the upper half plane. So the first vector is from zero to one, and the other one is an angle less than 180 degrees from that. And so when you start looking at the geometry of this plane, and you want to talk about well, okay, I want to look at lattices, but maybe I don't care so much about what basis I'm using for the lattice, you start to divide the plane up in certain ways. And what you discover is that the natural way to talk about this plane is using hyperbolic geometry, actually. And so all of a sudden, you're doing hyperbolic geometry. And I find myself doing hyperbolic geometry sometimes when I'm doing number theory, because when I want to look at these these objects and stuff, that's just the natural world in which they live. I mean, the mathematics kind of tells you what you have to do you. You know?

EL: Yeah.

KS: And so those are moments that I really enjoy, because you're doing something that you think is just some algebra, but all of a sudden, it turns out it's geometry.

EL: Very cool. Yeah. So we will include a link to your website, which I know has some of the cool illustrations that you've done available there. And to the illustrating math, there's an online seminar that meets monthly that is really nice to go to, if you can. And, yeah, it's a lot of fun. And I, yeah, just so many different fields of math represented with that in ways that I never would have guessed.

KS: That’s true. That's one of the nice things about that community is that there's people from all different areas that you wouldn't normally interact with, because usually you have a pretty narrow research circle, if you're doing research in mathematics. But there, you're talking to everybody. And it has a much more creative feel for that reason, you get surprised by people's ideas, because they come from just a little bit farther from your home base, you know?

EL: Yeah. And I think it also kind of pushes people to really think about how they're explaining things, where you have a shorthand when you're working with someone who is in, or talking to someone who's in such a close field, and since you don't necessarily have that same common background, people, I think, it seems like are very thoughtful about how they describe things and what they assume that you already know.

KS: Yeah, exactly. It's just good to get out of your little corner.

EL: Yeah.

KK: All right. Well, this has been great. I definitely learned something today. I did not know this connection between ideal classes and quadratic forms.

KS: Oh, I thought of one more, one book I'd like to plug.

KK: Okay. Please do.

EL: Great. Yes.

KS: Yeah, so Martin Weissman has written a book called — I'm going to get the title slightly wrong. It's An Illustrated Theory of Numbers, maybe? Oh, you have it. Oh, I got it right.

EL: Yeah. An Illustrated Theory of Numbers. It's been holding up my laptop, after I read it, I will say.

KS: Yeah, and so you were asking about illustrating number theory, and this is just a beautiful book. It's completely accessible. I used it when I was teaching an introduction in number theory for undergraduates. But it doesn't require any particular background because it starts from, you know, we’ve got the integers, we’ve got addition, we’ve got multiplication, let's do some stuff. And and he really looks hard for ways to turn theorems which are usually completely algebraic into something visual, and they're just lovely.

EL: Yeah, and really amazing illustrations, and full color, like everywhere, which I know is more expensive to make books, and that's why books so often have the color in the middle, like in a little section and not the whole thing. But I do think this is just much more pleasurable to read because it it is does use that aspect, too. And it's not as stark as every page being black and white.

KS: Yeah, it's so inviting. It's a wonderful book.

EL: Yeah. Great recommendation. Thank you so much for joining us. I really enjoyed talking with you.

KS: Yeah, me too. Thank you so much for having me on.

[outro]

For this episode, we were excited to talk to Kate Stange from the University of Colorado, Boulder about the bijection between quadratic forms and ideal classes. Below are some links you might find interesting as you listen.
Stange's website
The Illustrating Mathematics website and seminar, which meets monthly on the second Friday
An Illustrated Theory of Numbers by Martin Weissman
The Buff Classic bike ride in Boulder

Episode 91 - Karen Saxe

3 april 2024 | 34 min

Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined, as always, by my fabulous co-host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to remember how to do this. It's been a minute since we've recorded one of these. We kind of went dormant for the winter.

KK: Yeah, a little bit, a little bit. Yeah. But Punxsutawney Phil told us — I don’t, what did he say? Let's pretend he said six more weeks of winter.

EL: I think he usually does. I don’t know.

KK: I mean, objectively, there are always six more weeks of winter. Like, the calendar says so, right?

EL: Yeah.

KK: Anyway, yeah.

EL: And, you know, he probably is pretty good at seeing shadows if he's a prey animal because he'd be used to seeing, like, a bird coming overhead.

KK: That’s an interesting question.

EL: Do birds eat groundhogs?

KK: That’s what I was going to wonder. I mean, like, eagles, maybe, but groundhogs are pretty large, right? I mean,

EL: Yeah. What eats groundhogs?

KK: Well, that's something to investigate later.

EL: Yeah.

KK: So it is Pi Day, right?

EL: It is! Well…

KK: We’re actually, we're recording this on Pi Day. When our listeners hear this, it won't be, but we're recording.

EL: And, I always have to put in a plug for my calendar.

KK: That’s right.

EL: The AMS math page-a-day calendar on which Pi Day does not occur on this day.

KK: That’s right.

EL: There are other Pi days on this calendar, none of which is this day, my little joke here. So you can find that in the AMS bookstore.

KK: Right. Are you Team Pi or Team Tau?

EL: I’m Team whichever one works for the calculation that you’re doing. It’s not that big a deal.

KK: That’s right. That's right. Okay. All right. Enough of us, enough of our useless banter, although we did discuss what's the ratio of banter to actual talk, right, that there's, there's like a perfect ratio. But we are pleased today to welcome Karen Saxe. Karen, why don't you introduce yourself and let us know all about you?

Karen Saxe: Hi, there, everybody. So first of all, happy Pi Day. If listeners know who I am, I was a professor at Macalester College for about for over 25 years. And then about seven years ago came to work at the American Mathematical Society, where I am very happy to be the director of the Government Relations Office. So I work in DC with Congress and federal agencies. And could quite a bit about this. I'm also happy to be here because it's Women's History Month. And it will be appropriate that it is Pi Day when you hear what my favorite theorem is.

KK: Okay, good to know. So, I'm curious to know more about this government relations business. So I mean, I know that the AMS does a lot of work on Capitol Hill, but maybe some of our listeners don’t. Can you explain a little more about what your office does?

KS: Yeah, so we do a lot of things. So first of all, we communicate — I sort of view the work of our office as going two ways. One is to communicate to Congress why mathematics is important to almost everything they make decisions about, you know, our national security, health care, you know, modeling epidemics, thinking, like you’re in Florida, thinking about how to model severe weather and things they care about, and then why they should fund fundamental research in mathematics and all sciences. And then also you know, how they make decisions about education. So we tell Congress, we give them advice and feedback on our view about what they should do in those realms. And then on the sort of flip side, I tell the AMS community, the whole math community about what Congress is doing and what's happening at the agencies like the NSF, and Department of Defense and Department of Energy, that that they might care about things, things that would affect their lives. So that’s sort of it in a nutshell. I spend a lot of time on the hill. I just came this morning, I went to a briefing put on by the National Science Board, which is the presidentially-appointed board that oversees the NSF. And they put out a congressionally mandated report every few years on the state of, it's called the indicators report. I'm sure I found it more interesting than everybody else, but it's pretty fascinating. You know, it covers everything from publications around the world, like which countries are are putting out the most science publications, what the collaborator network looks like around the world, and that to sort of US demographic information about education, you know, who's getting undergraduate degrees? Who's getting two year degrees? Who's getting PhDs, that that sort of thing. It covers a lot, actually. Pretty interesting.

KK: Yeah, yeah. All that in like two hours, right, and then it's over.

KS: Yeah, all that in two hours. And then they give you the big report that you can. And I've got them sitting in front of me. But given that this is a podcast, showing things doesn't work.

KK: Well, we do it all the time.

KS: Here’s one of the reports I picked up this morning. Actually, one really, so they're, you know, they're one thing. And you might end up cutting this, but one thing that's sort of fascinating to me is they always list barriers for getting into STEM degrees. And you know, there are things listed, like college accessibility, things that — and even going back. So like, you know, school kids who say they don't have science teachers in their schools, they don't have math teachers, but they've added to this list. “I can't support my family on a graduate student stipend.” So this is something.

EL: Yeah.

KK: That’s real.

KS: And we are, we've endorsed a bill in Congress that would look that would help to improve the financial stability, I guess, you would say, or the ability to be a grad student or a postdoc. So it's looking at stipends, it's looking at benefits, you know, leave time, all that sort of stuff, making it a job that you can choose to take when you're 23, and have a family to support and could make a hell of a lot more money doing something else with a math undergraduate degree.

EL: Yeah, and not see it as something where it's like, you're kind of putting off real life for a little longer, which I think maybe in the past was more of the model, like, oh, yeah, you'll have a real career later. But you know, in your mid-20s, you'll just keep being a student and not have kids or, you know, things, you know, not have parents to support or things like that.

KS: Exactly.

KK: Yeah. Okay. That's, that's good to know. Thank you for all that hard work you do, Karen. So but this is a math podcast.

KS: Right.

KK: So what’s your favorite theorem?

KS: Okay, so first, I'm going to tell you about the three theorems that I didn't choose.

KK: Cool.

EL: Great.

KS: So — I'm sure everybody goes through this — and thinking about my research, it would probably have to be the Riesz-Thorin interpolation theorem, which basically tells you that if you've got a bounded linear operator on two Lp spaces, then it's bounded on every Lp space in between those two values of p, so I used that all the time when I did research on that sort of thing. Then, but I was primarily a teacher of undergraduates, and kind of my two favorite theorems to teach are always Liouville’s theorem and, and then the uncountability of the real numbers.

EL: Yeah.

KS: And Liouville, they’re the one that says, you know, that there's a bounded — if you have a bounded entire function function, it's got to be constant. And the result is so stunning, and it gives a great proof of the fundamental theorem of algebra, that every non-constant polynomial has a root. So I always love teaching that. And then of course, like, Cantor’s diagonal argument about the real numbers, nothing beats that proof in terms of like, cool proof, in my opinion.

EL: Yeah. All-time great.

KS: Yeah, all-time great, right. And I think it's been mentioned on your podcast before. But what I picked was this theorem that says that if you have a given fixed perimeter, then the circle maximizes the two-dimensional shape you can make, so the isoperimetric theorem.

EL: Nice! And as you said, very appropriate for Pi Day.

KS: Yeah, which, I hadn’t even thought about that, which is sort of also embarrassing. But until we started acknowledging Pi Day, I hadn't thought about that. So another way to say it, or the way you might see it in a textbook, is if you have a perimeter P and an area A, then P2−4πA is greater than or equal to 0, with equality if and only if you have a circle. So this theorem has a very long, fascinating history. Lots of great applications. And for all those reasons, I love it. I love history.

KK: Yeah.

KS: I love math.

KK: Yeah. Do you have a favorite proof of this theorem?

KS: I do, actually. Yeah. Well, I didn't know you'd ask that. So there are a lot of proofs. And the one that I like, and this comes from being an analyst probably, is in the early 1900s. Hurwitz gave a proof using Fourier series. I love that proof. And proofs are quite old, going back thousands of years to the Greeks. And then in 1995, Peter Lax actually gave a new short calculus-based proof. But I like the Fourier series proof, just because I like Fourier series.

EL: Yeah, that's a topic that I wish I understood better. Somehow I kind of missed really, ever feeling like I've really got my teeth into Fourier series. Maybe that's a little embarrassing to admit on a math podcast.

KK: I don’t know. I took that one PDEs class as an undergrad and, like, that's where you see it, you know, doing the — whichever, the wave or the heat equation, whichever one it is — maybe both? I don't know. And then that’s it, that shows you how much I remember, too.

KS: Yeah. Good. So you're not gonna dare ask me to give you that proof or anything?

EL: Yeah, generally, a proof like that on audio is not the ideal medium.

KS: It doesn’t work.

EL: Actually, you brought up these ancient proofs. So yeah. Yeah, I guess how long has humanity known this fact, do you think, or do you know?

KS: So it's considered that the Greeks knew the proof. And then it was proved around 200 BCE. It even features in Virgil's version of the tale of Dido, Queen Dido.

EL: Oh, that’s right.

KS: So yeah, I think that was around 50 or 100 BCE, after the Greeks knew the theorem. So can I say what that story is?

EL: Yeah.

KK: Yeah, please.

KS: So she apparently fled her home after her brother had killed her husband. Okay, so we're already in an interesting phase. She somehow ended up on the north coast of Africa after that, and she was bargaining to get some land. And they told her, oddly, that that somehow she could get as much land as she could enclose with an oxhide.

KK: Okay.

KS: And so she took this oxide and cut it into very thin strips, and then enclosed an area, that was the largest she could conceive of, with the given per perimeter.

KK: Okay.

KS: So there's that. So it appeared, like, 2000 years ago, or more, and then you sort of we sort of jumped into the early 1800s when Steiner gave geometric proofs. But what's kind of fascinating is his proofs all assumed that a solution existed. And I haven't looked at these proofs, at least not in a long time. But then later in that century, Weierstrass is credited with giving a proof that, well, first, he proves that a solution does in fact exist. And he did use the calculus of variations to get this proof. So that's, that's sort of the story of the, of the theorem.

EL: Yeah, this actually — you know, we say the Greeks knew this, but I kind of wonder if this is one of those things that humans would kind of intuitively know, even if they're not in a framework where they have language about proving mathematical theorems, even if that's not an aspect of, of their culture, but it seems like you're trying to get into the mentality of like, what is really intuitive or innate about mathematics for humans? And I wonder if that, you know, we kind of would understand, well, if I took a square or something, I could sort of bow it out a little bit, and get a little more area with the same string.

KS: Actually, I mean, one reason I love this theorem is you can give string to kids, and I used to do this, like in elementary schools, and tell them make the biggest shape. And you have to tell them what closed is, no, you have to describe that the string has to come back to where it started. And they all come up with a circle. And this is, you know, second, third grade kids. So it is really intuitive. Yeah. So what it's meant by the Greeks knew this theorem is not 100 percent clear.

KK: Because they didn’t even use pi, right?

KS: And then actually, Evelyn to what you just said, you know, there's something that's quite interesting to me, which is that, you know, if you think about, you know, shapes of constant width, you know what I'm talking about?

EL: Yeah.

KS: So, if you take the fixed perimeter, there's an infinite number of these, the circle’s the largest one and those Reuleaux, I think that's how you say his name, those triangles are the ones of smallest area.

EL: Okay.

KS: And you were just kind of alluding to that, like take a triangle and go puff out the sides, or something.

KK: And you can push in.

KS: Yeah. Right. And you can do it for any regular polygon.

EL: Yeah. Well, British money has a couple of these that are I think heptagons, Reuleaux heptagons? Are they all called Reuleaux? Or just the triangles? I don't know.

KS: No, but you’re right about that, they do. And so it's kind of funny, I saw something that was talking about these points, like, what possessed them to make those points? And if you have a machine that has a hole size, and you know, it could fit a circle, it has a diameter, right, but it can also obviously fit one of these other shapes. Yeah. So that works. And I think you're right. It's a heptagon, heptagonal version of those.

EL: Yeah. The first time I went to the UK, this was, I think, the most exciting things on my trip to me, was these coins. Like, who thought to make these? And I actually, I remember, I wrote a blog post about it and discovered that it was a little hard to figure out if I had the rights to use a picture because all the images of these coins are like, technically property of the Crown.

KS: That’s funny.

EL: Abolish the monarchy, man.

KK: Her Majesty relented in the end?

EL: Yeah, so strange. I was like, well, I'm not gonna beg the queen for the right to post this on my math blog. So I don't remember what happened with that. Hopefully, I'm not opening myself to takedown.

KS: I think you’re probably okay.

EL: Hopefully the statute of limitations has run out on that. Anyway.

KK: I recently came across, I was going through an old notebook, and I found — I don't know why I tucked it in there — from the late 90s. I had one of these 10 Deutsche Mark notes that had Carl Gauss on it.

EL: Oh, nice.

KK: And so I put it on Instagram. And I'm now starting to worry. Wait a minute. Will the German government come after me? Although it's not really legal tender anymore.

EL: Yeah, the pre-2000, whenever they went to the Euro, government.

KK: It was pre-Euro. Yeah, I think I’m safe too.

EL: But anyway, getting back to the math, Karen. So, has this been a favorite of yours for a long time? I guess to me, this is one that I don't think the first time I saw it, I would have been super impressed by it. So what was your experience? What's your history with this theorem?

KS: Right. So like, why did I decide I liked it? Because yeah, it's sort of like, okay, I mean, it's appealing, because everybody can understand it, it’s very intuitive. It's got this, the proof has this interesting history. But why I like it is because you probably know that I'm pretty engaged with congressional redistricting. And when they do measures of compactness of districts, this is the theorem that kind of motivates all their measures.

KK: The Polsby-Popper metric, right?

KS: Yes, exactly. And so you take the Polsby-Popper measure, which was come up in 1991. So like, different states, should I say something about redistricting?

KK: Sure, yeah.

KS: So yeah, I mean, just like the very brief thing is every 10 years, we have to do the census. This is mandated in our Constitution, for the purposes of reapportionment of the House of Representative seats to the state so then after the census is done the seats, which we now have 435 of them, they're doled out to the states. And how that's done is a whole nother you know, interesting math problem, more interesting, probably. But then once the states get their number of seats, like how many in Florida?

KK: We’re up to 27? [Editor’s note: It’s actually 28.]

KS: So let's pretend there's 27 for a minute.

KK: I think that’s right. [Ron Howard voice: It wasn’t.]

KS: Okay. Then, you know, the Florida Legislature, probably, I don't know who does it in Florida, but somebody.

KK: Let’s not talk about that.

KS: Yeah, let’s not talk about that. Whoever’s in charge has to carve up the state geographically into 27 districts, one for each representative, and how they do that geographic carving up is extremely complicated. And to answer the question, “Has this been gerrymandered?” there are certain measures of what's called compactness, and this is like a whole nother thing I could talk for hours on. And compactness sort of measures the lack of convexity, sort of, so like, are there long skinny arms going out? And this is where obviously, like a podcast is, is not the best. But in any case, you know, are there long skinny arms going out, or does the thing look like a circle? So the Polsby-Popper measure tells you how close to a circle, or a disk because it's filled in, but in any case, your district is. Well, that's kind of weird, because if you think about tiling any state with circles, it’s just not going to happen.

EL: Right.

KS: Yeah. So just to sort of fetishize circles is bizarre. But I guess, like, what are your other options? Well, there are lots of other options. But the Polsby-Popper is the most common. There's a handful of states that require specific compactness measures in their process, and many other states that require compactness, but they don't specify the actual measure. In any case, the Polsby-Popper is the most common. And the other common measure is called the Reock measure, and that also fetishizes circles. It's a similar type thing. So with the Polsby-Popper, it's kind of interesting, because they they first published it in a law journal in 1991, in this context for redistricting, but it has actually been mentioned, as far back as the late ‘20s. And can I read you a funny a funny opening line?

EL: Yeah, sure.

KS: So the it first appeared, as far as I know, in a 1927 paper in the Journal of Paleontology. Okay. And how's this for the start of a paper? In quotes: “How round is a rock? This is a question that the geologist is often forced to ask himself.” Okay.

EL: Nice.

KS: So that's a great opening sentence. And then it kind of carries on: “when he wishes to consider the amount of erosion that a stone has received.” And then the paper is actually about measuring the roundness of grains of sand.

EL: Oh, cool.

KS: So there's a lot to say here that the paper is filled with hilarious hand drawings, you know, but also, of course, that geologists seem to be male is another observation.

EL: Yeah, well, and the grammar rules of the time.

KS: Yeah, exactly. But even just this past January, I ran into a paper that was published, and uses this to measure the aggressiveness. It's in, like, a cancer journal. I can't remember which one. And I wrote it down, but of course, what do you know, I can't see it. Anyways — oh, Cancer Medicine is the name of the journal — and it used the Polsby-Popper measure to measure aggressiveness of tumor growth. So you know, it has a life.

EL: That's so so interesting. When you said Journal of Paleontology, I was just like, how is that going to come up in paleontology? But what do you say? Yeah, how round is a rock? It's like, yeah, you do need to measure that. I actually, just the other day watched this interesting video about sand grains and like, certain beaches, or, and certain dunes have different acoustical properties. Due to, like, if they've got a lot of the same sized sand grains and if they pack really well, or if they don't, sometimes there can be the squeaking effect, like when you walk on it, or in a dune, like when there's wind, there can be these like deep, deep resonances, like almost a thunder sound that happens.

KS: Oh, that is interesting.

EL: And this this video went and looked under the microscope at the sand on these different beaches, and kind of showed how some of them packed together better or worse, and some of them are more uniform. So they might secretly be using that metric.

KK: They might.

KS: That’s fascinating. I mean, I heard I've heard that squeaky sound on beaches.

EL: I never have I'm not a huge beach person. So I guess, yeah, but I'm curious about going to one of these beaches someday now.

KS: Yeah. And when you said that I was thinking of the packing, like how they pack, but that would have to do with their shape, and their size. Well, I don't know.

KK: So this is a sphere packing question now. And it's yes.

EL: Or a “how sphere-y is your sphere”-packing question.

KS: How spherey is your sphere?

EL: Not quite as catchy.

KK: Right. So the other part of this podcast is we like to ask our guests to pair their theorem with something, so what pairs well with the isoperimetric inequality?

KS: So naturally, you know, a mathematician would ask, are there analogs in higher dimensions? Right? And then back to how spherey is your sphere, so I play tennis quite a bit. So I'm going to pair it with tennis.

EL: Excellent.

KS: The shape of the ball abides by the theorem.

KK: Yes. Right.

KS: And works for so many reasons.

EL: Yeah. Well, and you are not the the first My Favorite Theorem guest to pick tennis, actually.

KK: That’s right. Yeah.

EL: Yeah, we've had Dr. Curto.

KK: Carina.

EL: Yeah. Carina Curto, paired paired hers with tennis. It was it was about linear algebra. That's right. Yeah. Yeah. Hers was about how this thing kind of goes back and forth. When you're doing this thing in linear algebra. So you picked different aspects of tennis to pair with your theorem.

KK: Yep. Do you play much do you, you play, you play a lot?

KS: I play — it’s embarrassing to put on a very well listened-to podcast — that I do play a lot, because I don't know how good I am.

KK: That doesn’t matter.

KS: But I play a couple times a week.

KK: I used to play quite a bit. So as a teenager, certainly. And then in my 30s I played a lot. I played a little league tennis. This is when I lived in Mississippi. And actually, my team won the state championship two years running at our level.

KS: Oh, wow.

KK: But I'm not any good. This was like, you know, I'm like a 3.5. Like, you know, just a very intermediate sort of player.

KS: Yeah, that's what I am.

KK: Yeah, my shoulder won't take it anymore.

KS: I still, I feel lucky. Because physically, I can do it. Right now. I'm in a 40+ league, and that's good. But next season, whatever you call it, or next season, I guess, I'm in an 18+ League, and I've done this before. It means the other players are allowed to be as young as 18. It’s a little humbling, even if we can serve, you know, we have the technical skills, like they’re, you know, like the shots you use in the 40s, like, lobbing is not a good strategy in 18+ because they can run.

KK: Back when I was in my 30s and played, I played a lot of singles still, and I could still do it. But when I would come up against the 20-year-olds, it'd be a lot harder. But then I also learned, I used to play a lot of doubles with with these guys in their 70s. And they destroyed me every time. They were just —because they knew where to be. They had such skill and good instincts for where the ball was going to be. It was humbling in that way.

KS: Yeah, it's it's fun. And I prefer playing doubles these days. It's just more fun and different strategy.

KK: Yeah, and less court to cover. That helps.

KS: Less court to cover. And it’s more social. It's a lot of fun.

KK: Yeah, so you haven't succumbed to pickleball, have you?

KS: I played once, on my 60th birthday. Because no one would play tennis with me. And I got invited to a pickleball thing. And I was like, Okay, we're gonna do it. And, you know, it was fun, but I haven't really. It’s a challenge in Minnesota playing pickleball because it's so windy and the balls are so light, and it’s like whiffle ball.

KK: That’s what they are, basically.

KS: The ball kind of blows around all over the place. So yeah, I haven't I succumbed to doing that. In DC I'm lucky to have enough people to play tennis with. There's a lot of them.

KK: Cool. All right.

EL: Yeah. Great pairing.

KK: Yeah, yeah. So we also give our guests a chance to plug anything they're working on. You sort of already did that. I mean, you're doing all the work. Anything else you want to pitch?

KS: I mean, back to what I do, one reason I love this new job is I get to go in and and make connections to any Congressperson. You know, they have their own interests motivated by their own history, their own life, their own constituents. And this can be — there are obvious things we think about, like people, congressional members who are interested in their electric grid, or ocean modeling for the Hawaii delegation. But it's fun. And it's a fun challenge to think of things. So there's one newish member who was a truck driver before he was elected to Congress. And, we went in and their office was like, we can't make a connection to math. And we started talking about logistics, you know, truck routing. And it was great. It turned into a great conversation where they hadn't really thought about that. So this is what I really love about my job, trying to connect math to anything they’ve got. What they’re interested in, I'm gonna I'm gonna try to connect math, and there are very few issues that that can't be connected.

EL: Yeah, well I actually have a question, something that our listeners might be interested in is like if a mathematician is listening to this, and wonders, how can I get more connected to what's happening? How can I understand what math and science, you know, representatives do on the hill? Is there a newsletter or a website or something that you have that they could look at? And, you know, maybe find ways to get more involved? Or at least more informed?

KS: Yeah, definitely. So first of all, I used to write a blog, but I don't do that anymore for the AMS. The AMS Government Relations page — so my office is the Office of Government Relations. And I believe if you search, AMS government relations, you'll get to my webpage, you know, the one that I call mine, and you'll see a lot of different things there. There are ways to get engaged. We offer felt three fellowships. Two are for graduate students, one is for a person with a PhD in mathematics to come and to come here physically and do things. One is a boot camp for graduate students, a three-day graduate boot camp to come learn about legislative policy. And then the the biggest one is a year long fellowship and working in Congress. I do hill visits with people. And you know, I'm pretty willing to bring almost any mathematician to the hill, and that can be virtual these days. So we have volunteer members through our committee work who fly in and do these hill visits. We did this last Wednesday, we had about 25 AFS, volunteers fly in, and that was a fantastic day. But I can do them virtually. I've done them with big groups of grad students from departments, and people can email me if they want. And I think you guys have my email.

EL: Yeah. Thanks.

KS: So those are the big ways. And then for AMS members who are a little more advanced in their careers, you can volunteer for AMS committees. And there's the Committee on Science Policy, which really focuses on this one. And then I'm also in charge of the Human Rights Committee for the AMS, which can be of interest to a lot of people.

KK: Sure.

EL: For sure.

KK: Lots going on there.

KS: Yeah, lots going on.

KK: Well, Karen, this is terrific. Thanks so much for taking time out of your day, and thanks for joining us.

KS: Thank you.

[outro]

In this episode, we enjoyed talking with Karen Saxe about her work as the director of the American Mathematical Society's Office of Government Relations and her favorite theorem, the isoperimetric theorem. Below are a few links you might find relevant as you listen:
Saxe's website and the homepage of the AMS Office of Government Relations

A survey of the history of the isoperimetric problem by Richard Tapia
The 1995 proof by Peter Lax
Evelyn's blog post about 50 pence coins and other British objects of constant width
The Polsby-Popper test to measure gerrymandering
A public lecture by mathematician Moon Duchin about mathematics and redistricting
The 1927 Journal of Paleontology article that first uses the Polsby-Popper metric (though not with that name)
An Atomic Frontier video about squeaky sand
Our episode with fellow tennis-enjoyer Carina Curto

The 10 Deutsche Mark note

Episode 90 - Corrine Yap

23 januari 2024 | 34 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right. Yeah, I was I was trying to think about what to say. And I was like, well, the most exciting thing in my life right now is that our city is starting a pilot program of food waste, like a specific food waste bin.

KK: Okay.

EL: But then I realized I also did an 80 mile bike ride last Saturday, and that's the first time I've biked that far. And that might be slightly more exciting than compost.

KK: Are you working up the centuries? Are you are you heading for?

EL: We’ll see. I felt pretty fine after 80. I also don't feel like I wanted to do 20 more miles. So we'll see. Someday, maybe

KK: The last century I did was, wow, it was 2003. It was 20 years ago. This is one called the six gap century in Georgia. And it goes over six mountain passes in the mountains of North Georgia, one of which has, like, a 15% grade, which is quite steep. It took me about eight hours. And then I hung up my bike and didn't ride it for like three months.

EL: Well, I mean, it would probably take me at least eight hours to do a flat century.

KK: Yeah, but back in my youth I could do a flat century in about five, but not anymore. Not anymore. So let's keep this banter going because I — so Ben Orlin, a former guest on our podcast, I saw Math with Bad Drawings today had various golden ratios. One of which was the golden ratio of hot fudge to ice cream in a hot fudge sundae, which he argues is one to one, but that's way too much fudge.

EL: That is so much fudge!

KK: But the podcast, like, substance to banter golden ratio, he claims is like two to one. So like a third of this should just be like us, you know, just shooting it.

EL: Saying nothing.

KK: Yeah.

EL: Well, I must admit, that's why I listen to fewer podcasts that maybe I would want to because I have a low banter tolerance. Which brings us to our guest today. Yeah, so we are very happy to welcome Corrine Yap today. Would you like to tell us a little bit about yourself?

Corrine Yap: Yes. So I am currently a visiting assistant professor at the Georgia Institute of Technology, Georgia Tech, in the math department. I'm also a postdoc affiliated with the Algorithms and Randomness Center. But I just got my PhD in the spring from Rutgers University.

KK: Congratulations!

CY: Thank you! I do a lot of, like, probabilistic combinatorics, and stuff around that. So that's sort of my main research work. I also do some performing and some playwriting as well. I actually just got back from a performance yesterday Worcester Polytechnic Institute in Massachusetts.

EL: Oh, wow.

CY: So very busy this time.

EL: Yeah. That’s actually one of the reasons that I've been wanting to invite you for a while. And I was like, well, I should wait until I've seen one of her shows. And then it just has not aligned to work out. Because I know you've done them at the Joint Meetings and things, and the times that I have been there and you have been there, it’s just not been a good time. So it's like, well, I'm not going to put this off forever. So even though I have not yet seen one of your shows, I'm very glad that that we could invite you and have you here and yeah, well, can you talk a little bit about the kinds of theater that you do, or kinds of — I don't know if it's mostly theater or more, like other? I don't know, speaking performances?

CY: Yeah. So it really started when I was a lot younger. And also in college, I primarily studied both mathematics and theatre, with no sort of vision as to what that would turn into in terms of a job or career or anything. I just really enjoyed doing both of them. And as an undergraduate I thought I was mainly interested in acting, but I started studying playwriting while at Sarah Lawrence College in Westchester, New York. And I started writing this play, which is the play that I continue to perform. It's called Uniform Convergence. And it's a one-woman play that's about math. It tells the story of Sofia Kovalevskaya, who is a historical Russian mathematician. She was born in 1850. And it tells a little bit about her life and how she faced a lot of obstacles to be successful as one of the first few women in academia. But it also has a portion that is sort of inspired by my experiences being Asian American, and also being a woman pursuing mathematics. And the setting is that of a real analysis classroom, a lecture where the character Professor….

EL: Hence, uniform convergence.

CY: Yeah, and she is lecturing to her students. So at one point, they do reach the point of the class where they do uniform convergence as a topic. So, you know, in the past, I did a lot more — like, in college, I did, you know, the auditioning for plays and being involved in rehearsals, and all this sort of stuff. But since going to graduate school, and now having an actual job, this one play is sort of the main way that I keep my ties to doing theater and the theater world.

KK: Very cool.

EL: Yeah. Well, that's cool. I didn't realize that Kovalevskaya was the subject of this. I actually just read Alice Munro's short story, Too Much Happiness, which is based on her life. And actually was not my favorite short story in the collection that it’s in, but it, you know, she is such a compelling figure and another woman who was interested in math, you know, at a time when it was a lot harder for a woman to have an academic career in any field, and was interested in literature. Wrote, I think both memoirs and fiction?

CY: She also wrote a play.

EL: Oh, wow.

CY: Yeah. But it wasn't about math. But yeah, she was very much also in both of these worlds in, you know, sort of a more artistic, creative mindset as well as a mathematical one.

EL: Yeah. Fascinating person. So yeah, that's really interesting. And hopefully someday I'll get to see it.

CY: Yeah, I'm still performing. I didn't think I necessarily would be. But it's been since 2017. I've been performing it at different college campuses, and sometimes at conferences at different parts of the country. And I still get invited places. So as long as that keeps happening, I'll keep going.

EL: Yeah, when I was still in academia and doing a postdoc, I did, you know, I'd started doing writing. And sometimes I would get invited to do both like a research seminar talk and a public engagement kind of talk. And so that that might be in your future as well.

CY: Yeah, maybe.

KK: Yeah. Broader impacts.

EL: Wearing both hats on one trip.

CY: Yeah, I actually, I forgot I am doing that. I think this is the first time I'm doing it. At Duke in October, when one day I'll be giving a seminar talk, and then the next day, I'll be performing the play.

EL: Yeah, cool. Well, we invited you on here to talk about your plays, but also to talk about your favorite theorem. So what have you chosen?

CY: Yeah, so I've chosen Mantel’s theorem as my favorite theorem. So this is a theorem that is in the area called extremal combinatorics. And I'll explain what that means. But the statement of the theorem is pretty straightforward. It says that if you have a graph, which I’m a combinatorialist, so for me graphs mean, collections of vertices with edges connecting pairs of vertices. If you have a graph on N vertices, then the maximum number of edges you can have without forming any triangles — so just three edges and three vertices connected to each other — the maximum number of edges you can have with no triangles is N squared over four with appropriate floor.

KK: Yeah, sure.

CY: And this seems like, okay, this is this is just a statement, maximum number of edges. What's so cool about that? You actually, we actually also know where the N squared over four comes from. It’s, the extremal example is the complete bipartite graph on parts of size N over two. So what that means is, you split your vertices up into two sets, each of size half the total universe. And all of your edges go between the two parts. So from one part to the other, not inside the vertices of the parts. So complete means you have all the possible edges crossing between the parts, and then bipartite because you have the two parts of the vertices, and that has N squared over four edges. And it has no triangles in it.

KK: Not even any cycles.

CY: Yes. Yeah, no odd cycles. Yeah.

KK: Okay, all right.

CY: Yeah. So, one reason I really liked this is because when I first learned it, I didn't really think much of it, I learned it in an undergraduate class in combinatorics. And there are, like, three, maybe four proofs that we learned that were all pretty short and straightforward. One of the most basic proofs is just via induction on the number of vertices, and there's nothing, there's no really heavy machinery that's needed at all. And I didn't think much of it. And I didn't have any context as to like, why do we care about this sort of thing. But every year, I learn more and more things that make me appreciate this theory, more and more, because it really was the foundation for this whole field that we call extremal combinatorics, which is really centered on these questions of, like, what are the maxima and minima of certain things that we want to count when we put certain constraints on the problem? So this is an example we want the maximum number of edges. And our constraint is we have no triangles. And you can, there are a lot of different directions you can go with this sort of theorem. One of the most sort of classical foundational ones is just to replace triangle with a different type of graph. Like you could say, Okay, if I want the maximum number of edges with no cycle of length four, or cycle of length 10, right, what can I say? Or if I want the maximum number of edges with no complete graph of size five, where complete means you know, the vertices, you have every possible edge between every pair of vertices. And this type of problem, sort of replacing triangle with other things. It's called a Turán type problem, because there's Turán's theorem that generalizes mantle's theorem to complete graphs of higher orders. And we basically know the answer of what the extremal number is, and the extremal constructions for almost every graph, except for when you consider a bipartite graph as your, instead of triangles.

EL: As the thing you're trying to avoid?

CY: Exactly. And there's a reason for this, there's a theorem where it basically fails, or it's trivial in the case that your forbidden graph is bipartite. And so there's been a lot of study, it's still a very active area of research. And what people are doing is sort of taking different flavors of this Turán type problem that sort of started with Mantel’s theorem. And my first paper in graduate school was on a topological version of this theorem, where we were looking at these higher-dimensional structures called hyper-graphs, which you can think of as a higher dimensional version of a graph, and looking at a more geometric or topological viewpoint on these hyper-graphs by making them into simplicial, abstract simplicial complexes. So we don't have to go into the details of that. But I found it, you know, when I did that project, I found it very cool that that we could take this seemingly purely combinatorial, graph theoretic statement about just counting edges, and somehow turn it into something that requires a little bit more of a geometric or topological point of view, which is not something I had spent much time with before. And so that's sort of one direction at the beginning of my grad school career, where I felt like I had suddenly a much greater appreciation for this theorem. And on the other end, where I am now, it's also connecting very heavily to the research direction that I'm currently pursuing, which is in statistical physics, which is for me an entirely unexpected application of this sort of thing. But if you think about it, this sort of characterization of the extremal structure saying, okay, we can achieve the maximum with a complete bipartite graph, you can view this as sort of a ground state, if you will, if you want to think of the vertices as like particles in some sort of distribution, and you can take a probabilistic point of view on these sorts of counting problems. For example, it turns out that the triangle-free graphs and the bipartite graphs, if you think of these two collections, triangle-free graphs and bipartite graphs on N vertices, they're very closely related to one another. In fact, almost all triangle-free graphs are bipartite. This is a theorem by Erdős, Kleitman, and Rothschild. So you can sort of ask how far does that behavior persist if you add more constraints to your problem? And you can think about it as in a probabilistic sense of thinking, well, what if I have a probability distribution on my triangle-free graphs? And I have a probability distribution on my bipartite graphs? How are those distributions related to one another? And what is the counting statement, say, in terms of the probability distributions when we when we consider a randomness point of view on these things. And the sort of magical thing is that when you go to a probabilistic point of view, there are very natural ways that you can put it into a statistical physics context, where in statistical physics, you are thinking inherently about probability distributions on certain particles, on particles in space, or different configurations of particles in space, where there's maybe a physics motivation underlying the distribution you define. But ultimately, you can distill it down into something that is, that is simply triangle-free graphs, or different discrete structures. So one thing that I'm really interested in right now is just exploring more of this somewhat mysterious, but somewhat really amazing connection between questions that arise in graph theory and combinatorics that, you know, for a long time, we have just thought of in that context, in the graph theoretic context, and how, looking at them from a more statistical physics perspective, can help us gain new insight into how to tackle these problems.

KK: Yeah, and hopefully, it'll go in the other direction. I mean, I think we have this idea that because we learn calculus, and we think about physics being based on calculus, but inherently, right, the universe has to be kind of discrete, so you can't divide stuff forever. So I mean, it sort of makes sense that the underlying business, when you get down to it, might have to involve some kind of graph theory questions.

EL: Yeah, that is remarkable that there's this connection. So this is maybe a naive question about what you're talking about doing. Like probability distributions on graphs, are you saying things like, the likelihood that that two vertices have an edge between them? Or are we talking about some other kind of probability distribution?

CY: Yeah, so there, I purposely didn't include too many details, just because there are a lot of a lot of actually interesting and all valid ways that you can think about imposing probability, you know, into this world into these problems, these extremal combinatorics problems. So one flavor is what you said, we can think of what's called the random graph model. The most common one is the Erdős–Rényi random graph model, where you simply have your N vertices, and for each pair of vertices, you flip a coin, and it can be a P-biased coin, independently, to decide whether you put an edge there. And you can analyze what happens in that graph. What are the likely properties that this graph might have, if you, for example, change P. And what's really cool about studying this model is that there are, for a lot of graph properties, you can find these thresholds with respect to P. And this is like a huge, very active area of research right now, there have been a lot of really cool things proven just this year, in the past few years with regards to a lot of open questions here. But you can sort of if you let your P, your probability that you're adding an edge, be a function of N, the number of vertices, and you imagine N going to infinity, then you can actually sort of chart what happens if you're trying to count, let's say, the number of triangles in your graph, the expected number of triangles in your graph, or other properties. And you can see how changing P changes the value of the thing that you're trying to count. And for a lot of things, they exhibit these thresholds where the probability of finding a particular structure is close to zero. And then past a certain threshold, it jumps up to something close to one, and it happens with high probability. And this is also mimicking something in the statistical physics world where we have things like phase transitions.

KK: Right.

CY: If you think of in physics, just like water’s liquid-gas sort of phase transitions. Where we're also interested in studying what happens to certain properties of your statistical physics distribution when you change the temperature of your different parameters of your model. Can you find the sort of phase transition where the behavior changes quite drastically?

KK: Yeah.

CY: And then so GNP is, is one of these ways you can sort of input probability into — you know, take a sort of probabilistic perspective on these problems. Another is simply something a little bit more physics motivated, is by just imposing a uniform, or nonuniform, or a weighted distribution on the things that you're trying to count. For example, if you want to study triangle-free graphs, you could consider the uniform distribution on all triangle-free graphs on N vertices. And then think about the uniform distribution on bipartite graphs and ask, like, are these distributions close in total variation distance? And you can conclude things about that based on what you know about how close are triangle-free graphs and bipartite graphs to one another? Well, what does that say then about the distance between the nniform distributions that you impose on each set? And that sort of thing characterizes the different sort of distributions that come from the perspective of statistical physics. There are things called, like, the Ising model, and the Potts model and the hardcore model that were defined by physicists. And it turns out that they are simply weighted distributions on things like graph colorings, and independent sets of graphs. And so you can study them in these two different contexts, in the context of the hardcore model from the physics world, or in the context of a distribution on independent sets from the graph theory world.

KK: The hardcore model, I love that name. That’s good.

CY: Yeah.

KK: Well, we’ve gotten pretty far away from triangle free graphs can have at most N squared over four edges. So you mentioned that there were like three or four proofs of this. Do you have a favorite?

CY: I have to say my favorite is the very straightforward induction proof.

KK: Okay.

CY: And the reason I like this is because it's a proof that I've done with high school students at a summer math program I teach at called MathILy-Er. And I do it as an hour-long inquiry-based activity, where I simply pose to them this question. I let N be something like six or five, something that they could start drawing examples for them, then say, how many edges can you have before you start having to find triangles? And they often come up with the extremal construction, the complete bipartite construction first. And then I asked them how can we prove that this is actually true that this is the maximum. And they've learned induction at this point, when I do this activity. And so it's a nice lesson in induction, because it requires strong induction. And everybody wants to do weak induction, first of all, and they always want to what I call induct up instead of induct down. They always want to start with an extremal example with N vertices and try and build something with N+1 vertices. And it doesn't work.

KK: Right.

CY: And I always have to remind them, you have to start with something that has N+1 vertices, and remove something and see what happens. Yeah, yeah.

KK: And the base case of one vertex is super easy, right?

CY: And then there's also some argument about how many base cases we need and whether we need one or two or three, or where do we start? And so I think it's just a really nice exercise and practice. And it's simple, but I get to give a little, tiny spiel at the end, not nearly as much as I have said here in this podcast so far. But a tiny hint as to like, you know, what's cool about this theorem, and what more could you do? And some of the students have been interested enough to try and generalize to complete graphs or higher orders, you know, a complete graph on four vertices and try and mimic the same proof. And yeah, I think it's a really nice activity.

KK: Cool.

EL: Yeah. So a complete graph on four vertices includes a complete graph on three vertices so therefore you're trying to avoid something more, so like some of these ones that have triangles could still not have the four. Sorry I'm thinking out loud here because I have very little graph theory intuition. So okay, just like which direction are we going, and how many of these are we avoiding?

KK: You and I are probably the same, Evelyn. Like, we probably took one undergrad graph theory course and then yeah, and then then became topologists.

EL: Right. It’s kind of like it came up in, my introduction to proof class, but never a specific class dealing with graph theory things. Although the times that I've taught in high school programs or stuff, it is the kind of thing that can be quite accessible because the idea of drawing a graph, it's not hard to explain to anybody.

CY: Yeah, to answer your question. So there are actually lots of triangles in the extremal example for the complete graph on four vertices.

EL: Okay.

CY: Just to give you a sense of how it generalizes, the extremal example is the complete tripartite graph where you take three parts now sides and over three, and you have all the edges between the parts, so it looks like a giant triangle.

EL: Yeah. This kind of makes me want to go think about graphs a little bit. Yeah.

KK: Well, so the other part of this podcast is we ask our guests to pair their theorem with something. So what do you think pairs well, with with this theorem?

CY: So yeah, this this question was actually harder for me.

KK: It’s harder for everybody!

CY: I thought of something right away. And then I thought, no, I can't say that. I have to say something cool. And my pairing has to be something neat that makes me seem like a cool person. But I just couldn't think of anything better. So bear with me.

KK: Okay.

CY: My pairing is tofu. Okay. And here's why.

EL: Oh, tofu is great!

CY: Yeah. Okay, great. Great. So I thought of this because I think tofu is also somewhat of an underrated ingredient. But it is also so versatile, and you can use it in so many ways. So I grew up eating a lot of tofu because I grew up in a Filipino-Chinese household. And it was just sort of a staple of the things we were eating. But then I realized that not everybody knows or appreciates tofu. The first time I met someone who had never heard of tofu before, it just sort of shocked me, but then I realized it's not a common thing everywhere. But it's used in so many ways. And so I have been vegan since 2015. And also, every year I gain more and more appreciation of tofu as an ingredient. Like, you can use it in stir fries. There's now cheese that's made of tofu, you can make eggs using tofu. You can make a pie using tofu. There are so many ways you can use tofu. And there are so many more vegan options at restaurants and grocery stores and everywhere. So I feel like you know, for anyone who hasn't had tofu before, I would recommend at least giving it a shot.

EL: Yeah, yeah. And yeah, I mean, I grew up in a household that did not eat tofu much, my parents don't eat too much. But yeah, I'm not vegetarian or vegan, but like eat a lot — we have recently been enjoying this vegan Korean cookbook, I mean, it's called Vegan Korean. You might have seen it. [Editor’s note: It’s actually called The Korean Vegan.]

CY: Yeah. I have that!

EL: And just checked out this vegan Chinese cookbook that of course, it's like I think multiple sections are tofu because it's like the tofu tofu part and the tofu skin part, and all of this stuff.

KK: And, you know, do you use silken or what.

EL: But yeah, we’re a high-tofu household now.

CY: Nice. Yeah, there are so many different levels of tofu that you can have.

EL: Yeah, so many different textures, like, the Korean soft tofu is different from like the soft tofu in the cardboard package. Yeah, and so I finally found a Korean grocery store in Salt Lake that I could get to and got, like, the real stuff and oh man, great. That soft tofu soup, so good. And I can actually eat the kind I make because when I get it at a Korean restaurant, it's way too spicy. So I cut — in that cookbook, I think I cut at minimum, sorry, maximum spiciness is, like, a third of what the recipes start with, sometimes a sixth and see if I can work up.

KK: The correct answer level and there was you know, N squared over for, the floor.

EL: Yeah. I am impressed by the spice tolerance of Koreans.

KK: Asian cuisine in general, we once years ago, I was director of the University Honors Program, there was this place in town. It was an Asian place, and they have various stir fries. And you could ask for your spice level from zero up to no refunds, right? And so we had a student worker who was from Bangladesh, we went to lunch there one day, and he got the “no refunds.” And we were like, how is it? And he just went, eh. Like, it's just not very hot. And it's just an interesting cultural thing. Because, you know, I grew up in the Midwest, my mother's family was German, you know, we ate a lot of fried potatoes and sausage, like no flavor, you know? And it's just all what you get used to. Right?

CY: Yeah.

KK: All right. Well, this is we like to give our guests a chance to plug anything. Where can people find you online? Or you've talked about your plays, so that's good.

CY: Yeah. I mean, you can find my website. I recently updated it. And now it's got an all-purple background, which I'm very happy with. And it's corrineyap.com. That's Corrine with two R’s and one N, in case you forget. And, yeah, I continue to perform my plays. So if you're ever interested in bringing me out somewhere to perform, I am always happy to consider doing that. And I've done it at a lot of math departments. I did it at some conferences, but I don't have any conference performances coming up. So mainly like seminars and colloquia slots, things like that. So Evelyn, if you have any universities around you in Salt Lake City who might be interested in hosting a performance, you can let me know.

EL: Yeah. I'll make you vegan Korean food.

CY: Oh, amazing. But yeah, I mean, I just do this for fun. So it's not something that I'm trying to, I'm not trying to schedule, you know, 100 performances on my show this year. I just do it whenever someone is interested in having me there, but I'm always open to new inquiries. So yeah, that's, I guess, the one thing that I'll plug.

EL: Okay, great. Well, it was lovely to have you I'm so glad we finally got to at least meet online.

CY: Yeah, you as well. Thank you. Thank you so much for inviting me. This is a lot of fun.

KK: This was great.

[outro]

On this episode, we enjoyed talking with mathematician and playwright-performer Corrine Yap about Mantel's theorem in graph theory. Below are some related links you may find interesting.
Yap's website
MathILy-Er, a summer math program for high schoolers
Wikipedia on Turán's theorem, the generalization of Mantel's theorem
The Korean Vegan

Episode 89 - Allison Henrich

12 november 2023 | 35 min

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined as always by your other and let's be honest, better, host.

Evelyn Lamb: I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And tomorrow is my 40th birthday. So everything I do today is the last time I do it in my 30s. So, like, having my last mug of tea in my 30s, taking out the compost for the last time in my 30s, going for a bike ride for the last time in my 30s. So I'm, I'm kind of enjoying that.

KK: Well, congratulations. Let's not talk about how long ago I passed that landmark. I will say there's a switch that goes off when you turn 40. So riding your bike will be more difficult tomorrow, I assure you.

EL: Well I’d better get one in then.

KK: Any big plans?

EL: I’m actually going to the Janelle Monae concert. She's in town on my birthday. I'm sure that's a causal relationship there.

KK: It must be.

EL: So yeah, I'm excited about that.

KK: Okay, so fun fact, my Janelle Monae number is, is two. So I have a half brother. Very long story. I have a half brother, who also has a brother by — his mother has two children with — my dad was one of them. And then another man was the other one. So this other one, his name is Rico. He was a backup dancer for Janelle Monae.

EL: Wow. So yeah, brush with celebrity there.

KK: I mean, of course I've never met Janelle Monae, but you can — actually if you look him up, so there's a style of dance, sort of Memphis Jook, it’s called. Dr. Rico. He's something else. Amazing dancer.

EL: Wow. Interesting life.

KK: That's right. That's right. So anyway, enough about us. We have guests on this show. So today we're pleased to welcome Allison Henrich. Allison, introduce yourself, please.

Allison Henrich: Hi. Yes. I'm Allison Henrich. Happy birthday. I'm so excited for you.

EL: Yes, you get to be on my last My Favorite Theorem of my thirties!

AH: Yes, awesome! I feel so special. So I'm Allison Henrich. I'm a professor at Seattle University, and I'm also currently the editor of MAA Focus, which is the news magazine of the Mathematical Association of America.

KK: I have one on my desk.

AH: Woo-hoo! Is it one of mine?

EL: Yeah, and when we were chatting before we started recording, you made the mistake of mentioning that you've done some improv comedy. Is that something you do regularly?

AH: So I wasn't an improv artist. This is such a cool event. This science grad student at the University of Washington started this type of improv comedy where they have two scientists give short five-minute talks. And then this improv comedy troupe does a performance that's loosely based on things that they heard in the science talk. And so I gave a talk about some basic knot theory ideas, and it was so funny. I wish everyone could have the experience of an improv comedy troupe doing a whole set about your like research or your job. Yeah, it was so amazing.

EL: Cool. But also, it sounds a little stressful. A little bit. Yeah.

AH: Yeah. You want to not be too boring. And you gotta, like — it's really interesting. The other speaker tried to work in things for them to make jokes about, and they totally didn't take the bait. And they found like more interesting things to make jokes about, but I definitely tried to work in some things that would help them riff off of my talk, and it worked pretty well. Like just referring to knots with quirky names and making jokes about knot theorists and whatnot.

KK: Sure. What-knot. Hahaha.

AH: There are a lot of good knotty puns.

KK: Sure. Okay, so this podcast does have a theme. And the question is, what's your favorite theorem?

AH: Yes! This is a hard question.

KK: Of course.

AH: I’ve decided to tell you about my second favorite theorem. Should I admit that?

KK: Sure.

EL: I’m sorry, that’s a different podcast, My Second Favorite Theorem. It has two slightly worse hosts.

AH: It’s the cheap knockoff.

KK: No, it's the sequel, once we get rid of this one, we're gonna move on.

AH: Just, we're all out of mathematicians, we’ve got to go through them again. So my, let's call it my favorite theorem.

KK: Sure.

AH: My favorite theorem is the region crossing change theorem. So I have to tell you a bunch of stuff before I can explain what this theorem is.

KK: Sure. But it must be about knots.

AH: It is about knots. So, you know, knots we represent, typically, with two-dimensional pictures called knot diagrams, where you have ways of representing when a strand is going over and when a strand is going under at a crossing. And so every type of knot that there is has infinitely many diagrams you can draw of it. But no matter how you draw a diagram of whatever your favorite knot is, it can always be unknotted if you're allowed to do a special kind of move called a crossing change. So if you have your favorite knot diagram, and you're allowed to switch the over and under strands on whichever crossings you want, you can always turn that knot diagram into the diagram of an unknot, which is like a trivial knot that'll fall apart if you unravel it a little bit.

EL: Basically just a circle, right?

AH: Yeah, a circle. I mean, all knots are circles, so I have trouble. Like, a geometric circle.

EL: A boring circle.

AH: Yeah, a boring circle.

EL: And so this theorem, does it come with like, a number of how many of these crossing changes?

AH: Ah, so this is not my favorite theorem. This is a theorem that's going to help us understand my favorite theorem.

EL: Okay.

AH: So this theorem has a really interesting proof that Colin Adams calls “proof by roller coaster.” So the the theorem that says you can unknot any not diagram by changing crossings, you can accomplish unknotting using a certain algorithm where you choose a starting point to travel around a knot, and you decide that every time you encounter a crossing for the first time, you're going to go over it. So the fact is that you're kind of like always traveling downwards. And then when you get to the very end, you take a little elevator back up to where you started. So this will always create an unknot. So it's not that surprising that this is true, that if you're allowed to change whatever crossings you want, you can unknot things. What is surprising is my favorite theorem, which is that region crossing changes can unknot any knot diagram. So let me tell you what a region crossing change is. So you have your knot diagram in the plane. A lot of us kind of imagine that this plane is on a big sphere. So can we picture not diagram on a ball? Is that okay?

KK: Sure. Make it a big enough ball, and it looks like a knot diagram.

AH: Exactly. Yup. So we've got a knot diagram on a ball, and the knot diagram basically separates the surface of the ball into different regions, right? So this amazing theorem uses this operation called a region crossing change. And what a region crossing change is, is you choose a region in the diagram, and you change every crossing along the boundary of that region. So in my head, I'm picturing kind of like a triangular region in the diagram. And if I do a region crossing change on that region, I'm going to change all three crossings that are kind of around that region. So this is the amazing result: every not diagram can be unknotted by region crossing changes. So you no longer, seemingly, have control over individual crossings, you can only change groups of crossings at a time.

KK: Okay.

EL: But you can still do it.

AH: Yes, you can still do it.

KK: Right. That seems less likely. The other one, you told us and I thought, Well, yeah, I can kind of see, before we even saw the proof, I could sort of imagine, well, yeah, you just lift them up basically.

AH: Exactly. You lift it up, and then if it gets stuck, you know, change that crossing. But you can only change groups of crossings with the region crossing change. But amazingly, it's still an unknotting operation. So that just blew my mind when I heard that.

KK: Okay, so now I have questions. So, more than one, right? You can't expect to be able to just do one of these, right?

AH: Right. I mean, so if you have a region that just has one crossing on it, it's like a super boring region, because it's just a little loop.

KK: Yep.

AH: And that's actually called a reducible crossing.

KK: Sure.

AH: If you just have a little loop, it doesn't matter which way, which is going over and which is under.

KK: No, but I guess I meant, so you know, you've got one region, right?

AH: Yeah.

KK: So there might be multiple regions, you might have to change many of these, right?

AH: Yes, yes.

KK: What if two are adjacent, then you do one flip on one and one flip on the other, then you're undoing some of the flips from the other.

AH: Exactly.

KK: Is that why it works, maybe?

AH: That is why it works. So it’s a really cool proof. It's actually a proof by induction, which is so cool, that you can have like a proof on knot diagrams that's a proof by induction. But it's by induction on the number of reducible crossings. So the number of these crossings that you could sort of flip out of the diagram. They're not really necessary for the knotedness of the knot. But the base case is the most interesting part of the proof, where you have a knot diagram that has no reducible crossings. So no extraneous little loops or flips on it. But it's very constructive, and it uses things like checkerboard colorings, and it uses splices, or smoothings, which is where you take a crossing and you turn it into — like, you basically get rid of the crossing by cutting it and reattaching ends so that it's just — I’ve got this picture in my head, how do I say it? What's the best way to say that? So you have a crossing, and you want to get rid of it by cutting it and reattaching ends so that there's no crossing anymore. Does that make sense?

KK: Well, it’s sort of like a braid, right? I mean, so you imagine sort of a braid cross, you just clip the string above and below and then you just reattach, then you don't have it, right? Is that what you’re doing?

AH: Okay, yeah, what you just said totally makes sense because I could see your fingers.

KK: This would be a better video podcast, I suppose.

AH: I know. Yeah, at least for topology, or geometry. But the proof basically creates a checkerboard coloring that tells you how to find a collection of regions where you can basically control which crossing you're going to change. So I can change just one crossing, by carefully selecting a group of regions where exactly one, or exactly three of the regions involved in that crossing are going to get changed, but every other crossing in the diagram is next to either zero, two, or four regions that are being changed. So if it gets changed, it'll get changed back and look like it like it started.

KK: Right. Okay. All right.

EL: So I have not thought about knot theory, probably since we talked with, like, Laura Taalman on this podcast years ago. It's not something I think about a whole lot. And so I was not expecting this induction to be on the number of reducible crossings because they're so silly, you can just undo it, and then your diagram doesn't even have it anymore. So, yeah, why not the number of crossings or the number of regions or something?

KK: Yeah.

AH: So the reason reducible crossings are annoying for region crossing changes is because at a reducible crossing — you know, at any crossing, if you zero in on it, it looks like there are four different regions involved in the crossing, but with a reducible crossing, two of those four regions are actually the same region.

KK: Right.

AH: So it can look locally like you're changing two regions, so that you know, the crossing shouldn't flip. But you're really changing one, so the crossing does flip. So that's why reducible crossings are the annoying thing that you need to carefully control.

KK: Okay. All right.

AH: Yeah. And so once you get into the inductive step, you basically want to take a reducible crossing, change it so that you have two pieces, one has one fewer reducible crossings, and you know how to deal with that. And then one is a totally reduced diagram of a knot.

EL: Yeah. But the base case is the hard part, it sounds like.

AH: Yes, yes, yes. Totally.

EL: Interesting.

AH: Yeah. So yeah, so one of the reasons I love this is because I love unknotting. In general, I find unknotting questions really interesting. And I highly recommend everyone go listen to Laura Taalman's My Favorite Theorem podcast because she talks a lot about unknotting problems. But also the woman who proved this result is named Ayaka Shimizu. She’s a Japanese mathematician, probably my age, maybe a little bit younger, maybe she's about to have her 40th birthday or something, I don't know. But she is one of the coolest mathematicians I've ever met. She's definitely the cutest mathematician, and her talks are so cute that you're like, oh my gosh, I'm watching such a cute talk! And then you realize, oh my God, this result that she just proved is really amazing! So she's just super, super cool. I love her so much, and I think it's amazing that she proved this result that, you know, the Japanese math community wondered about for a long time, but no one came up with a proof before her. And she must have, maybe she was even a grad student at the time, or she was definitely a very young mathematician when she proved this result. So I love it.

KK: So here's a question: why would you want to allow such operations? I mean, because physically, changing the crossing, I mean, that would be great when your shoes are knotted, right? Like, you could just go Oh, snap, that's unknotted. Right. Is there a practical reason? And by practical, it could be including things like, it doesn't change the knot invariants or something? Or I don't know, it must if you get to the uknot, but I mean, is it… or is it just fun?

AH: Well, so the other thing — yeah, it's just fun. The other thing you need to know about me is that I study games that you can play on knot diagrams.

KK: Okay.

AH: And this result enabled this Lights Out-type game, they actually have a website. You can search for this game called Region Select. It's a really fun solitaire game that's a lot like Lights Out if you've heard of that game. And basically, the fact that the region crossing change is an unknotting operation basically means that any lights out game that you can think of or any region select game you can think of is playable, so you can have a knot diagram. Basically, at each crossing, instead of a crossing, you have a light. So it looks a lot like a graph, actually. You have a light and the lights are, some of them are on and some of them are off, and you need to select regions to try and turn all of them on or turn all of them off. And it's a really fun solitaire game that comes from this.

KK: Okay.

AH: But I’ve actually use the region crossing change to invent one of the many games that I've studied. It's called the region unknotting game. And basically, I'm super interested in these types of two-player games, where you start with a knot diagram, or maybe the shadow of a knot diagram. And you have two players doing something to the diagram, and one player wants to create the unknot and the other player wants to create something knotted. And so we have many games of this variety we've invented. One is the knotting-unknotting game. There's the region unknotting game. I’m about to publish a paper with some students called the arc unknotting game. And there are more. I could go on and on listing games.

EL: Kind of like you know that you can always unknot these things, but it's like, can you unknot it faster than someone can knot it? Is that sort of what's hard about playing this game?

AH: It doesn't have to be faster, necessarily. So the game, these games always are of the form, each player is going to move and they're going to go back and forth until everything is completely determined. And then at the very end, you see whether you have a knot or an unknot. And so you could be playing the long game, like, Oh, I'm just gonna wait it out playing on these little crossings over here to force the other player to play in this region of the knot diagram first, so that I can, you know, have the last move and turn it into a knot at the very end. So yeah, they're combinatorial games, topological combinatorial games, which is cool, because then you, then there is a player who has a winning strategy. And so your goal is to figure out which player is it? And what is a strategy that will always allow them to win?

EL: You said that the proof is constructive. So does that mean that given a knot diagram, you — someone who knew the proof — could actually say, okay, I can, you know, look at this knot diagram and do some sort of wizardry on it and say, Okay, the second player is definitely going to have a way to win this game. Or first.

AH: Yes, it can help. But of course, when you're playing two-player games, the other player can always thwart it. Like, let's say, I have to change these three regions in order to make this unknotted. Well, the other player knows that too. And so they're going to make it so that I can't change one of those regions. But actually, the the constructive way that the proof goes for region unknotting, the region unknotting operation, like basically, there's a complementary set of regions that will have the same effect. You can either do all the moves on this set of regions, or you can do all the moves on this other set of regions, and it will have the same effect on the diagram. So that does help inform game strategy, although we haven't looked at the types of diagrams that are terribly difficult to see how to unknot, because those are already hard enough to figure out strategies for.

KK: Right, right, right. Maybe it's like NIM, right? Like, if when you're playing, and if you have a huge numbers or piles of toothpicks, or whatever, you just kind of play randomly for a while, right?

AH: Yeah.

KK: And then when it gets small enough to where you can kind of analyze it, then you start to do it.

AH: Yes. Actually, I was giving a talk on not games at the Canada-USA math camp. And John Conway was in the audience. And he and all the students who were obsessively playing with him got really excited about calculating numbers for these topological combinatorial games. It's kind of a funny story. He said, he doesn't usually come to talks that speakers give at the math camp, or he didn't. And he said, normally he would leave before the speaker started speaking, because he was afraid to make speakers nervous. Like he didn't want them to be too nervous with him in the audience. But he was so intent on thinking about some problem that he was thinking about with a student there, that he just accidentally ended up in the room until it was like too late to leave. And so he told me afterwards about this dilemma he had, like, would it be worse for him to stay? Or worse for him to get up in the middle of my talk?

EL: Oh yeah, I’m glad he stayed. It would feel like a snub.

AH: Yes, he did stay. And we had a nice conversation about it afterwards, which was amazing. Because if you don't know about John Conway, he was, like, the king of knots and games and all of these things that I care about. So it was very cool.

KK: Yeah, that is cool. All right. So part two.

EH: Yes.

KK: What does this theorem pair with?

AH: This is it was such an obvious answer to me. With this paired with, because are you familiar with Nancy Scherich and her math and dance work?

EL: No, I don't think so.

AH: Nancy Scherich is a knot theorist. Actually, she works with braids. And she's also an amazing dancer, aerial acrobatics person. Acrobaticist? Acrobat. Aerial acrobat. And when she was a grad student, she won the Dance Your Ph. D competition, representing cool things about braids with dance. And since then she has recorded a number of other videos demonstrating mathematical ideas. And my husband is a musician, and he makes the music for her videos.

KK: Okay.

EL: Ao she just had a video that came out within the last month that is showing the proof of Alexander's theorem, which is a theorem about braiding. And the music that my husband composed for her dance piece, my husband's name is James Whetzel, was just beautiful, and just beautifully went with this performance that shows how the theorem works. And so I would recommend James Whetzel’s music.

KK: Unbiased, of course.

AH: I’m totally biased, and he has a new, actually, so I always forget if it's under Whetzel or James Whetzel because he has two different music personas. Right, so he has a new EP under Whetzel, W H E T Z E L, and the title track is “I want to go about my day,” and I think “I want to go about my day” would pair very well. Oddly enough, he also has songs called “Reidemeister Moves” and “This Is what Topology Sounds Like” and “Mama Proves a Theorem.” So he has some various songs with mathematical titles.

EL: So interesting that he came up with those and you also have done with things with these. What a weird coincidence.

AH: I know. How strange, isn't it? Yeah. So, but anyway, I think that everyone should go check out Nancy Scherich. I mean, you could probably just go to YouTube. Scherich is S C H, E R I C H. And check out Alexander's theorem. It's so beautiful. She does pole dancing to it.

KK: Okay, cool.

AH: Because it's about how you can turn any projection of a knot into a projection that always revolves in the same direction around a pole. So it works really well with that medium.

EL: That is so neat. So we have some things to watch and listen to after we're finished with this episode.

AH: Yeah.

KK: So we would like to give our guests a chance to plug anything that they're working on, or where we can find you on the intertubes.

AH: Yes, this is very timely because I'm trying to get out the word about an interesting event that I'm cohosting at the Joint Math Meetings.

KK: Okay.

AH: So, last Joint Math Meetings, a bunch of folks associated with Center Minorities in the Mathematical Sciences put on a storytelling event at the Joint Meetings. And it was so amazing and lovely. And they're doing it again, this Joint Meetings. But my friend Aaron Wootton and I were so inspired by this that we decided to also host a storytelling event at the upcoming Joint Meetings. And the theme is, people will be telling stories about some professional rejection that they experienced that was pretty crushing that ended up turning into something even better. So Aaron and I realized we both have stories like this where we didn't get something, we were totally feeling awful about it. And then it ended up being like a way bigger, more awesome thing. So we have a number of speakers lined up, but we need more. And so we have a web form that I created a bit.ly URL for. If you're interested in in telling a story no more than five minutes in length of the Joint Meetings, go to bit.ly/JMM2024STORY, all uppercase. Well, the bit that l y is lowercase, uppercase, JMM2024STORY, and we'd love to have people submit requests to speak, and I really hope that we have a good turnout for the event itself. It's going to be on Friday afternoon at the Joint Meetings. So mark your calendars. And let's see, the session is called Inspiring Stories: How an Academic Rejection Led to Something Amazing.

KK: Okay. In San Francisco, here we go.

AH: Yes. Can I add one more thing?

EL: Of course, you also, you also host a podcast, right?

AH: Yeah.

EL: I don't know if that's the thing you wanted to plug, but you should plug it too, and whatever you were about to say.

KK: You can plug as many things as you want.

AH: Okay, I have a lot. Okay. So we're just finishing up a book that's a handbook for math majors, called Navigating the Math Major: Charting Your Course. And it's going to be published through MAA Press by the AMS, and that should be coming out by MathFest of next year. So be on the lookout for that, especially if you're at a university that has one of these one- or two-credit freshmen seminars for math majors, or, like, an intro to the math major course. But also, it'll be good for just advisors and mentors to recommend to students and for students who might just be starting out in their college career, and they need some advice about, you know, what communities they should try and be a part of, how to apply for an REU, what kind of weird jobs are available for people. And this is what Evelyn was talking about earlier, because she did a wonderful interview for us about science writing, and that career path for math majors. So I definitely want to plug that. And regarding the podcast, it's a collaboration that I do with my friend who's an artist, Esther Loopstra. The podcast is called Flow into Authenticity. But what it's about is, if you're stuck in your life, it could be professionally or it could be personally, how can you use creativity and intuition to get unstuck? And we're actually writing a book on this called Think Like an Artist, Create Like a Mathematician, that's going to be published by 619 Wreath, which is Candice Price and Miloš Savić’s new publishing company. So that might be coming out in 2024, as well.

EL: Yeah. Well, and that sounds like something all of us can probably use it at some point.

KK: Sure.

EL: We always feel a little stuck.

AH: Yes. It's designed — we’re kind of aiming it professional stuckness in the book, but it's really broadly applicable. So very excited about that. And Esther Loopstra is amazing. She's a fine artist. She used to be an illustrator. You know, she used to work for, like, American Greetings and Target and all these places doing illustration. But now she's a fine artist and creative coach, and just super insightful about how we can use creativity to get unstuck.

EL: Cool.

AH: So, yeah, so the podcast is flow into authenticity. And the book is Think Like an Artist, Create Like a Mathematician.

KK: Cool. All right. Well, we'll try to link to everything that we can find links to.

AH: Okay, thank you.

KK: All right. Well, Allison, this has been terrific. Thanks so much for joining us.

AH: Thanks. Thanks for having me. This is such a fabulous opportunity, and I really appreciate getting to talk to you about today.

KK: Sure.

EL: Yeah. It was a lot of fun.

[outro]

On this episode, we talked with our delightful guest Allison Henrich, a mathematician at Seattle University, about the region crossing change theorem in knot theory. Here are some links to things we mentioned that might be interesting for you.
Henrich's website
MAA Focus magazine
Ayaka Shimizu's paper about the region crossing change theorem
Region Select, a game you can play where you try to unknot a knot using region crossings
An article Henrich coauthored about the region unknotting game
Nancy Scherich's YouTube channel, where she shares videos of her dances about math
James Whetzel's song I Want to Go About My Day on Bandcamp
The signup form for the mathematics storytelling event Henrich is cohosting at the Joint Mathematics Meetings in January 2024
Flow into Authenticity, the podcast she cohosts with artist Esther Loopstra

Episode 88 - Tom Edgar

9 oktober 2023 | 26 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. It's Friday. Hooray!

EL: Yeah, yeah.

KK: Long Weekend. Yeah.

EL: It’s the start of a new month. Everything — anything is possible.

KK: Right.

EL: Including a great conversation with our guest.

KK: Yeah. I think it will be good. It's been an okay day so far.

EL: Great.

KK: The hurricane notwithstanding.

EL: Yeah.

KK: But yeah, that went by. But yeah, Hurricane Idalia really did some serious damage. And it’s, yeah, it's rough.

EL: Yeah, and there was recently the tropical storm on the other side of the country that actually kind of affected our weather, and today, I am hoping that the gale of wind outside my window isn't too much, too hear-able on the audio.

KK: I don't hear it, so it must be okay. Yeah.

EL: Great. Well, anyway, we are here today to talk with Tom Edgar about his favorite theorem. So Tom, would you like to introduce yourself?

Tom Edgar: Yeah, sure. Hi. Thanks for having me. It's fun to be here. I love your podcast, as you both know, but now everybody knows I love your podcast. I'm Tom Edgar. I'm a professor of mathematics at a small, comprehensive university in Tacoma, Washington called Pacific Lutheran University, just south of Seattle, about 35 minutes, maybe. Depending on traffic, like an hour and a half. I'm also currently the editor of Math Horizons, which is the undergraduate-level periodical from the Mathematics Association of America. And spend a lot of my time on those two things right there and just getting ready to go back to teaching here starting next week.

KK: Oh, you guys start after Labor Day. Okay, good for you.

EL: Oh, yeah. That is nice. Yes. And I think we've worked together a little bit on various Math Horizons things.

TE: Yeah, both of you have. So I mean, Kevin's on my editorial board, and he's written a couple of things. And then, Evelyn, I met you I think it in person at ICERM back forever ago. And I remember you were nice enough to do a piece about your awesome calendar, which I still have. I actually have a second copy now because I just have two now.

EL: Excellent. Yeah. Well, I would recommend getting one for every room.

TE: It doesn't hurt: one for the office, one at home.

EL: I’m not biased at all.

TE: No, one for your for your classrooms for your students. It's a great idea.

KK: Right. And it's universal. It's not year-specific. So reminder to all of our listeners, go to the AMS bookstore where they seem to be having a sale all the time, right?

EL: Yeah. Can’t afford not to! That's right. Anyway, Tom, now that you've so kindly plugged my calendar for me, what is your favorite theorem?

TE: And just that wasn't planned either. Right? That was just, you know, it's a nice thing that you've done. It's really cool. Yeah, so my favorite theorem is a hard thing. Because I've been listening your podcast for a number of years, and I was like, hey, if I ever get a chance, I wonder what I would talk about. And I had one that I was going to talk about, but I I've changed recently. There have been some projects that I've done in the past few years that kind of have changed my viewpoint. And so the theorem that I want to talk about is a pretty elementary theorem, in some sense. Most mathematicians will have seen it, a lot of, any math-adjacent people will have seen it. And it's the formula for the sum of the first N positive integers. So if you were to add up, say one plus two plus three plus four plus five, right, you can do this addition problem. My son, who's eight, can do this addition problem. But is there a quick way to get to the answer? And so the result is that if you add up one plus two plus three plus four plus five, you can actually get that in sort of fewer computations by multiplying five by six and dividing by two. And so the general formula is, if you were to add up the first N positive integers, pick your favorite number to stop at, N, then the theorem says that that sum should be N times N plus one divided by two. So the number that you stop at, multiplied by the next number, and then take half of that. So I really love this theorem for a variety of reasons.

KK: So there’s the apocryphal, probably apocryphal, story about Gauss, right?

TE: Yeah, for sure. So I definitely enjoy this aspect of it because most people think, oh, there is this story. So the story is, I'm not even going to tell the story because I've read — Brian Hayes has an article where he tries to get to the bottom of this actual story and where it came from, but the general idea is that, you know, some teacher of Gauss gave this as an exercise, to find this sum and expecting it to take a long time and Gauss produces the answer almost instantaneously. I like talking about this because a number of people have changed that story over the years. And so it gets more dramatic, or things like that, or a lot of people think that this is Gauss’s sum formula, that Gauss was the very first person to come up with this, like in the 1800s, like, nobody knew that, you know, this was it. But this has certainly been known — you know, one of my favorite proofs is the picture proof where you imagine the sum of the first N integers is sort of almost like a staircase diagram, one box at the top, two boxes below that, three boxes below that, and so on. And you take two copies of this staircase diagram, rotate one 180 degrees, and stick them together, and you have an N by N +1 rectangle. And Martin Gardner attributes this to the ancient Greeks, right? So presumably, people been drawing this in sands, and all sorts of things, for as long as people been thinking about counting, right?

EL: I must admit, I do — like, that story always bugs me because people, I don't know, people will use it as evidence of like this amazing genius. And I'm sorry, if this is, I don't know if I sound like I’m bragging or something. But like, I figured this out when I was in school, and I'm not a Gauss, by any stretch.

KK: Don’t sell yourself short.

EL: And it's like, you sit around playing with numbers a little bit, then, you know, you can figure this out, it's figure-out-able, which I think is good for people to know, rather than think, Oh, you have to be, you know, some native genius to be able to figure something like that out.

TE: Yeah, for sure. And, and I think, like, I don't know if you've read Brian Hayes’s article on it or not.

EL: I think so.

TE: Yeah. He brings up the point that maybe the reason people like it is because it's sort of, like, the student having this victory over the the mean classroom teacher. And somehow we just love this idea, not necessarily the genius myth, but this idea that like, oh, the the student won, or something like this. But yeah, but it's fun to talk about too. And just that always opens up the conversation with people about all the misattribution that we have in mathematics, right? Theorems named for people that maybe don't even have anything to do with that theorem, for one reason or another.

KK: So let's talk proofs. So you mentioned the one that Martin Gardner did with the picture. Okay. What's your favorite proof? Do you have one?

TE: Yeah. I mean, that one's pretty amazing, if you ask me. You know, I mean, another reason I like this is that this is sort of, if not the, it's probably the standard first induction proof that any undergraduate sees, right? So you learn about induction, and then you prove this formula by induction. I dislike that proof in one sense, and I love that proof in the other, right? So it's nice from learning induction. On the other hand, it's like, man, it's induction. I didn't get anything out of that. Whereas that picture proof from the ancient Greeks, right, just tells you exactly what what to do, right?

EL: Yeah. And I'm trying to remember is there a book or something called, like Proofs without Words or something like that? And it's a great proof without words, because it doesn't take a whole lot of scaffolding to show this picture and the numbers and to see exactly what's going on.

TE: For sure. Yeah, yeah. So Roger Nelson has three compendia now, like Proofs without Words, right? So this is three books, maybe almost a total of 600 pages of diagram proofs. And that one is in the first edition. And it's definitely — I mean, there's a couple iconic proofs without words, and I would put it as one of the top four iconic proofs without words. There's the Pythagorean theorem with a couple, and a couple of other ones that go along with it. But that's that. But my favorite proof actually — well, so, back in, like 2019, right at the end of 2019. Right, the beginning 2020 Before the before, sort of all the craziness, a mathematician named Enrique Treviño, who's a professor at Lake Forest College in Chicago, he was posting some things on Twitter about different proofs of this theorem and I knew a couple and I sent it to him, he's like, Hey, we should write these all up. So we got together and wrote these all up. And so we have a compendium that's online of 35 proofs so far, of the of the fact. And we finished that just before — I think it was end of January 2020, we sort of finished it. We've been working on it here and there ever since. But one that came out of there that's my favorite — and it's hard to describe, so I'll see what I can do — but it's also a picture proof. But instead of taking two triangular diagrams, so two staircase diagrams, you take eight staircase diagrams. The same kind of picture, instead of two and you just glue them together and you get a rectangle, you take eight. So again, the visual here should be sort of a right triangular stack of squares, N squares on the bottom, one square on the top, and then it's right oriented. And when you put eight of these together, you get a perfect square, except there's this one missing cell in the middle. And so it tells you that eight times this, this number, which these are called the triangular numbers, because they fit into these triangular arrays. So eight times the Nth triangular number is basically the Nth odd square. So (2N+1) squared, except missing one, missing one cell, so minus one. And this proof to me, it's much more complicated, in some sense. Like, why don't you just use the real picture proof, the easy one with two? But this one indicates that there are a lot of other things going on. So you can use this proof essentially, to prove that odd squares are congruent to one mod eight and these kinds of things right here. I mean, it sort of falls right out of that. And then this was key to Gauss’s — what's it called? — three triangle theorem, which says that every positive integer can be written as the sum of three triangular numbers. And so this fact plays a role. This visual proof plays a role there.

KK: Okay.

EL: Oh, nice.

KK: Very cool.

EL: Yeah, I'll have to draw that out later. I'm not quite sure I believe you, but I'll take your word for it for now.

TE: You’re going to have to draw it out, for sure. I was like, Oh, should I? Kevin asked my favorite. I wasn't going to necessarily going to talk about that one, but for some reason, I liked that one because it opened my eyes to a lot of other things going on in math as well. So it just has a connection, you know, thinking about what are called figurate numbers. So these are numbers that can be arranged in certain geometric patterns. So the triangular numbers, the squares, these are familiar ones to us, but there are just so many cool mathematical ideas that somehow I never picked up as an undergraduate or a graduate student about these, like Euler’s pentagonal number theorem, or Fermat’s polygonal number theorem, just amazing facts out there that I just never would have come across.

EL: Yeah, well, I guess that one is kind of an overpowered proof for that particular formula. But like you said, yeah, it kind of opens the door to a few different things, a sledge hammer for a mosquito.

KK: I like that.

TE: Those are some of my favorites of the ones that that Enrique and I compiled. One of the ones that sort of blew my mind that we came across was this idea that you can use Euler’s polyhedral formula for planar graphs, right? So the the planar graph version, you can use this and it proves the sum of the integers formula if you just find the right graph, and that's like a sledgehammer!

KK: Oh, nice.

TE: But it’s a beautiful, really powerful theorem for topologists. I think both of you somehow are topologists or topology-adjacent. Am I wrong about Evelyn? Not you?

EL: Yeah. Oh, yeah. Why not?

KK: No, it's true, Evelyn.

TE: The fact that you can use you know, this Euler’s polyhedral theorem, which I know has been featured on your podcast before, and maybe even recently, you know, to me was really powerful, like, oh, you're using something really strong. But it's also a way that you can introduce people to a cool idea with this relatively simple fact, elementary fact that they might be encountering as early undergraduate-level mathematicians, or even earlier than that.

KK: Very cool. All right, so I know visual proofs are kind of your thing. So have you animated this one? I know you like to animate these things. I see them on Twitter occasionally.

TE: Yeah, so I spend my time animating. For the past year and a half, two years this, this arose out of the pandemic, right, we all went online, and some of us were teaching online and kind of upset with maybe some of the digital content that we could produce. And so I spent some time trying to figure out how to how to do some animations. But yeah, so this one I animated, I animated 12 of them, so a dozen of the proofs from Enrique and I, that we compiled I animated a dozen of them last year. This was part of, I submitted as part of Three Blue One Brown, Grant Sanderson, runs this summer of math exposition stuff. So I submitted that last year as my video, the idea being that you should think deeply about simple things because you can encounter a lot of things along the way. And this is not my quote, this is a quote from Ken — the person who started the Ross program, and I'm forgetting the Ross program, I'm forgetting the founder. His last name is Ross but I can't necessarily remember the first name. Okay. So yeah, so I have animated some of them. And I believe I've animated, I think I've animated Euler’s polyhedral theorem, Pick’s theorem, the classic visual proof, there's combinatorial proofs. So there's like, a double counting proof. And then there's one that uses bijective proof. So just some really cool ones out there to see and explore.

KK: On YouTube? They’re on YouTube, right?

TE: Yeah, that’s on YouTube. Yeah. Mathematics Visual Proofs is the name of the YouTube channel at this point. Who knows? It changes if you have to change it, right?

KK: Well, we'll link to it. We'll find it.

TE: Okay. I appreciate that. Thank you. All right. Cool.

EL: Yeah. And so you said maybe this isn't the theorem you would have picked, if we had asked you, you know, three years ago or something. So how, how did this theorem get — Was it this project with Enrique that got you interested in it?

TE: Yeah, I mean, I've always loved the theorem, but sort of seeing all of all of the available proofs and the ways that it could open me up to things. It’s given me well, a couple of things. So when you teach a discrete math course, you can essentially teach the entire discrete math course using this theorem. You can talk about so many different discrete mathematical ideas using this and so it can be fun that way. So I've done that in a discrete math class and really enjoyed that experience with students as they see the connections being made. It's maybe a little more fun to talk about than some of the the others, I mean, the other theorem that I probably would have talked about is called Kummer’s theorem. And that one is fun to talk about, but it requires a little bit more knowledge, or a little bit more technical detail sometimes. So I like the accessibility in this one. I like that I get to speak with people — whenever I get to talk about this, I speak with people about the fact that mathematicians are looking for other proofs sometimes, right? I think mathematicians know this, we know this, that you're not always just looking for one proof. Some people say you're looking for the best proof, the so called proof “from the book.” I don't know if I agree with that. I just like the idea that we're looking for other proofs, other ways to try to understand these things to give that broad picture. And somewhere along the way, before or after, I came across this quote, It's my absolute favorite quote from a, from a mathematician, maybe ever, it's from Bill Thurston, who was a Fields medalist in the late 20th century and passed away only about a roughly a decade ago, maybe. He says, what did he say, “we're not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.” And I just, I wish I had said that. If I could have said that, I think I could die happy, like that was my quote. But I like the idea that we're not just — mathematicians aren't just sitting in the room trying to pump through more results, that we are actually interested in understanding and communicating and trying to get those ideas out.

EL: Yeah. And that, you know, what insight can we get by looking at this problem in a different way even if we already know the answer?

TE: Exactly. I think a lot of people just don't think that way about mathematics. People who are not, who haven't been around mathematics long enough, think that it's just one and done, right? You do this problem, and you move on to the next.

KK: Right, right. Or that we're just sitting around, like, doing arithmetic with really big numbers, right?

TE: Yeah, that's kind of what — that’s actually what that's what this is. This is arithmetic with really big numbers!

KK: That’s right. But clever arithmetic! They think we would just sit there and add it all up. It's like, why would I do that? I don't want to work that hard.

TE: Yeah. Yeah. I'm kidding. That's good.

KK: All right. The other thing we like to do on this podcast is ask our guests what it pairs with. What pairs well, with this formula?

TE: Yeah, so this is the greatest part about your podcast, not that there not other good things about your podcast, right? I think you two are great together. And it's fun, you know, but I think the idea of this and I was — this is the challenging part with with the other theorem I was thinking about. I was like, wow, what would I pair it with? I don't know. Presumably, I would come up with something. But this one was fairly easy for me. When I was younger, a movie came out, and over time, I guess it's become somehow I read online, that it's one of the greatest comedies of all time. I'm not sure if I agree with that. But I watched this movie a lot. And this movie is called Groundhog Day.

EL: Oh yeah!

TE: Have you seen Groundhog Day?

KK: Many times!

TE: Exactly.

EL: My thing about Groundhog Day is like watching it once is like watching it several times. Right. And then if you watch it more than once you've just like really increased your your volume of Groundhog Day.

TE: Right. So you you have no idea, exactly, you have no idea how many times you've seen this movie, you're sure you've seen this movie 30 times, but maybe you've only seen it twice. Right? But for people who haven't seen the movie, the premise is Bill Murray is a weatherman from Pittsburgh, Pennsylvania, and he's tasked with covering Groundhog Day and Punxsutawney Phil and he doesn't want to go there and essentially ends up in sort of a time loop where every morning he wakes up and it's exactly the same day and he's the only person who thinks he's reliving the day and everyone else is treating the day as the same. And so he does various things to try to, I guess the idea was to sort of “get it right,” sort of be the best possible person. But from my perspective, this is exactly — what would a mathematician do if they ended up in the Groundhog Day situation? Well, which is every single day I would just find a new proof of the sum of the integers formula and I would maybe never be bored. Maybe I'd never get it right and get out of the time loop. But I liked this idea because essentially in the movie, he learns a lot about himself, he learns a lot about the people around him. And this is sort of what happened with me working with Enrique and learning a lot of the things that come along with this theorem. You learn a lot of stuff and like, oh, this is stuff I didn't know, and it's led me to a lot of other things that I didn't know and connected me with other people. And so it's kind of like that movie, I guess. So, you know, sit down and watch that movie and figure out a couple of new proofs of the sum of the integers formula.

KK: And remind yourself of the genius of Sonny and Cher.

EL: Yes.

TE: A song that you probably probably can't listen to ever again, without automatically thinking about the movie.

KK: No, probably not.

EL: Yeah.

KK: No, that's a great pairing. I like that.

EL: Yeah, that's a nice one. I think. So I think in the movie, one of the things he does is he becomes this great piano player, right? Because he has so many times through the day. And you know, he goes, at some point, I think goes to his lesson and is like, oh, yeah, I've never played piano before and just busts out something. I always thought, like, oh, that would be — what would I have the dedication to do something like that if I got this time?

KK: What else do you have to do?

TE: Well, it's a great, that's what's so cool about the movie is, like, really, if you put yourself in that situation, you could do whatever you want. Right. I think that was what was so good about it in the end, he learned to play the piano, he learned to be a good person, I guess as well. But you know, like, you just learn a lot of things. He

KK: He learned to do ice sculpture!

EL: Yeah, that’s right.

TE: Yeah. The end scene, like, the last day when he does everything right, it’s just it really puts it, it brings it together so nicely. Like, oh, he saves that person's life and builds his ice sculpture and he's really filled himself out, right? I mean, there are some dark parts of the movie as well, but it ends nice. I can see why people might say it's the greatest comedy of all time.

EL: It’s up there, for sure, I think.

TE: And from the mathematics — there's this one scene, like from mathematics point of view, mathematicians, they famously love their coffee. And there's this one scene when he's kind of at one of his low points, and he's just eating all of the foods at the diner and he grabs this thing at coffee, and he just drinks it straight like that. I'm like, oh, okay, I could see a mathematician doing this in Groundhog Day.

EL: Yeah.

KK: All right. Well, this has been great. We always like to give our guests a chance to plug anything they want. So you've plugged the YouTube you've, you've plugged a little well, we plugged it for you.

TE: Yeah. Thank you. Oh, yeah. Plug the YouTube I appreciate.

EL: And Math Horizons, which I'm still involved with for one more year. And then there'll be someone taking over there. Yes. Yeah. It's been a long time. I don't know if I have anything else to plug otherwise, I appreciate you all having me on. It's fun to come and talk about these things. I guess I could plug — No, I don't know, for mathematicians interested about this favorite proof that I mentioned of the sum of the integers formula, this somehow told me that there's a connection between, there's sort of three famous proofs that you see as an undergraduate math major, would be the sum of the integers formula for induction, the fact that the square root of two is irrational. And then maybe the arithmetic mean, geometric mean inequality, you might learn as a first inequality type proof in a in a real analysis course or something. But somehow, there's a visual proof for all of these and the visual proof is somehow the same. So I think that possibly those theorems are somehow the same, in some realm. And so I spent a little time trying to prove one of those theorems using different techniques. So I recently had an article if people want to check in Math Magazine about the arithmetic mean, geometric mean inequality, where you prove it using moments of mass and centers of mass. And I was inspired to do this because David Treeby proved the sum of integers formula using moments of mass and centers of mass.

KK: This one? [Kevin holds up Math Magazine.] It happens to be sitting on my desk.

TE: That’s a different one.

KK: That’s not you?

TE: I didn't — I didn't know that — No, that is me, and I wasn't going to plug them both. But that's where I use the centers of mass to prove that the square root of two is irrational.

KK: Okay, that's what it is.

TE: So somehow this proof allowed me to connect those things together. And so it's been fun to play around with ideas that I that I don't know. So if you're interested in how balance plays a role in pure mathematical ideas, I would check those out. So that's one thing I can plug.

EL: Yeah, we’ll link to those. Those sounds really interesting.

TE: Thank you.

KK: All right. Well, Tom, thanks so much. It's been terrific.

TE: Yeah, thank you both. I know it's hard work, the work that you all do, but I think the community needs it and we appreciate it and it's great for my drives to work.

KK: Okay, thanks.

EL: Well thank you.

[outro]

On this episode of the podcast, we chatted with Tom Edgar of Pacific Lutheran University about the formula for the sum of integers between 1 and n. Here are some links you may enjoy:
His website and Twitter profile
Math Horizons
His collection, with Enrique Treviño, of proofs of the sum formula
His YouTube channel, Mathematical Visual Proofs, including his video on the 8-triangle proof of the sum formula
His article about proving the square root of two is irrational using centers of mass
His article about using centers of mass to prove the arithmetic-geometric mean inequality

Also, Brian Hayes’s article about Gauss: https://www.americanscientist.org/article/gausss-day-of-reckoning

Episode 87 - Tatiana Toro

7 september 2023 | 24 min

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I am one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and your other host is…

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we sadly are past our beautiful, not too hot spring and fully into summer. So we enjoyed it while it lasted. I didn't have to turn on any air conditioning until after the start of July.

KK: I think we started air conditioning in March.

EL: Slightly different.

KK: Little different vibe down here in Florida, but that's where we are. So anyway, it's summertime here, which means that there are tumbleweeds rolling through my department and I'm answering a few emails a day and trying to work, trying to do math. And boy, sometimes it's hard, you know, but sometimes it isn't. So. Anyway, so today, though, we are — this is great — we are very pleased to welcome Tatiana Toto, who will introduce herself and let us know what she's all about.

Tatiana Toro: Thank you very much for the invitation. I'm very glad to be here. And in fact, I'm very glad to see Evelyn's cloud that I had heard about in other podcasts. So I'm Tatiana Toro. I'm a mathematician at the University of Washington, where I have been a faculty member since 1996. And currently I am the director of the Simon's Lab for Mathematical Sciences Institute, formerly known as MSRI. And I'm in Berkeley, California, and summer hasn't arrived yet.

KK: It never will.

EL: Yeah, that’ll be November, right?

KK: I had actually forgotten that the name of MSRI had changed to the Simon's business. That’ll take some getting used to. I think I mentioned before we started talking, I spent a semester there, way back in 2006, and my son came with me, and my wife did too, and he was seven at the time. And now he's an adult living in Vancouver. It's weird how things change. I love that building, though. And the panoramic view you have the bay, and you can watch the fog roll in through the gate at tea time. It’s just a really wonderful place. So congratulations. How long have you been director? Has it been a year yet?

TT: It’s almost a year, a year August first.

KK: Yeah. That's fantastic. What a terrific position. And I'm glad that you're willing to take it on. Do you split your time between Berkeley and Seattle? Or are you mostly in Berkeley these days?

TT: I am mostly in Berkeley. My students are still in Seattle, so I see them mostly on Zoom. But once in a while on a Friday, in Seattle.

KK: Oh, so you go there. You don't fly them down?

TT: Some of them have come, actually one of them this year to the summer school.

KK: All right. So what is this podcast about? Favorite theorems. And you told us yours ahead of time, but we'll let you share. What is your favorite theorem?

TT: Okay, so my favorite theorem is the Pythagorean theorem, and I know that everybody's gonna say what on earth are you talking about?

EL: No, I really, really love this choice. And, you know, I've said this on many other iterations of this podcast, but I love that, you know, we'll get things that span the gamut from Pythagoras theorem, or the infinitude of primes, or something like that, all the way up to something that you, you know, you need to have been researching for 20 years in some very ultra-specific field to even understand, and so, you know, it just like shows how math connects with us in different ways at different times in our lives, and how we can appreciate some maybe things that seem very simple about math, even when we have had math careers for for many years. So yeah, tell us about how did you end up settling on the Pythagoras theorem?

TT: So, actually, it has played a very important role in my career. Like, when I describe it to my students, when I'm teaching a graduate class and I talk about the some of the theorems I'll describe in a minute, I tell them, you know, one of the key ideas in my thesis was the Pythagorean theorem. So let me explain. It appears in many other results in this area of geometric analysis. So for example — let me give you two examples. So what was my thesis about? You have a surface, a blob in space, and you're trying to — two dimensions in R3 — and you're trying to understand if you can find a parameterization, which means a good way to describe it in terms of the plane. So can you deform the plane in a nice way so that it covers the surface? And a nice way means that distances are not changed too much. So I had some specific conditions for this surface, and the answer, the key, is in the situation I was looking at, yes, you could do it. And when you go and deeply look at what makes this possible, it is the Pythagorean theorem because the basic point is that if you can control how distances are distorted, you can control how the whole shape is mapped from the plane. And at the time, it looked like a curiosity. You know, I graduated many years ago. At the time, a few years earlier, Peter Jones had solved the analyst’s traveling salesman problem, which I'm gonna — just in general terms, let's imagine you have a lot of points in a square, and you're trying to understand whether you can pass a curve of finite length to all of these points. You're going to tell me, “If they’re finite, of course you can.” But you want to do it in an efficient way, in a way that doesn't depend on the number of points. And so he had found the condition that told you if this condition is satisfied, then yes. And there's not an algorithm, that doesn't exist yet, that tells you what's the best curve, but there's a curve, and he tells you that the length is no more than something. And what's behind that is the fact that if you have a straight triangle that has sides, A and B, and the other one is B, A squared plus B squared equals C squared. And it really is understanding that. And there's another important thing, the fact that the square root also plays an important role in these, but really, really, if you ask me, “What are the tools you need in this area?” I'll tell you how the square root behaves in the Pythagorean theorem, and then a couple of good ideas and you're able to reconstruct the whole thing.

KK: I’m now curious about this traveling salesman problem. So there's no algorithm though?

TT: No, there's no algorithm. I used the word analyst’s traveling salesman problem because the analyst wants to know whether you can pass a curve of finite length. Maybe you can say you're not ambitious enough. You don't want the shortest possible curve. To build the shortest curve, there’s no algorithm. And the construction of Peter Jones builds a curve, but it's not necessarily the best one.

KK: Sure. Yeah.

TT: It doesn't tell you it tells you the length is no more than D. But it's not. Yeah, no.

EL: Yeah. I'm trying to remember if, like, I think there probably are some algorithms or some, like results that say like, you can get within a certain percentage of something. But yeah, the algorithm for the actual fastest path doesn't exist yet. Which is, you know, it's one of those things, it's like, huh, that's kind of surprising that we don't have a way to do that yet. Just means that there's still work to be done. Still jobs out there for mathematicians.

KK: Well, because the combinatorial on the graph theory one is, is NP complete, right? I mean, yeah. So that that are NP-hard, or whatever. NP-something. I've never been clear about the differences. But is this one known to be that too?

TT: I believe.

KK: Okay. All right.

TT: But you can construct — you know, so this was what was interesting about the problem, the result of Peter Jones, is that — the result of Peter Jones, and I have to say, I was very ignorant of that result, which had just happened a few years prior to my thesis. I have to remind the young audience that at the time, there was no internet the same way, and there was no arXiv, and you know, there was no Zoom. And then Peter Jones had a couple of postdocs at Yale, Stephen Semmes and Guy David, who started working on this. And the truth is, may I tell story about my thesis?

EL: Yeah.

KK: Please do.

TT: So my thesis came out of misunderstanding. I went to my advisor, and I showed that these surfaces that I was looking at, which were some that he had looked at, that there was this property about distances over the surfaces, like if an ant traveled on the surface between two points, you know, taking the shortest path, it was comparable to the Euclidean distance. And so I went to my advisor, Leon Simon, and I told him, you know, I've been able to do this about these surfaces. And then he told me, oh, then I guess they have about they admitted bilipschitz parameterization, which is this good description. So okay, so I went to the library, and I looked through every possible book that I could find, and I couldn't find that. So I went back two weeks later and asked if he’d mind giving me a reference for this results, and he said, oh, I don't have a reference. That must be true.

KK: It must be true.

TT: And that became my thesis problem. And then, oh, there were many iterations of attempts. And I could do specific cases, but I could not do the general case. And on May of my fourth year, finally, somebody gives a colloquium where he talks about good parameterizations. And he talks about things like what I was thinking. I was thrilled. I mean, I thought, oh, I'm going go read everything this guy has written and my answer will be there. And then I told my advisor afterwards, I think I'm going to go read this guy's work. And this guy was Stephen Semmes, and he comes from harmonic analysis. And my advisor says, no, stop reading, I don't want you reading anymore. You just prove that theorem and that’s it. I don't want you reading. But one good thing, you know, harmonic analysts use squares, rather than balls. That's the most useful comment my advisor had.

EL: Huh!

TT: And what's interesting is that Stephen Semmes was talking about a broader class of surfaces than mine. And for those, he was asking, “Do bilipschitz parameterizations exist?” And for those the answer still is not known. And if I had gone and read everything that he had written, I mean, he was the big shot, I was the student, I might not have gotten my result. And I remember when I told Stephen at some point in the fall, oh, you know, I proved this, his first question, his first reaction, was, “I don't believe you.” And he said, “How did you do this?” And I said, “Using the Pythagorean theorem.” And so that's why the Pythagorean Theorem really is very dear to my heart.

EL: Yeah. So I imagine that you saw the Pythagorean Theorem many years before you were in grad school. Do you remember, did it make a big impact on you when you saw it in school for the first time? I don't know what what year that would have been, elementary or middle school or whatever it was?

TT: So I remember, I think I remember when I saw it because I remember the book. I had a beautiful — I went through the French system. I'm Colombian, but I went through the French system, and in the French system at the time, they tracked us very early on. And so we had these beautiful math book that, you know, I still remember how it smelled, and it was in there. But I remember the book, not especially the theorem. I never thought much about it until I got to graduate school. I used it other times.

EL: Right. I mean, I think maybe the beauty of that kind of thing isn't necessarily what you're looking at, when you're a kid and first seeing math. You’re more like, okay, how can I use this to do the problems on the homework or something like that? So you were tracked into math pretty early on? You knew very early on that you were interested in math?

TT: Yeah.

KK: It’s nice they let you just do math. I think in the US what happens, I think, is students who are good at math are told they should be engineers. As if they're kind of the same thing, and they're not.

TT: But that, you see, now, you feel free to remove this if you want. That's what the boys were told. The girls — since math was roughly like philosophy, and I come from a South American country, it was okay.

KK: That’s fascinating. Okay, interesting.

EL: Yeah. Well, I mean, there's a lot of different, you know, philosophies about whether tracking that early, you know, kind of deciding on what direction you want to go that early, is good or not. You know, it works for some people and not others, definitely.

TT: Absolutely. I think it worked for me very well. And it didn't work on any of my classmates who were in the same class. I mean, I thought everybody loved it the same way I did and had as much fun. And then, it's interesting. Later on, I've learned that that wasn't the case. And then some of them suffered through it, you know. But to me, it was great.

KK: So this is a French system in Colombia? Okay, this is a bit — okay, let’s get there. How did that actually happen? Why were there French schools in Colombia?

TT: Well, I'll explain why there were French schools in Colombia and how I got into a French school. So there's something that's called a cooperation agreement between France and developing countries, where they have schools. The primary reason to have them is so the kids of their diplomats can continue their studies, but then they also offer them to the general population at a very reasonable price. They are private schools, but they are not as expensive. They're a fraction, or they used to be a fraction, of what the other private schools were. And at the time, so Colombia for a long time was what was called a Sacred Heart country. And so the ties with the Catholic Church were very strong. And so in terms of education for the girls, it was most girls went to nun school. But I am not Catholic, and therefore I couldn't, that was not an option for me. And so we needed a coed school. I mean, my parents wanted a coed school. The girls schools were all nuns. They wanted a coed school, and we needed an affordable coed school, and public schools were not good, and still unfortunately are not good. That's how I landed in the French school.

KK: Fascinating.

EL: Wow. Okay. Yeah.

KK: Our listeners are learning all kinds of stuff, right?

EL: Yeah, yeah, we've wandered a little away. But luckily, we know, thanks to the Pythagorean theorem, that we can walk back in a certain amount of time. So yeah, the other things that we like to do on this podcast is have you pair your theorem with, you know, some food, beverage, sport, you know, whatever, delight in life you would like.

TT: So I actually will pair it with walking. So I'm going to give myself the title of urban hiker. I do walk long distances around town and in cities on a regular basis. I mean, I walk about two hours a day, at least. And so I pair it with that, because most often when I walk, I'm actually doing exactly the opposite of the Pythagorean theorem. I want to go the longest possible way, not the shortest possible way. But once in a while, I take the diagonal. And now that I'm living here in Berkeley, there's a beautiful diagonal that I take. And so I think about that here often.

EL: Yeah. Do you like the hills?

KK: Yeah, I was about to say, do you actually hike all the way up to the building there? Because that is quite a hike.

TT: Not when I'm coming to work. But sometimes on weekends I do. You know, I want to crease and it depends. It depends what I'm doing while I walk. I use walking as a way — if I am listening to a book, then I can go up the hill. But if I want to talk on the phone, I need to go down the hill, because the reception here is terrible! I know exactly at what point on the hill, you lose AT&T.

KK: That’s true. Yeah, like I said, I was there some time ago and cell phones weren't quite as good as they are now. But yeah, my reception was terrible at the institute.

TT: Well, your cell phone might have improved, but the reception hasn’t.

EL: Yeah. I love this pairing I love walking and biking as like, ways to, you know, see the city on a human scale instead of when you're in a car or something and you just almost teleport from point A to point B, you don't like see the — you kind of don't get the same environment around you, that kind of effect. So I like that. Even though walking is also a great time to sort of, like, let your mind wander and not think about what's around you, listen to your book, or talk on the phone with someone, or think about proving that next theorem, or anything like that. So it's kind of that, it has both of those things.

KK: You live in two great cities for walking.

TT: Ye.s. With respect to, you know, seeing things differently, one thing I find amazing is that depending on what side of the street you walk, you see things differently.

KK: Absolutely. All right, this has been terrific.

EL: Gotta be some metaphor in here.

KK: I’m sure, I’m sure. Yeah. So it's always nice to get another perspective on the Pythagorean theorem. So we didn't even — it's one of those things that everyone knows so much, we didn't even tell them what it was. I think it was embedded in there somewhere.

EL: Yeah.

KK: But the idea that it is still vital, like still important in modern research mathematics, you know, is a really interesting thing to know about. We all just sort of take it for granted. Right?

EL: Yeah, this theorem that has been known by humans for millennia. And, you know, still is important.

TT: One of the things that these ways of building parameterizations, so they developed into a whole field, and then they moved to other areas. So there are some recent results by Naber and Valtorta trying to look at a singular set of minimizing surfaces, varifolds, you know, that minimize some sort of energy. And they have been able to give a very good description of the singular set by using these type of parameterizations. And they're all basically, the basis is always the Pythagorean theorem. It's really, that's how distances change.

KK: That’s right. It’s completely fundamental.

EL: Thank you so much. This was really fun.

TT: Thanks for the invitation.

[outro]

In this episode, we were happy to talk with Tatiana Toro, mathematician at the University of Washington and director of the Simons Laufer Math Foundation (formerly known as MSRI), about the Pythagorean theorem. Here are some links that you may find interesting.
Toro's website and the SLMath website
Our episodes with Henry Fowler and Fawn Nguyen, who also love the Pythagorean theorem
The analyst's traveling salesman problem on Wikipedia
Naber and Valtorta's work on singular sets of minimizing varifolds

Episode 86 - Sarah Hart

20 juli 2023 | 41 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. Or perhaps today we should say the maths podcast with no quiz at the end. My name is Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. It's Juneteenth.

EL: It is, yeah.

KK: And I'm all alone this week. My wife's out of town. And yesterday was Father's Day and I installed cabinets in the laundry room. This is how I spend my Father's Day, something we've been talking about doing since we bought the house.

EL: That’s a dad thing to do.

KK: 14 years later, I finally installed some cabinets in the laundry room. So it looks like you had a good time in France, judging from your Instagram feed.

EL: Yes, yeah. And I'm freshly back, so I'm in that phase of jetlag where, like, you get up really early. And so it's 9am and I already went for a bike ride and did some baking and had a relaxing breakfast. At this point, I'm always like, “Why don't I do this all the time?” But eventually my natural circadian night owl rhythms will catch up with me. I'm enjoying enjoying my brief, brief morning person phase.

KK: Yeah. Never been one, won’t ever be one as far as I can.

EL: Yeah. Just keep moving west, and then you’ll be a morning person for as long as you can keep jetlag going.

KK: That’s right. That's right. Yeah.

EL: So yeah. Today we are very happy to have Sarah Hart on the show. Sarah, would you like to introduce yourself? And tell us a little bit about, you know, what you're all about?

Sarah Hart: Ah, yes. So my name is Sarah Hart. I'm a mathematician based in in London in the United Kingdom. I'm a professor of mathematics, but my true passion is finding the links and seeing them between mathematics and other subjects, whether that's music or art or literature. And so I think there's fascinating observations to be made there, you know, the symmetries and patterns that we love as mathematicians are in all other creative subjects. And it's fun to spot them and spot the mathematics that's hiding in all of our favorite things.

EL: Yeah. And of course, just a couple of months ago, you published a book about this. So will you tell us about it?

KK: Yeah,

SH: So this book, it's called Once Upon a Prime: The Wondrous Connections between Mathematics and Literature. And in the book, I explore everything from the hidden structures that are underneath various forms of poetry, to the ways that authors have used mathematical ideas in their writing to structure novels and other pieces of fiction and the ways that authors have used mathematical imagery and metaphor to enrich their writing, authors as diverse as you know, George Eliot, Leo Tolstoy, Marcel Proust, Kurt Vonnegut, you name it. And then I also look in the third section of the book at how mathematics itself and mathematicians are portrayed in fiction, because I think that's very, very interesting and shows us the ways in which those things at the time the books are written, how is the mathematics perceived? How has it made its way into popular culture? And how mathematicians are perceived as well, that tells us something fascinating, I think, about the place of mathematics in our culture.

EL: Yeah, definitely.

KK: We’re always portrayed as either mentally ill. Or just, like, absurd geniuses, you know, when really, you know, we're all pretty normal — most of us are pretty normal people, right?

SH: Yeah. Well, we are, as everybody, there's a range. There's a range of ways to be human. And there's a range of ways to be a mathematician. But yeah, we're not all tragic geniuses, or kind of amoral beings of pure logic, or any of those things that you find in books. So yeah, and there are some sympathetic portrayals of mathematicians out there, and I know I talk about some of those, but yeah, it's very interesting how these these tropes, these stereotypes can creep in.

EL: I must confess I'm about three quarters of the way through, I haven't quite finished that last section. But the first few sections that I've read, I've definitely — I keep adding books to my “Want to Read list,” so it’s a little dangerous.

SH: Oh yeah, it should have a little warning, the book, saying “You will need a bigger bookcase.” Unfortunately, you know, you will want to go and read all of these books. And yeah, “Sorry, not sorry,” I think is the phrase.

EL: Yes, definitely. I downloaded — so I don't need a bigger bookshelf because I put this one on my ereader — but I downloaded The Luminaries, which sounds like a really interesting book and excited to get to that, you know, in the neverending list of books that I'd like to read.

KK: Right, we were talking about talking about our tsundoku business before [tsundoku is a Japanese word for accumulating books but not reading them]. So I actually I did, with a friend in the lit department, or in the language department, we taught a course on math and literature a few years ago.

SH: That’s fantastic.

KK: It was. It was so much fun. It's the best teaching experience I've ever had. But I was glad to read your book because we missed so much. Right? I mean, of course, we only had 15 weeks, you know, we and we talked about Woolf, like To the Lighthouse is kind of an interesting one. And yeah, I did finish the book. So sorry, Evelyn, I won. But no, it's it's actually, you know, it is spectacularly well written and, and I'm glad you're having success with it. Because it's — again, I like this idea, that you're sort of humanizing mathematicians and mathematics and showing people how it's everywhere. Isn't that part of your job? Aren’t you the Gresham professor, is that correct?

SH: Yes, I’m the Gresham professor of geometry. So Gresham College is this really unique institution, actually. It was founded in 1597 in the will of Sir Thomas Gresham, who was a financier at the Court of Queen Elizabeth I in Tudor times. And in his will, he left provision for this college to be founded that would have seven professors, and their whole job was to give free lectures, at the time to the people of London. Of course now it's all livestreamed and it goes out and is available all over the internet. And anyone could go and it was just, you know, if you wanted to learn these subjects — and he thought there were seven most important subjects at the time that he said, I still say, geometry and mathematics more broadly, very important — but it was geometry, music, astronomy, law, rhetoric, physic, which is the old word for medicine, and I perhaps I’ve forgotten one. But yeah, these subjects, and so still today, this is what Gresham College does, free public lectures to anyone who wants to come. Now, you used to have to give them once in Latin and once in English. Now, you do not have to do it, thank goodness.

KK: Yeah. Who would come?

SH: I don’t know. Yeah, if I had to suddenly give my lectures in Latin, that might be slightly more of a challenge. My role there is to communicate mathematical ideas to anyone who wants to listen, so a general audience. And some of them will have mathematical training, but many will not. And they they're just kind of interested people who find things in general interesting, and mathematics is part of that. I love that idea, that mathematics is part of what a culturally interesting person might want to know about. And that is something that perhaps used to be more so than it is today. And I really would like mathematics to somehow be rehabilitated into what the cultural conversation involves, rather than it seems to be perhaps in a little bit, sometimes it's pigeon holed or put to one side, you have to be a geek to like mathematics. You have to be unusual. And it's really not true. It's not the case.

EL: Yeah. Wow, that sounds like a dream job. I’m writing that down and putting it on my dream board? It's yeah.

KK: I seem to remember, so I read the review of your book, I think by Jordan Ellenberg, who's also been on.

SH: Yes.

KK: It mentioned that the first person who held your chair invented long division. Is that right?

SH: It’s true.

KK: That's what used to get you a university job, is you invent long division.

SH: Yeah. So that's, you know, what a lineage to be part of. I really feel honored and humbled to be in that role. And, actually, I'm the first woman to do this job in its 400 and whatever year history which, yeah, okay, you could say, yes, we might be a bit late with that one. But I feel it's a real privilege to do it.

EL: Yeah. Well, that's wonderful. So we have invited you on this show to tell us what your favorite theorem is. So have at it.

SH: Okay, so, my favorite theorem, I guess it's could be called a collection of theorems really, but the properties of the cycloid. So the cycloid is, it’s my favorite curve. And it's my favorite curve that probably unless you're a mathematician, you may not have heard of it. So people have heard of ellipses and circles and parabolas. And they've heard of shapes like triangles and things, but cycloids, people tend not to have heard of. And for me that's a surprise because they're so lovely. And the history of the study of the cycle of which, you know, we can we can talk about, is so fascinating and fun, and so many of the most famous mathematicians that people have heard of, like Isaac Newton, and Leibniz, and Mersenne, and Descartes and Galileo, and Pascal and Fermat, all of those people worked on the cycloid and were fascinated by it. And so there are these beautiful properties that it has, which we can bundle up into a theorem. And that would be my favorite thereom.

EL: That’s great. And yeah, in case anyone listening to this doesn't know about the cycloid, it’s a cool curve. And it's actually, you know, it's a curve that a lot of people haven't seen as such, but it's one that does kind of arise sort of in everyday life, kind of. So yeah, do you want to describe what a cycloid is?

SH: You can make a cycloid quite easily. It’s a fairly natural idea, I would say. Imagine a wheel rolling along the road. And now somewhere on the rim of the wheel, you paint the put a little blob of paint, or something like that, or if it's in the dark, you can put a little light. And then and then as the wheel rolls along, that blob of paint or little light will be following a particular path, as the wheel rolls.

EL: Going up and down.

SH: Kind of up and down. And eventually, sometimes it'll touch where the ground is. And then we'll go up and down again. And what you get is a series of arches, they look like arches. And that's what the cycloid is, normally you take one arch and call that the cycloid.

KK: Right.

SH: So this is quite a natural idea, what kind of shape will that be? And what is this arch shape? And the first thing you can say is, yeah, is it something I already know about? So early on in the study of this curve, which is first written down as a question, what is this shape? About 1500. Marin Mersenne, who is famous for Mersenne primes, among other things, so he thought maybe it's half an ellipse. And that's not too bad an approximation, but it isn't quite that. And so that's sort of question one. Is it something we already know? And it wasn’t. So then, people like Galileo started to ask, well, what do we what do we like to know about shapes and curves? So there are two questions really, at the time, they were called the quadrature question and the rectification. So quadrature is what's the what's the area? So if you make this arch, what's the area underneath this arch, between the arch and the road, I guess. That's question one. And the other one is the rectification: what's the length? So how long is this arch in terms of the circle that makes that makes the arch, the cycloid. And Galileo didn't know how to calculate either of those things. But he actually made, he physically made a cycloid. So he got a piece of sheet metal, and he rolled a circle along it, and he got the path. And then he cut it out and he weighed, he weighed the bit of metal that he had.

EL: Oh wow!

SH: To find an estimate for the area. Okay? So this is a real hands on thing.

EL: Yeah, that’s commitment.

SH: Because he did not know. So he physically made it and weighed it. And he got an answer that was around about three times the area of the of the circle that makes it, roughly speaking, and he said, Okay, if we all think, what’s a number that's roughly three, that's to do with circles, right? And so he wondered, could it be pi times the area of the circle? It isn't. It isn't pi times the area of the circle! Galileo never managed to work out exactly what it was. But this guy Roberval, Gilles de Roberval, did manage to work out what the area is. He didn't tell anyone how he'd done it because at this time in history, there were all these priority disputes, who sorted this thing first, who has done what first? People would sometimes go to the length of writing their solutions in code. So Thomas Hooke, who was another Gresham professor, when he worked out what we call Hooke’s law now, he wrote Hooke’s law down as an anagram in Latin, before he told anyone else. And then if anyone else came up with it, he could say, look, here's my anagram that I did earlier to prove that I thought of it first. So there were all these weird and wonderful things that people did at that time to establish priority. But Roberval, he had this incentive for not telling that he knew the area under a cycloid. And the incentive was this — it was not a good idea for them to do this — the job he had at the time, Roberval, was renewed every three years. And to get the job every three years, there were some questions that were set. And if you could answer those questions the best out of all the people who tried to do it, you could get that job for the next three years. But the person setting the question was the incumbent professor. So if you're the incumbent professor, you need to set questions that only you know the answer to, and then you get to keep your job. So for a few years, Roberval could say, you know, what's the area under this cycloid, and no one else knew. So he worked it out. And his proof was quite nice, but it wasn't published until 30 or 40 years after his death. But it actually — and this is the first lovely thing about the cycloid — the area, if you have a circle that's making this cycloid by rolling along road, the area underneath one of these arches is exactly not pi times, exactly three times the area of the generating circle. So a lovely whole number, simple relationship between the arts.

EL: What are the odds? It’s almost miraculous.

SH: Fantastic. So here's another equally miraculous thing that kind of adds to the first one. Then people try to work out what's the length of this cycloid? And the person who managed to solve that was, in fact, Christopher Wren. So he's well known as an architect, and he designed St. Paul’s, the wonderful dome of St. Paul's in London, and many other churches in London. But he was also a mathematician among many other things. So he solved the rectification problem, what's the length, and if the circle that makes this, the cycloid has diameter d. So we know that the circumference of that circle, the length around the circle would be pi times d. Well, another beautiful whole number relationship, the length of the cycloid arch is exactly four times the diameter. A beautiful whole number relationship. It's fantastic. So you've got these two lovely properties of the cycloid. And people were fascinated by it. So it had this nickname, the Helen of geometry, as in Helen, you know, face that launched a thousand ships.

KK: Right.

SH: It was a very beautiful curve with beautiful properties. But there's another reason why it was called the Helen of geometry. And it was because, like Helen of myth, it started lots of squabbling. So I mentioned Roberval, who had proved the area formula for the cycloid. Someone else came along a few years later, and found out this this result, and Roberval immediately accused him of plagiarism. And this guy was like, No, I didn't do that. But they argued about it. I think it was Torricelli. And and When Torricelli died a few years later, team Roberval said he's died of shame because of being a plagiarist. He may have died of shame. But he also happened to have typhoid at the same moment. So you know.

KK: Sure.

EL: Shame-induced typhoid?

SH: But you know, so that was one squabble, but then Fermat and Descartes had an argument because they both proved something about the tangents to the cycloid. And they hated the way each other done this. So I think it was Fermat did have a particular method. Descartes said that this method was ridiculous gibberish. So you know, he's not mincing his words, he’s not saying “I prefer my method” but “Fermat is speaking gibberish nonsense.” So they argued. But, you know, this beautiful curve has other exciting properties. And this is where it goes for me from, “Okay, nice whole number relationships, cute.” But then one of the things that we all love in mathematics is where something you've studied over here, reappears in a completely different context. And this is what happens with the cycloid. So it comes up to in connection with trying to make a better clock. So there's this mathematician, Christiaan Huygens, who is trying to make a better clock. And he comes up with a pendulum clock. And so pendulum clocks improved timekeeping dramatically. Before the pendulum clock came along, basically, it was a sundial or nothing, really. There were no good mechanical clocks. And the ones that existed would lose about 15 minutes a day or something of time. The pendulum clock comes along. And so you can do kind of the mathematics of a swinging pendulum, and if you make a little approximation, so the approximation that you make is that for a small angle, theta, the sine of theta is approximately theta. So you can make that approximation. And it's pretty good for small angles. And if you do that, then when you work out what the forces are acting on the pendulum, you find that, roughly speaking, it'll take the same time to do its swing wherever you release it from. So it has this kind of constant period, basically. And that's why pendulum clocks are useful for telling for time. But they're not perfect, because we had to use an approximation to get to that point. So Christiaan Huygens is wondering, is there actually a curve that I can make, that will really genuinely have this constant period property, that wherever I release a particle from on this curve, it will reach the bottom in the same time?

KK: Right.

SH: Because that's what the pendulum almost does, but doesn't quite do. And so he said — and this problem is known as the tautochrone problem, because it's “the same time” in Greek. And it turns out, guess what, the cycloid solves the tautochone problem. It's precisely — so we have an arch, you've got to turn the arch upside down. So now you can roll, your particle can roll down. And wherever you release a particle from on the cycloid, it will reach the bottom in exactly the same time.

KK: Remarkable.

SH: I mean, assuming you know, it's smooth, no friction or whatever. It's just rolling down under gravity. And I mean, it's not even clear that such a curve could exist, right? It's quite a thing to ask. And yet, the cycloid has this property, and it's fantastic. So that's an amazing thing. And few years later — so Huygens worked this out. A few years later, a different problem was posed. It's kind of a related question, or it's something to do with particles anyway. And the question here is called the brachistochrone problem. And it was proposed by Johann Bernoulli, one of the Bernoulli brothers. And he posed this kind of publicly in a journal saying, Okay, if you now have two points A and B, A is above B, and you want to have a curve such that when a particle rolls down that curve from A to B, it will reach point B in the quickest time, so what might that be? Is it sort of a parabola, maybe a straight line, what's it going to be like? And this problem was posed to the mathematicians of Europe as a challenge, and quite a few big names enter this competition to see if they could do this. So Leibniz was one, Gottfried Leibniz, Bernoulli himself solved it, his older brother solved it, and then they got this anonymous entry. And it was so beautifully done, and elegantly produced, the solution to this, that, even though it was anonymous, when Bernoulli he saw it, he said this famous phrase, “I recognize the lion by his claw.”

KK: Right.

SH: And it was Isaac Newton, who had solved this problem. And guess what? It's the cycloid again. The cycloid solves this problem as well. So you've got this amazing curve, which is a natural idea. It's got these lovely whole number relationships about its length and its area, and then it suddenly also can solve these totally different questions about particles rolling down in the quickest time or constant time. And so that is why I love the cycloid so much. Everybody’s worked on it. It's got this amazing history, it's really beautiful.

KK: This sounds like a good public lecture.

EL: Yeah.

SH: I just get really.

EL: The cornucopia of the cycloid.

KK: Yeah, so question, the original area calculation that Roberval did, did he use calculus? Or was this a geometric argument?

SH: So he used something that isn't quite calculus yet, Cavalieri’s principle. If you're comparing areas, if you have got two shapes where if you slice through, the length of those slices is the same at every point, then the areas are the same. So he used that principle, which you can extend to volumes as well. And he kind of did a particular, so he managed to do this. And he had the curve that you make for the cycloid, he made it up from three different pieces. And he did this sort of slicing argument to compare it to with things he already knew, one of which was the sine curve, although I don't think he noticed it was a sine curve at the time, but we can now see that. So now, you would make that argument with calculus. But it's the same basic idea. You're slicing something very finely.

KK: Right. You could almost imagine Archimedes figuring this out.

SH: Yeah. Yeah, exactly.

EL: Yeah. So I mean, you've made a very compelling case that this is a very cool curve that has all these properties, So like, why is this your favorite? Or I know it's hard to pick a true favorite. But yeah, can you talk a little bit about, like, how you encountered it and what makes it so appealing to you?

SH: Well, there's at least two things. There might be three. One is, I love the simplicity of the results about the area and the length, that they are just lovely, simple relationships there comparing to the circle that makes this this curve, which itself is easy to think about what it is. So it's not contrived at all. It arises fairly naturally from just thinking about wheels rolling along roads. You get this curve, and then these relationships are very simple. The second reason I love it so much is because of this unexpected appearance of the cycloid in this totally different context from from how you imagined it. When it's generated by just, you know, a wheel, but then a curve that has these other properties, that’s very surprising. There are other things we could talk about to do with it. involutes, and other kinds of things where it crops up, but that for me, it encapsulates why it's such an exciting thing. And it's like when you first encounter pi or something, or you see the e to the i pi plus one equals zero, it gives you that same kind of feeling, that thing's from over here, and this other constants from over there, you know, that they're linked together seems really surprising. But the final thing, I suppose this kind of links in again with what we were saying about mathematics and literature, is how the cycloid has caught people's imagination over time. And it's both of mathematicians, but outside. And there are several books that mention cycloids. So Moby Dick is one. That's got a lovely little passage about cycloids. But also, Gulliver’s Travels mentions cycloids, Tristram Shandy by Laurence Sterne, this amazing, crazy 18th century book talks about cycloids. And those are just three that are really classic books. It was in the air at the time, and perhaps we don't necessarily — like, a modern and modern person may not have heard of cycloids. But certainly if you were educated in the 18th, 19th century, you may well have heard about cycloids. And that, to me, is very interesting too.

EL: Yeah, do write a little bit about this in your book that Moby Dick part, I have gotten to that part. And apparently, did you say that Melville apparently had some amazing math teacher in high school. And so, you know, kind of was able to really capture his imagination about math and then bring that into literature later, which is just kind of a cool thing to think about as math teachers, people who teach math. It's like, yeah, even if your your students don't end up in math or something, they might, you know, hopefully bring some of what you teach them that direction.

SH: Yeah, absolutely. I mean, it’s the value of having a great inspirational teacher. Just look at with Melville. So he had a teacher. He went to a school called the Albany Academy, and he was good at school in some areas, mathematics was something he was particularly good at. And he actually won a prize for being the first best at ciphering, was what it was for. Cipher, the old word for calculation.

KK: Right.

SH: His prize was a book of poetry, which I liked, because for me, that's absolutely a natural prize, but it wouldn't necessarily be thought so. But his teacher was a man called Joseph Henry. And Joseph Henry was no ordinary schoolteacher. He was a very good scientist in his own right, he went on to become the first secretary of the Smithsonian. So you know, pretty impressive. But physicists will know the name Henry, because the Henry is the scientific unit of inductance. And that's for Joseph, that is Herman Melville's maths teacher at school. So he was by all accounts an exceptionally good teacher, to the extent that some of his classes were actually, members of the public were allowed to come in and attend as public lectures. So there's a record that says, a request of his that he wants to have additional books for the more advanced students to entertain them beyond the normal curriculum. And so I don't know, and we can't know for sure, how Herman Melville learned about cycloids. But I could very easily imagine that a lesson on Friday afternoon, let's just talk about this fascinating curve because it's really interesting. And Melville did have a love, then, of mathematics, which just comes out in his writing. You can just see it, the way he chooses metaphors and imagery, they're often mathematical. And you can just see it's, it's not thinking “I must include some mathematics.” It's just the sheer pleasure of it. The delights, the joy of mathematics just comes out in his writing, which is wonderful to see.

EL: Well that was such a cool story that I read in there. And I have loved to revisit this. I don't think I've actually thought about cycloids since I taught calculus, right, which, it's been quite a while since I taught calculus. It is a fun, it’s a very common example in calculus books now. You'll kind of go through and solve some of these, these things. And I think, when you do parametric curves, maybe?

SH: Yes.

EL: So yeah, lots of fun, but I don't think I had really appreciated it as this whole whole thing before. So the other thing we like to do on this podcast is ask our guests to pair their theorem, or their bouquet of cycloid facts, with something else in life. So what have you chosen for your pairing?

SH: Well, so I've chosen Moby Dick.

EL: Okay.

SH: Because, I mean, he does talk about cycloids in the book. It’s not just because of that, but with the cycloids is this lovely passage where Ishmael, who is, you know, traveling as a deckhand on a whaling ship with Captain Ahab, who perhaps is not entirely sane, and we discovered that through the book, but there are many — Ishmael sort of has these wonderful meditations, he's just thinking about things. And some of them are mathematical, and some of them aren't. But there's one particular point where he is cleaning the the try pots. A try pot is something you had on a whaling ship, where there's great cauldron like pots where they kind of render the whale blubber down, and then you have to clean them. And so he says, you know, this is a place for wonderful mathematical meditation. And he and he talks about, as his soapstone is circling around the inside of the try pot, he says, I was struck by the fact that in mathematics, the cycloid is the curve where you can you can release something and it falls to the bottom in a constant time. And so he's just sort of drops in, the cycloid, just mentions it while he's daydreaming about something else. But Moby Dick, it's full of mathematical ideas. And it’s, you know, they are interested in numbers, to the extent that Ishmael keeps, he has the data or information about whales, measurements and statistics about whales, he has them tattooed onto his body, because as he says, you know, I didn't have a pen to hand, kind of thing, there was no other way to record. So he just has them tattooed on his body. Ahab is doing calculations on his ivory leg, you know, there are all these discussions about number. But there are lovely pieces of imagery around the infinite series of ripples in waves in the sea. There's a metaphor about loyalty where Ahab says to the cabin, boy, you are loyal as the circumference to the center, you know, the circumference always stays the same distance from the center. And it's just lovely little pieces of mathematical imagery throughout, and throughout all Melville's work. So I thought, yes, Moby Dick would be a very good pairing.

KK: Yeah. And so you actually have a paper about this in the Journal of Humanistic Mathematics, right?

SH: Yeah.

KK: Ahab’s arithmetic?

SH: Yes. And that itself is a little bit of a reference to a discussion that happens in Moby Dick, which is where two of them were talking about a book called Daboll’s Arithmetic, which was the kind of classic text in American schools, I think, at the time, which had all these rules about how to do calculations. And you could do mysterious things with with this book because, you know, if perhaps the mathematics hadn't been taught by a teacher like Joseph Henry, perhaps you learnt you've learned these rules off by heart, you don't quite understand them. And so they talk in the book about cabbalistic contrivances of producing these things. And at one point, someone says, “I have heard devils can be raised with Daboll’s arithmetic.” So, you know, this is the other side of mathematics, where people sort of hold it in or but also, perhaps, they have some suspicions around what do all these symbols mean? And it's very interesting, if you look at that book, Daboll’s Arithmetic, it isn't like a mathematics book would now be. So when he talks about how to find the areas of circles, for instance, pi is not mentioned at all. He says, you square the radius, and you multiply it by 22/7, or if you want a more accurate thing, you could multiply it by what's that other approximation right? 355/113? But he doesn't say “because these are approximations to pi,” it's just like, you can do this or you can do that.

EL: Here’s a number.

SH: Tust where's that come from? So that's a very interesting thing. And so there are mathematics books discussed or mentioned in Moby Dick as well. And if you know a little bit about them, so Euclid, of course, is mentioned a little bit. Yeah. So the book is full of mathematics. And I really wanted to think about in the article I wrote, why — how did Herman Melville know all this stuff? Why, you know, where does it come from? Because, you know, he's not a mathematician. And this is why, you know, nowadays, we're sort of taught to believe, or somehow we come to believe, quite often, that you're either a mathematics personal, or you're not. And if you're not, then you don't know any and you don't care. But this is absolutely not the case for one of our greatest writers, Herman Melville. And so you know, yeah, where did that come from? And, you know, it was just lovely to, to find out a little bit more about what he knew and how he knew it, and where it all came from.

KK: Very cool.

EL: Yeah, that's great. I have confessed to you already, but I will confess to our listeners that I have not read Moby Dick, but it is on my list that I hope to get to this year. It's a little daunting.

KK: You’d better get cracking.

EL: I know. I’ve only got six months.

KK: I read it at bedtime. That's when I tend to read, and so I read it, you know, maybe 10 or 12 years ago, and it took me quite a while. Yeah, yeah. It's pretty dense too.

SH: It is. I mean, I didn't read it till I was older. Because, you know, you hear this is the “great American novel,” and you should, everyone should have read this book, and then you feel bad that you haven't read it, and then you feel annoyed that you feel bad that you haven't read it. So there's all these barriers that you put up for yourself. And, you know, I'm so glad that I did eventually read it, because I loved it. It's so rich. And there is, you know, many many, many layers of interpretation and depth in the writing, but it is a great book. So yeah, I hope you will enjoy it when you read it.

EL: Yeah.

SH: You know that there are books that we all — I haven't yet read, I don't know if I will ever read, maybe one day, Finnegans Wake. I do mention it in the book because James Joyce, I talk about Ulysses a little bit and Dubliners in the book. But Finnegan’s Wake for me, I tried and I didn't quite quite get there. All I can say is in the middle of Finnegans Wake, there is a picture which could have come straight out of Euclid’s Elements. It's got equilateral triangles, two circles intersecting. But yeah, that for me, maybe one day, maybe I'll have a sabbatical one day and that will be what I do in that sabbatical.

EL: There just, there is so much. There's so many good books published now, you can't you can't read them because you’ve got to read last year's good books. But I mean, it's just you — Yeah, anything you read is great. And you’re never going to get to all of it. Enjoy what you read.

SH: Exactly. Amnesty of all our unread books. It's fine. We forgive ourselves.

EL: Yeah. Thank you so much. for joining us. This has been a lot of fun. You know, we do like to give our guests a chance to plug things but we've already talked about your book quite a bit. Is there anything else that you'd like to to mention about what you're working on or other things that you've published that you'd like us to share?

SH: Oh, no, I think I'm alright. So coming up. I mean, not for US listeners, but I've got an event coming up in a couple of weeks is going to be really fun because we're going to watch a classic B movie from the 1950s, which is this film about giant ants terrorizing the New Mexico desert. It’s called Them! with an exclamation point.

KK: Yeah, I've seen the posters.

SH: Yeah, right. Yeah. Which is super fun. But that's about, yeah, something has happened. Who knows? But there are giant ants. They have a lot of fun with it. But we're going to watch the film at the Barbican Centre in London. And then we're going to talk about, yeah, what does mathematics tell us about what life is like? Could giant ants exist, could giant spiders exist? Or giants, or tiny people like Lilliputians. And so that's a kind of fun thing that's coming up. But yeah, you've already, if you look at my book, you will already have a reading list that’s like 100 new books that are gonna be fun, fun to read and explore. So yeah, there's plenty to go on.

EL: Great.

KK: Thanks so much, Sarah, this has been great fun.

SH: Thank you for having me. Yeah. I’ve loved it.

EL: Bye.

SH: Bye.

[outro]

In this episode, we were delighted to talk with Sarah Hart, the Gresham Professor of Geometry at the University of London, about the serendipitous cycloid. Below are some links you might enjoy as you listen.
Hart's website and Twitter profile
Her book Once Upon a Prime and its review in the New York Times
Hart's article Ahab's Arithmetic about mathematics in Moby-Dick
The Wikipedia entry for the cycloid, which has links to many of the people we discussed

Episode 85 - Matthew Kahle

2 juni 2023 | 31 min

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined today as always by my fabulous co-host.

Evelyn Lamb: Well, thank you. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City. And anyone who's on this Zoom, which is only us and our guest, can see that I am bragging with my Zoom background right now. We just got back from a trip to southern Utah, and I took possibly the best picture I've ever taken in my life. And 95% of the credit goes to the clouds because they just — above these red rock hoodoos outside of Bryce Canyon, I turned around and looked at it while we were hiking, and I was like, Oh, my gosh, I have to capture this.

KK: It is quite the picture.

EL: My little iPhone managed.

KK: Yeah. Well, they're pretty good now. Yeah. Anyway, so yeah, I'm getting ready to — I have three trips in the next three weeks. So lots and lots of travel, and I'm gonna make sure I mask up and hopefully I don't come home with COVID, but we'll see.

EL: Yes.

KK: Anyway. So today, we are pleased to welcome Matthew Kahle. Matt, why don’t you introduce yourself?

Matthew Kahle: Hi, everyone. Thanks for having me, Kevin and Evelyn. I'm a mathematician here at The Ohio State University in Columbus, Ohio. I've been here for 11 or 12 years now, and before that, I spent a good part of my life in the western United States. So those clouds look familiar to me, Evelyn. I miss the Colorado sky sometimes.

EL: Yeah, just amazing here.

KK: You did your degree in Seattle, right?

MK: I did. I did my PhD at the University of Washington.

KK: Yep. Yep.

EL: Great. And what is your general research field?

MK: I work a little bit between fields. My main interests are topology, combinatorics, and also probability and statistical physics. And I think I usually feel most comfortable, or maybe I should say most excited mathematically, when there's sort of more than one thing going on, or when it's in the intersection of more than one field.

KK: Yeah, lots of randomness in your work. He’s got this very cool stuff with random topology. And I remember, some paper you had few years ago, I remember really sort of blew my mind, where you had some, you're just computing homology of these random simplicial complexes, and, like, some four- or five-complex had torsion of order, you know, I don't know, 10 to the 12th, or some crazy torsion coefficient. Yeah.

MK: Yeah. So we were really surprised by this too, and we still don't really have any way to prove it, or really understand it very deeply. Kevin was mentioning some experimental work I did with some collaborators a few years ago. But yeah, that is the gist of a lot of what I think about, is random topology, which I sometimes try to sum up as the study of random shapes. And one of the original motivations for this was as sort of a null hypothesis for topological data analysis, that if you want to do statistical methods — if you want to use topological and geometric methods, and statistics and data science, you need a probabilistic foundation. But one of the things we've discovered over the last 15 years or so is that these random shapes are interesting for their own sake as well. And sometimes they have very interesting, even bizarre, properties, where we don't even know how to construct shapes that have these properties at all, but they're they are there. And we know they exist, because of the probabilistic method. Yeah.

EL: So let me be the very naive person who asks, like, how do you, I guess, come up with — like, what do you randomize about shapes? Or you know, if I think about, I don't know, randomly drawing from from some sort of, I don't know, bucket of properties, is it that or is it… Just what is random? What quantity or quality is being randomized?

MK: Right. So a lot of the random shapes or spaces that I've studied have have been on the combinatorial or discrete side. So for example, there are lots of different types of random simplicial complex that people have studied by now. And typically, you have just some probability distribution, some way of making a random simplicial complex on n vertices. And n can be anything, but then the yoga of the subject is that typically n goes to infinity. And then we're interested in sort of the asymptotic properties as your random shape grows. So one of the early motivations, or early inspirations, for the subject of random, simplicial complexes was random graph theory. So you can create random networks various ways, and people have been studying that for for a bit longer, probably at least 60 years or so now, with new models and new interesting ideas coming along all the time. For example, there was originally the Erdős–Rényi model of random graph where the edges all have equal probability, and they're all independent. This is a beautiful model mathematically, and it's been studied extensively. We really know lots and lots about that model of random graph now, although surprisingly, people can continue to discover new things about it as well. But in today's world, some people have studied other models of random graphs that they say may have made better model real world networks, for example, social networks, or what we see in epidemiology, and so on. The Erdős–Rényi model is something that's tractable, and that we can prove deep math theorems about, but it might not be the best model for real world networks. But, you know, I think of the random simplicial complexes that I study sometimes as just higher-dimensional versions of random graphs.

EL: Okay.

MK: So as well as as well as vertices and edges, we can have higher-dimensional cells in there, and and that starts to sort of enrich the space. It's not just one-dimensional now, it could be two-dimensional, or it could be any dimension.

EL: So you might not know. You've got some large number n, and you might not know what dimension this random — you’re, like ,attaching with edges with some sort of probability between any two things. And so you might not know what dimension your simplicial complex is going to be until after you randomly assign all of these edges and faces and, you know, and n- whatever the word is for that, n-things. [Editor’s note: it’s n-simplex.]

MK: Yeah, absolutely. That's right. It could be that the dimension of the random simplicial complex is itself a random variable. And you know, that we don't ahead of time even know what the dimension of it is.

EL: Cool!

KK: So there's lots to do here. This is why Matt has lots of students and lots of lots of good projects to work on. But anyway, we invited you on not just to talk about this really interesting mathematics, but to find out what your favorite theorem is. So what is it?

MK: Okay, so I've been thinking about this. Well, I have to admit, I think I asked myself this just knowing of your podcast in case I ever got invited on. And then I've been thinking about it since you invited me. I would say my favorite math theorem, probably the one I've thought about the most, the one maybe that affects me the most, is Euler’s polyhedral formula, which is V−E+F=2. Right? So let's just start out saying, well, you know, what do we mean by this? I think my understanding of the history of it is that it was something that as far as we know, the Greeks didn't observe even though they were interested in convex polyhedra. And sometimes people consider the classification of the perfect Platonic solids is one of the peak contributions of Euclid’s Elements. But we don't know that they recognized this pattern that Euler noticed thousands of years later. If you take any convex polyhedron, a cube or an icosahedron, or a pyramid, a bi-pyramid, any kind of three dimensional polyhedral shape that you can imagine that's convex, V, the vertices is the sort of number of corners of the shape and E is the number of edges. And then F is the faces. It always is the case that V−E+F=2. So Euler noticed this. And it's not clear if he gave a rigorous proof or not. I don't even know if he felt like anything needed to be proved, maybe it was obvious to him. And nowadays, we have many, many beautiful proofs of this fact. But one of the things that strikes me about it is that, it’s sort of in hindsight, is that this is just sort of the tip of a very big iceberg. There's a much more general fact that we are just kind of getting our first glimpses of, and nowadays, we would think of this as not just a phenomenon about convex polyhedron, 3-dimensional space, that it’s just a general phenomenon in algebraic topology, or you can say =more generally, in homological algebra. It's just sort of a feature of nature somehow.

KK: Right, right.

EL: I think something that I really enjoy about this fact is you can present it at first as a theorem or as a fact. But then this fact kind of leads you to this new definition that you can observe about all sorts of different shapes, you know, this number that is the vertices minus the edges plus the faces, hopefully, I got it in the right order, yes. Then you can assign that, you know, you can say, like, what does, you know, if you've got a torus, like a polyhedral torus, or, you know, a higher-genus object or a higher-dimensional thing, you can sort of use this, and so it's like a fact becomes a definition or a new thing to observe.

MK: That’s right. Are you saying, for example, you know, we have the Euler characteristic is an invariant of a space?

EL: Right.

MK: And that might, if you're introduced to a new topological space, that might be one of the first things you might like to know about it. And so yeah, it becomes its own invariant. It’s a way of telling some different spaces apart, for example.

EL: Yeah. So do you have a favorite proof of this favorite theorem?

MK: I do. I present it and the graduate combinatorics and graph theory course when I teach this course. So already, we're looking at a little bit more general formulation than what Euler looked at. We don't just have a convex polyhedron in 3-dimensional space, what we have is a connected planar graph. So we have some kind of network with nodes and connections between them, and it's one that you can draw on the plane without any of the edges or connections crossing. And in this case, the faces now are just going to be the connected components, or the regions, in the complement of the graph that then comes with an embedding into the plane. And then V is the number of vertices of the graph, and E is the number of edges. So V−E+F=2 in this case, so for for just any connected planar graph, this might seem totally unrelated, but it's actually a more general version than what we just saw with convex polyhedra because you could take any convex polyhedron and unwrap it, or stereographically project it into the plane and get a planar graph. But planar graphs could have lots of other features. So when I present this in class, I tend to give three or four different proofs of it. There's a beautiful proof that I've heard attributed to John Conway, where he says something about, like, letting in the ocean or something. So your graph is connected, but there may be some cycles in it. And anytime you have cycles, the Jordan curve theorem tells us there's an inside and outside. So John Conway wants to let the ocean in. The ocean is the sea, is the outside of the graph, let it in until it touches. So what he's saying is if there's any cycle, delete one edge from it, and so what this does is it reduces the number of edges by one because you deleted an edge, but it also reduces the number of faces by one because that two regions that were inside and outside of that cycle are now the same region, so V−E+F has stayed the same.

KK: Right.

MK: And then eventually, you've just got a tree. There's no more cycles left, but your graph is still connected, so it must be a tree. And we know that every finite tree with at least two vertices has a leaf, has a vertex of degree one. And again, you can prune away that, and then you've reduced the number of vertices by one and the number of edges by one. And V−E+F is again not changed. So at the very, very end, we're just left with a single vertex in the plane. There's one vertex and there's one region, which is everything except that vertex. So at the very end, V−E+F=2. But through all those steps, we know that it never changed. So it must have been V−E+F, it must have been 2 at the very beginning. So I love that proof.

EL: Yeah.

MK: There's another proof that I think I like even better, which is that you consider the dual graph and a spanning tree. You pick a spanning tree on the original graph and a spanning tree and a dual graph at the same time.

EL: So the dual graph being where you replace, you swap vertices and faces.

KK: Yes. For every face there’s a vertex and you join two when the two faces share an edge.

MK: That’s right, exactly. So one thing that's tricky about that is that now the dual of even just a nice planar graph might be a multi-graph. So just imagine a triangle in the plane. The dual graph has two vertices, one inside the triangle and one outside, but there's three edges connecting those two vertices, because there's three edges in the original graph. And the edges in the dual graph have to correspond to edges in the original graph, and that's important. They cross them transversely. So then you choose a spanning tree on each one, and you and you realize that — you count the number of edges in each and you somehow — now I'm getting a little stuck remembering the proof, but the punch line is in the original graph, I guess the number of edges is V−1. And then the dual graph, the spanning tree, the number of edges is F−1. And these have to be in correspondence. So you just immediately just write down V−1=, sorry, no, I don't remember exactly how that the end of that proof goes. But there was something about it I liked. It seemed like the other proofs, you're kind of doing induction on either the number of vertices or the number of edges or the number of faces, and that you have to make some arbitrary choices. And this proof by duality doesn't use any induction and doesn't require any choices. It just kind of comes for free. And you sort of immediately see where the 2 comes from, because there's a V−1 on one side and an F−1 on the other side, so the 2 just sort of pops out immediately from the proof. There’s — I think it’s Eppstein? — some mathematician collects proofs of Euler’s polyhedra formula on his website, and he has at least 10 or 20 different proofs. And when you read them all, some of them start to remind you of each other, and who knows what counts as the same proof or different proofs.

KK: Sure.

MK: But there are some neat contributions in there. One of them he attributes to Bill Thurston in the middle of some very influential notes that Thurston had in differential geometry. And he's talking, he's giving his own proof, I think, that the Euler characteristic of a differential manifold really is an invariant of the manifold, for a smooth manifold, let’s say. You could triangulate it, and then the Euler formula, the Euler characteristic, you could just say is the alternating sum of the faces of every dimension. But why doesn't that depend on which triangulation you pick? And Thurston gave a really beautiful kind of almost physical argument with, like, moving charges around. I like to show the class this one also. At that point, we leave — I don't know how to make that proof work for planar graphs, but it works beautifully for polyhedra, for convex polyhedra, like what Euler first noticed. And apparently, it works also for higher dimensional manifolds, too, although I've never gone through that proof carefully.

KK: Yeah. Right. Well, the proof that you said might be attributed to Conway is sort of the one that I always knew, and I never heard it attributed to him, but that's good. It's sort of nice. You can explain that one to just about anybody right? You just sort of imagine plugging away an edge and a face at the same time basically, yeah.

EL: Yeah. A proof that proof that is of something that is so visual, but you can really understand over a podcast, is a special proof. Because I do think that it doesn't take a whole lot of you know, imagination, to be able to follow this audially.

KK: Audially, is that a new word?

EL: There is a real word that is embedded in that word. Aurally, that’s the real word I was trying to say.

KK: Yeah, so is this sort of a love at first sight theorem? I think I first learned this theorem in the context of graph theory.

MK: I think for me, too.

KK: And then I became a topologist kind of later. And then of course, now I think of it as, oh, it's the alternating sum of the Betti numbers, but that those two quantities are equal is an interesting theorem in its own right.

MK: Right. Yeah. So I was trying to think about this. When did I learn about this theorem? And I think I first learned it in graph theory also. But then I know now that it's much more general, and I don't even know if I ever remember anyone telling me that specifically in a class or reading it in a particular book or paper. I think this to me, maybe part of what I like about the Euler formula is that I feel like my understanding of it has just deepened over time, and that there’s kind of a series of small revelations. At some point, I started thinking of it as the alternating sum of the Betti numbers, and things like that. And since I like the combinatorial side of topology, and have simplicial complexes or cell complexes, also the alternating sum of the number of faces of each dimension. But then even in the last couple of years, my understanding has continued to develop because now I think, you know, well, you could just have a chain complex, and all you know is the dimensions of the vector spaces, but it makes sense to ask what's the homology of the chain complex, so they're the Betti numbers again, and again, the alternating sum of the Betty numbers now is the alternating sum of the dimensions of the vector spaces of your chain complex. But I think I probably first saw, you know, the graph theory version of it, maybe in an undergrad or a first graduate combinatorics course.

KK: All right. So the other thing on this podcast is we like to ask our guest to pair their theorem with something. So what have you chosen to pair Euler’s formula with?

MK: Well, you know, I've been stumped by this. But you know, thanks for the warning that I am going to get asked this question. So I had a little time to think about it, and I'm not totally stumped on the spot. But the thing that keeps coming to mind the most when I ask myself that question is some of Bach's music. Johann Sebastian Bach is really known for his four-part harmonies and for counterpoint, and it feels a little bit like this: You're listening to a beautiful piece of — it could be anything, you know: a fugue on an organ, or four-part harmonies that were written for choral music or something like that. And when you listen to it, you can listen to a recording of it two or three times and each time pick out a different voice to follow along. And there are just these independent melodies, harmonies that he's somehow weaving together. You can also just relax and just let the whole thing wash over you. And honestly, that's most often how I listen to music. But it's completely fascinating to just hone in on one particular thread. And so I think a lot of people feel like Bach's music has maybe a mathematical feeling to it, or that it's mathematically perfect or precise. So you could say that Bach, pairs with mathematics already, but the reason I want to try to connect it with the Euler formula that I like as my favorite theorem is that there are these sort of different layers. And just the same way you can kind of listen for one voice, and then tune your ear and listen to a different voice and emphasize that, I feel like this is one of these areas, of one of these kinds of mathematical phenomena, that’s just sitting there in, you know, platonic space, or wherever it lives. And you can look at it. So if you look at it from the topological point of view, it's the alternating sum of the Betti numbers, the number of holes in each dimension. But if you look at it through a combinatorial lens, then it's the alternating sum of the number of faces of each dimension. Or you can just step back and it's just its own thing. It's just an invariant of the space, the Euler characteristic, and these just happen to be different ways to compute it. But it has that feeling to me that you can look at it different ways. But you're really always looking at the same thing. Just we're putting on different glasses or looking at it through different lenses, and so it reminds me of that sort of, I don't know, counterpoint and music or something.

EL: Yeah. Oh, I love this pairing! I'm also, I play viola and I sing, and when you get to a point when you’ve learned a piece that you've learned it enough that you don't have to be just concentrating on, like, am I singing the right note at the right time, but you can actually start hearing like, oh, I didn't originally hear that the parallel that the bass and the soprano line has right here, or the way we come in and then the altos come in and something like that. I've been singing a lot of, you know, things that have these fugal sections in them, which is — I haven't actually sung much Bach recently, but similar things — and I just love that pairing and how seeing the same thing, or singing the same music over and over again, you hear something different every time. You know, just a little easter egg that you didn't pick up the first 20 times you practiced this piece, and then now you hear and you say, oh, next time I really want to make sure that I, you know, do that crescendo with the tenors just perfectly or something. I love that.

KK: Yeah, so I see the edge of a keyboard there in your Zoom, Matt. Do you play?

MK: A little bit. I mean, nothing to write home about, but it's something I enjoy. I took it back up during the pandemic as a hobby. And I've been practicing a little bit. This over here, I have a little portable keyboard, and then I have an electric piano out in the living room. But I've been practicing music with one of my friends. We get together, like, once a week and and just play some cover songs. And I like what you're saying, Evelyn, about hearing different things. Even, you know, we'll be playing some song by REM or somebody that I've known, I don't know, it seems like my whole life, it’s very familiar. But once we start to play it, once we start to sing it, then I hear all kinds of different things in it that just listening to the same recording that I've listened to before all of a sudden, I'm like, wait, Mike Mills is actually doing some really interesting harmonizing and this track, and not only is he harmonizing, like singing different notes, than what Michael Stipe is singing, he’s actually singing different words. He's saying something in that song I never even noticed he was saying. So anyway, music and mathematics, I think that's probably another big thing that they have in common, is that, you know, a little bit can go a long way, and even just entry-level, you can already start to appreciate the beauty of it, but that it's sort of almost inexhaustible how deep it goes and that you can always, there's always more to learn. There's there's always more to notice.

EL: Yeah, with Bach specifically, you know, as a viola student, I think I started playing the Bach cello suites, an octave up on the viola, I was probably 10 or 11 years old? And it's like, I will still play those same suites that I started learning when I was in fifth grade. And it's like, it always has something to teach me. It's something that I can always get something more out of.

KK: Yeah. I think we can all agree that that Mike Mills is REM’s secret weapon. I took up the guitar about 10 years ago, so I'm terrible, and I play alone. But it's still something that I enjoy to do. It's certainly, it's a good way to exercise your — what was it Leibniz said? That music is the pleasure the brain derives from counting without knowing that it's counting?

EL: Oh, yeah. That’s a good little quote, to file away for us math-musician-type people.

KK: That’s right. All right, well, so we always like to give our guests a chance to plug anything. Where can we find you on the interwebs?

MK: Yeah, I don't have anything particular to plug, but you can find, you know, all my mathematical work on my professional webpage, matthewkahle.org. There's links to all my papers and everything there. And, you know, if my friend and I get our REM cover band off the off the ground, we’ll keep you posted.

EL: All of our Columbus area listeners can find you.

KK: I can play rhythm guitar on some of the tracks if you need somebody.

MK: All right. We'll have to all get together if you come out and visit in Columbus.

KK: Well this has been great fun.

EL: Thanks so much for joining us.

MK: Thanks for having me today.

[outro]

On this episode, we were delighted to talk with Matthew Kahle of the Ohio State University about Euler's polyhedral formula, also known as V−E+F=2. Here are some links you might find useful as you listen to the episode.
Kahle's website
His paper about torsion in homology groups of random simplicial complexes
The Erdős–Rényi model of random graphs
Euclid's Elements, book 13, is devoted to the classification of Platonic solids. Also found herestarting on page 438.
The Jordan curve theorem has made a previous appearance on the podcast in our episode with Susan D'Agostino.
David Eppstein's website with 21 different proofs of Euler's formula. Thurston's proof is here.

Episode 84 - The Students of TCU

2 maj 2023 | 50 min

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knutson, professor of mathematics at the University of Florida. And today I am flying solo while I am at Texas Christian University in Fort Worth, where I'm serving as the Green Honors Chair for the week. And I've been given some talks and meeting the fine folks here at TCU. And today, I have the pleasure to talk with some of their students. And they're going to tell us about their favorite theorems and what they pair well with. And we're just going to jump right in. So my first guest is Aaryan. Can you introduce yourself?

Aaryan Dehade: My name is Aaryan. I'm a sophomore computer science major at TCU, and I'm from India. And I've chosen to go with the fundamental theorem of calculus.

KK: Okay. The fundamental calculus.

AD: Yeah.

KK: Okay, so now there are two parts of the fundamental theorem of calculus. Do you have a favorite part?

AD: So that's what I like about it. Honestly, I can't choose a favorite part because one of the parts is very important, and the other is interesting.

KK: Yes.

AD: So the first part, basically, it tells us the relationship between the integral and derivative. The second part tells us — like, basically, you can use that second part supply and to get solutions for questions in calculus. Does that makes sense?

KK: Sure.

AD: So what I like about this theorem is that it's not like other theorems where it has two parts. So it's kind of interesting how everything in calculus is based off of these two theorems. And if you didn't know what the relationship between the derivative and the integral was, probably you wouldn't be able to do anything with mathematics with it.

KK: Sure, maybe. Yeah. So you said one part was useful, and one part was interesting. Which part do you think is interesting?

AD: I feel like the first part is interesting, because it's almost intuitive. Like, you know that should happen, like the relationship between integrals and derivatives should be, like, one is an inverse of the other. And that is intuitive. So it's almost given. And it's interesting that we have to say that.

KK: Really, you think that’s intuitive? I mean, I’m not — I don’t know, when I first learned the fundamental theorem, I thought it was kind of shocking that somehow this this thing, this integral, which is sort of defined as in terms of these Riemann sums, somehow that went, you know, if you let x be the upper limit there, and you differentiate that function, you get your original function back. That’s intuitive? That’s amazing.

AD: Yeah, basically what it is, it's just an area of a rectangle. So I just thought of it as just decreasing the width of the rectangles, and then you get smaller and smaller rectangles, so you get the area. And then if you take the function at that point…that’s what I think.

KK: You’re cleverer than I am. I was just kind of dim, I guess, and I didn't think it was so intuitive. I mean, I saw the proof and believed it. But then, yeah, then the second part is how you actually evaluate integrals.

AD: Yeah, so that's what you use to calculate the area between two points.

KK: Yeah. Although I guess the problem is, right. So the theorem says that, you know, if you want to find the, the integral, the value of this definite integral, all you have to do — and our listeners can't see me doing the scare quotes — “all” you have to do is find an antiderivative of the function. Right?

AD: Right.

KK: Yeah. And then you spend all of Calc II learning how to find antiderivatives.

AD: Yeah.

KK: And even then, if I hand you an arbitrary function, you can't even do it, right? Like that's the sort of disappointing part of that theorem, is that most functions, you can't find a closed form antiderivative for. And so what do you do? But you’re a computer science major. You know what you do, right? You do it numerically, right?

AD: Yeah.

KK: Okay. Very cool. So you've known this theorem for quite some time, I guess.

AD: Yeah, I've done it since high school.

KK: So you love the theorem.

AD: I really, yeah, I do. Because when I was in high school, I used to sit at my dining table and study because I wanted to have some snacks at the same time.

KK: Sure. As we all do.

AD: I would just spend hours just doing sums on integrals, or basically just integrals. That was difficult at that point.

KK: Sure.

AD: And yeah, it was interesting, because I got used to that at some point. And then it got easier. And I just started liking the satisfaction of being able to do this. That was fun.

KK: Cool. All right. So on this podcast, we also like to ask our guests to pair their theorem or something. So what pairs well with the fundamental theorem?

AD: So as I said, I used to sit at the dining table and have snacks. And there's this really, really popular biscuit in India called Parle-G. And I used to have that with tea while doing my sums. So that was the highlight of it, that's why I used to look forward to studying, just for those biscuits.

KK: Okay, so I assume there's an Indian market in town somewhere, right? Can you get these?

AD: Yeah, I do have them in my dorm right now. Yeah. I have them every day.

KK: All right. So what are these called again?

AD: Parle-G.

KK: Parle-G. Okay. So I'll have to go to the Indian market when I get back home and see if I can find these because I am always on the lookout for a good new biscuit.

AD: Yeah, they’re amazing. Okay, so you should you should know, our very first episode of this podcast, aur guest, who was Amie Wilkinson, who is on the faculty at University of Chicago, chose the fundamental theorem as her favorite theorem. So you're in very good company, because she's a phenomenal mathematician. And okay, thanks.

AD: Awesome. Thanks so much.

KK: All right, up next, we have Toan. So why don't you tell us about yourself and what your favorite theorem is?

Duc Toan Nguyen: Okay. My name is Duc Toan Nguyen. People usually call me Toan. I’m an international student from Vietnam, and I'm a sophomore majoring in math and computer science

KK: Okay, great. So, favorite theorem. What’ve you got?

DTN: Yeah. So as my peer Aaryan, he chose the fundamental theorem of calculus, right?

KK: Yes.

DTN: But I want to bring another fundamental theorem in analysis, which is the mean value theorem.

KK: Oh, okay. So I have a theory, okay. I call the mean value theorem, the real fundamental theorem of calculus.

DTN: Yeah, me too!

KK: So why do you like it so much?

DTN: Yeah, I think I have the same idea with you of why it is called the real fundamental theorem. I think, because to prove the fundamental theorem of calculus, you need the mean value theorem.

KK: You absolutely do.

DTN: Also for analysis, the most popular and common tool in calculus, which is a derivative test, also has the mean value theorem behind it.

KK: That's right. Yeah, that's right.

DTN: So when I first so I first approached the mean value theorem when I was in high school. I took the Math Olympiad in Vietnam. So I had to prepare for that, and there is a section about that.

KK: Okay.

DTN: So it's called the Lagrange Theorem, it was kind of very fancy. Yeah. And it usually applies to — so you know, in the exam, we had some of the problems related to the continuous version, and f(a) minus f(b), something like that. Most of time, we used the mean value theorem. So yeah, it was kind of cool at the time, but I really enjoyed that until last semester, I took real analysis. So I could see the whole process was using the mean value theorem. That's why it can be taught in one lecture or one unit. Even today, this semester, I’m taking multivariate analysis. And it's also a very fundamental thing in proof, everything from differentiability on. It also even has a mean value theorem in it higher-dimensional space.

KK: So maybe we should remind our listeners what the mean value theorem actually says,

DTN: Oh, okay. So, let f be a function defined on an interval [a,b], so that f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). So the theorem say that there exists a point c between a and b that is not is not inclusive so that f(b)−f(a) is equal to f’(c)(b−a). So I think the mean value theorem, the name comes from the quantity f(b)−f(a) divided by (b−a).

KK: Yeah. Right. So the average rate of change over the interval is equal somewhere to the instantaneous rate.

DTN: Yeah.

KK: Yeah, that's right. That's how you prove the fundamental theorem, too, because it's just a telescoping sum when you write it out correctly. And the mean value theorem sort of pushes everything away, and then you're done.

DTN: Yeah, that's also my favorite part. Because it can tell you the relationship between the integral of functions and their derivatives.

KK: Right, right. So what do you want to pair with your theorem, what pairs with the mean value theorem?

DTN: Yeah, I want to pair with something really weird. Which is a phone with FaceTime. Okay, so I'm here. I study. I'm far from my home. My home is in Vietnam, which is on the other side of the Earth.

KK: Almost exactly opposite, right?

DTN: Actually it takes 20 hours from this time to my country time. So it's actually like, opposite, it's more. So yeah, and the FaceTime, why? Because through FaceTime, I can see what people in my home are doing and they also [can see me]. So it's kind of a bridge or relationship that connects what I'm doing here and what my family is doing there. And, you know, my family is always wants the best for me and hopes everything is good for me here. And me too. So that's a very meaningful thing for me.

KK: Yeah. That's great. And I'm glad that technology exists. When I was in college, the internet didn't exist. So you know, I had to call people on the phone and phone calls to Vietnam, I imagine, would be — I can't imagine what that would cost. It was expensive enough to call my girlfriend who lived four hours away.

DTN: It’s kind of more romantic. And you can give them a love letter.

KK: We wrote letters too. All right. Well, Toan, thanks so much. That was great.

DTN: Yeah. Thank you.

KK: Up next, we have Maiyu Diaz. You can introduce yourself.

Maiyu Diaz: My name is Maiyu Diaz. I'm a second-year graduate student here at the department of mathematics at TCU.

KK: Cool, and you have the best shirt. You win the shirt contest today.

MD: Oh, thank you.

KK: I’m actually kind of wishing you would like give me that shirt. [Editor’s note: How dare you say that and not send us a picture?!] All right, so you’re a second-year grad student here in the math department. Okay, great. And so yeah, so what your favorite theorem?

MD: So my favorite theorem is — I don't see it really formally presented, but Stirling's formula where n factorial can be approximated by n^n e^−n times square root of 2πn.

KK: Yes.

MD: That is my favorite theorem.

KK: Yeah. So that’s a really interesting approximation for factorials. So okay, where did you come across this?

MD: I first came across this, I want to say, when I was first studying the factorial back in grade school, and it was just more of like, looking up on a Wikipedia page, which was something that I really relied on when it came to writing essays for my English classes. And when I found out that it could also be used as a resource for mathematics as well, I thought, oh, factorial, let's learn interesting things about this. And that's where I found an example about Stirling's formula, under that. It wasn’t until much later that I was able to understand how that was derived, and there were a variety of proofs for proving Stirling's formula. One of them relies on probability distributions, which are looking at how the factorial works. And then there's another way of finagling a little bit with the integral formula for n factorial that comes from the gamma function, and that's another way of deriving Stirling's formula.

KK: Do you have a favorite proof of your favorite theorem?

MD: I do, but it's a very uncommon proof.

KK: Okay.

MD: The proof relies on a contour integral involving the derivative of the Riemann zeta function.

KK: Oh! I don't know if I know this proof, but what contour do you use?

MD: So you're going to go ahead and do a contour that is on the half strip, so you're just going to fix a number like σ and since the derivative of the Riemann zeta function, you just want to pick a σ just a little bit to the right of where it converges. So real part greater than one, and that's going to be a line integral from σ − i infinity to σ + i infinity, okay? And what you want to do is that you kind of want to start pushing that back a little bit so you can start picking up the residues of the derivative of the zeta function. You’re going to go ahead and possibly the non trivial zeros you could possibly hit, though. You’re definitely going to hit the pole at s=1. That's where the n log n minus n term comes from, when you look at the log of n factorial.

KK: Right, okay, sure.

MD: You keep pushing that more, and then you're going to be picking up the rest of the terms from the Stirling’s formula as well. It's pretty interesting.

KK: Yeah. All right. I haven't seen this proof. That's very cool. Yeah, okay. Cool. All right. So what do you think pairs well with Stirling's formula?

MD: I am going to say chicken tikka masala.

KK: Chicken Tikka Masala. Okay. I do like chicken tikka masala. What in particular makes you want to link those two things together?

MD: So Stirling's formula, I would say is a little bit spicy. And I underestimated it at first.

KK: Right.

MD: Because that was just something I would not have been — I just, if you’d asked me if the formula was intuitive, it’s just like, absolutely not. Where's where does this e term come from? Where does the square root of two pi come from?

KK: Sure.

MD: It’s not until you start familiarizing with the proof more. It's like, okay, at this point, it's not that I'm used to that, I’ve just seen this too many times.

kk: Sure. Sure. Yeah.

MD: But I want to say it’s spicy because of a paper, a PhD thesis, that was published in 2014 by Matthew Lamoureux, whose advisor is Keith Conrad, who wrote a paper on Stirling's formula, just devoting a series of notes on it, you can see all the things he has compiled and I like reading through his notes. And this PhD thesis, he was looking at a generalization of the factorial, which is the factorial for number fields. So instead of looking at the derivative of the zeta function, what it was looking at was a modification of the derivative of the Dedekind zeta function because they had a function defined over number fields. And that's the same technique as well. You want to keep pushing it along the line, be able to pick up non-zeros, possible poles, and it just more or less the same outline as well, and spicy because number one, there is a ton of information coming about the factorial just from the location of zeros and poles of these zeta functions. I was like, Whoa, this is some pretty advanced stuff. I don’t want to be messing with this. But then it's also really delicious because one, I think chicken tikka masala is very delicious.

KK: It is. Agreed.

MD: But also delicious in the context of these formulas, because the approximation formula for factorials is reliant on the poles and zeros of these zeta functions. So it's like, whatever I want to know about on the left side over here, the approximation for the factorials I'm looking at, all I’ve got to know is the information about the zeros and poles of these zeta functions.

KK: Right.

MD: That’s the spicy and delicious part.

KK: Very cool. All right. That's a good pairing. I like that. Cool. Thanks, Maiyu.

MD: Thank you.

KK: Up next, we have Hope Sage. Why don’t you introduce yourself?

Hope Sage: Hey, I'm Hope. I'm a junior physics major at TCU.

KK: Cool.

HS: My favorite theorem is Bell's theorem.

KK: Okay, I don't think I know this theorem.

HS: Okay. It’s kind of Physics-y. It's from quantum mechanics. And John Bell wrote it as a response to the EPR [Einstein-Poldosky-Rosen] paper, which is kind of a famous paper in quantum physics, where it talks about how quantum physics is probably incomplete, and there's probably some hidden variable that's underlying it. And Bell uses this inequality to calculate the probabilities based on what you would expect classically. So if the quantum particles are not entangled, then you would get this expected probability. And then he shows that experimental results kind of conflict with that. And so the underlying assumption is that from that, local realism isn't a thing. So then the universe is like, super wacky.

KK: I think we knew that, right?

HS: Yeah.

KK: So okay, so Alright, so my quantum mechanics is — well, calling it rusty would be, like, an insult to rusty things. So the probability of what? The state that a particle is in?

HS: Yeah, so whenever you have two entangled particles, one might be spin up, one might be spin down, okay? And you can run an experiment. They have a beam splitter and two detectors, and you measure the different states. It's kind of a traditional example. And the probabilities depend on the angle that everything is situated at. So you have nine, and then you were to take all those probabilities, he basically proves this inequality that just shows that mathematically, it can't be possible for there to be hidden variables, which means that things are paired and they would have to be, like, communicating at faster than the speed of light, which doesn't happen. So then there are all these theories about what could theoretically be the underlying nature of the universe from that okay, in different interpretations.

KK: Okay. So which is your favorite interpretation of what might be going wrong here? Or right, whatever the right word is.

HS: There’s this interpretation kind of extended from this called the many worlds interpretation of quantum mechanics. I don't know if I’d necessarily say it is the most likely to be correct. But I think it's the most fun one. It's also fun because you can read cool science fiction books about it. Like, every action you take, there's a different universe, different paths. I think it's kind of fun.

KK: Sure, right. So so right, so like, right now, what we're doing, we could, like, split into any number of paths. And there are all these weird different outcomes that could happen, depending on whether or not some quantum state is what it is or not.

HS: Yeah, basically. One of my favorite books is Dark Matter. It's by Blake Crouch. And it's about, like, every action that you take, there’s a different universe. And then there's infinitely many possibilities based on every single decision you make, which is kind of interesting, because every decision that you make does create a different next possible decision.

KK: Sure.

HS: But yeah.

KK: Okay. Well, all right, so our minds are getting blown. All right, what pairs well with this theorem?

Well, sometimes it's called “spooky action at a distance” and so I was going to go with Halloween candy. Because spooky.

KK: Yeah, yeah. All right. Okay, so what what's your favorite Halloween candy though?

HS: Probably Reese’s.

KK: What? Okay, I have strong Reese’s opinions. So which, like the full size or the miniatures or what?

HS: Okay, a Reese's Peanut Butter Cup, but the dark chocolate version.

KK: Okay. All right. I can respect that. I am team miniature. I think that's the correct ratio of chocolate to peanut butter. But the dark, I get it, I understand. Okay. All right. That was great. Thanks, Hope. Jonah, why don’t you introduce yourself?

Jonah Morgan: All right. Well, I'm Jonah Morgan. I'm a freshman engineering major here at TCU, and my favorite theorem is Gödel’s incompleteness theorem.

KK: Gödel’s incompleteness theorems. So that's more than one theorem. All right, so let's remind our listeners what what at least one of them is.

JM: Sure. So the first one: The first of Gödel’s incompleteness theorems is effectively any — I'll call it interesting, okay — any sufficiently interesting or complex set of axioms, it fundamentally has theorems or statements that cannot be proven, but which are true.

KK: Cannot be proven inside the system, right?

JM: Yeah, cannot be proven inside the system. Right.

KK: And okay, you hedged around, but I think “sufficiently complicated” just means, like, you can do arithmetic.

JM: Yeah, you can add numbers. Because if you just can't do anything, then well, you can’t say anything.

KK: Okay. Yeah. So that's the first one. What's the second one? I think I don't even remember the second one.

JM: So the second one was, I think — I'll make some background. I just think background’s kind of fun to know.

KK: Sure.

JM: His first theorem kind of says, Okay, well, if you have a set of axioms, you can have effectively a statement that says this statement cannot be proven by the axioms. So if you have that statement, then well, if that statement is true, then it's a true statement within the system that cannot be proven by the axioms. But if it's false, then it is a statement which cannot be proven. Or it is a false statement, which cannot be proven. I'm a little rusty on that aspect of it. But effectively, the idea is, so there's this weird statement that you can have in any system, which makes it he says incomplete, where incomplete is the word for it. But then, so mathematicians are like, “Okay, well, we want to prove things. And you gave one example, it's a bit of a weird example. I don't want to prove that a statement is unprovable.” So then he has a second theorem that says, well, also there are true statements which are unprovable which we cannot prove or unprovable.

KK: Yes.

JM: And so that's the second theorem. And that's like, okay, so you can spend your life working on a theorem or working on a problem, and then you can't even know whether or not you can know the answer to this problem within within the set of axioms that you're working with.

KK: Right. So it's hopeless, in some sense. Yeah. You can't fix this issue.

JM: No. It’s — some people say math is broken. But really, it just incomplete. There are certain things —

KK: Yeah, I mean, I think it caused a crisis amongst certain elements of the mathematical community, but I'm with you, I just sort of view it as, well, okay, so there are unprovable statements. It doesn't mean the bridges that we build are going to fall down,

JM: Right. Everything we have proven still stands. You can still prove a lot of things.

KK: All right, still lots more to prove. Where did you come across this?

JM: I think some time in grade school, middle school, high school, I got really into just watching videos about mathematics. And at first they were just little conjectures and little fun things and then Gödel’s Incompleteness Theorem stuck out because it's like, I was diving into the world of math for the first time. And there are so many things that you can prove, and even these, like, crazy things that I never thought were provable, or like things that are so complex. Fermat’s last theorem took 300 years to be proven.

KK: Right.

JM: And I don't understand any part of that proof, to be honest with you.

KK: Same.

JM: But we did it. And it took a long time, but sort of my idea after seeing all of this was that, well, anything can be proven if you have a sufficiently — or maybe we're not smart enough to find the proof, but everything should be provable. Mathematics, it’s a language, you should be able to explain things in that language. And then Gödel’s incompleteness theorem says no. And also, there are things that you just — it kind of changes your perspective on that.

KK: Right.

JM: And so then you get questions like, well, you know, the Riemann zeta hypothesis, one of the most famous unsolved hypotheses. And a lot of people see this and they're like, Oh, well, does this mean that this this million dollar problem, one of the millennium problems, could just be unsolvable? And we can't even know that it's unsolvable? And I think that was the first thing I looked up when I heard about this theorem. And something that was even more interesting to me was that if the Riemann zeta hypothesis is false, it’s provably false. Because it is equivalent to saying that there exists some number on the real part one half line, yeah. So you can write an algorithm, and given infinite time, you will find sure if there is one, you will find it. So it is provably false, which also means that proving that the Riemann zeta hypothesis cannot be proven means that it must be true.

KK: Okay. Yeah.

JM: So I can't give you the formal explanation on that. I don't know what it is. But the general idea is that you can prove that something is true by proving that you cannot prove it.

KK: Yeah, that's a little mind-twisting. But I can see why this would appeal to you, as you’re coming into your own intellectual being sort of state and moving out of being a kid. Yeah, that's really great. Okay, so. So what do you think pairs well with the incompleteness theorems?

JM: So, this isn't sponsored, but GrubHub, or Uber Eats or whatever.

KK: Okay.

JM: Because, occasionally, I order food there. It's easy, it's convenient. Most of the time, you get what you ordered. And sometimes your driver takes a nugget and you don’t — and, you know, that's gone. And sometimes he just doesn't show up at the door, and you're left wondering, where's the stuff that I paid my money for? And I think the feeling is similar there that you can you can pay for something, like you can spend your time working on this theorem, and it just unprovable, and you'll never know where it went or where it goes. And also you can pay for your food and just have no idea where.

KK: Yeah, well, that's a good pairing right there.

JM: Thank you.

KK: Thanks a lot, Jonah.

JM: Thank you.

KK: All right, up next, we have Anna Long. Anna?

Anna Long: Yeah, so my name is Anna. I'm a senior math and French double major. Actually.

KK: Nice!

AL: I’ll stick with English for you.

KK: Je ne parle pas bien le français.

AL: Tres bien!

KK: So I can say I don't speak the language really well in several languages.

AL: Well, that’s all you really need.

KK: That’s right. It was only that and “toilette,” you know, yeah.

AL: Yeah, so my favorite theorem I picked is the invertible matrix theorem from linear algebra.

KK: Okay.

AL: And it's really a pretty big theorem. It's 24 equivalent statements for a square matrix, that'll call A. So just a few of my favorite little statements in there. So obviously, we have that the matrix is invertible, that the columns then form a linearly independent set. So there's always a solution. And then that, for it to equal zero, it's only the trivial solution, that your vector is zero. That it has n pivot positions, or that it has full rank. And then that the linear map is both one-to-one and onto, and that the determinant is nonzero and that zero is not an eigenvalue.

KK: Right. And that's only, like, six of the equivalent conditions. So yeah, if you open up a linear algebra book, there will usually be at some point, some page where they list all these, I had forgotten there were 24. I can probably — so I'm teaching our senior-level analysis course this year. And part of it, we do some stuff with operator theory, and they remind, in our text that we wrote ourselves, they use of 11 of the equivalent conditions, but not all of them. So yeah, I can't imagine. I don't even think I know what some of the other ones are. I'm sure I would if you told me.

AL: A lot of them are, like, jumbled together. So, like, n pivot positions and full rank I've seen defined as two different things, but they’re really the same. But essentially, because of the theorem, they're all the same.

KK: Yeah. So yeah. So why do you love this theorem so much?

AL: I just love it because it's so useful. I’ve had two linear algebra classes now, and it just makes life so much easier trying to prove that any various things in class. So yeah, it's just great being able to find the easiest statement and prove that one, and then you just know all of them are true.

KK: Right. Yeah, I do like those things when, like, 50 things are equivalent. that's really nice. Yeah. Okay. So I guess you learned this in your linear algebra course. You had a second linear algebra course? What’s the second one?

AL: I’m in applied linear algebra right now.

KK: Okay. All right. So you're doing, like, singular values and things like that?

AL: Yes, we are.

KK: Okay. All right. So that's super useful stuff. Linear algebra, of course, I think is one of those things that we don't teach enough of. And basically, any problem in math comes down to either making some estimate, like analysis, or some linear algebra problem, it seems to me. so yeah, the more you learn, the more you know, then the better off you'll be. So you're a senior. What’s next for you?

AL: Graduate school.

KK: In what? In French?

AL: No. In math. Yeah. So I haven't decided on a school yet, but I'm looking at PhD programs in applied or computational math.

KK: Excellent. That’s great. Well, good luck to you. So what pairs well with this theorem?

AL: Yeah, so I picked chicken tortilla soup, mostly because it's my favorite soup. But you kind of have all sorts of different things piled in there. You've got your spices, you've got your chicken, you got your, you know, pieces of tortilla, you've got maybe chives or your different little vegetables in there. So you have a whole bunch of things that may or may not look very similar or different to each other. But you get one scoop of it, and you have the whole thing. So it's kind of like a 24-for-one deal.

KK: It’s like 24 equivalent soups in one.

AL: Right!

KK: Okay. Very cool. Yeah. So are you from Texas?

AL: I’m from Oklahoma.

KK: Okay. So this region, yeah, that sort of makes sense. That’s a popular sort of soup. Yeah. Okay. All right. Excellent. Well, thanks so much, Anna, that was great.

AL: Thank you.

KK: Up next, we have Matthew Bolding. Matthew, welcome!

Matthew Bolding: Hey there. Thank you. Yeah. So my name is Matthew. I am a senior dual degree student for mathematics and computer science, and I'm from around here as well.

KK: Okay. Cool. All right. So what's your favorite theorem?

MB: My favorite theorem today, or really, I guess all time, is the four color theorem.

KK: The four color theorem. Oh boy. Okay.

MB: Are you familiar?

KK: Oh, yeah, I am. So there's a lot to be said about this theorem. So yeah, tell us what it is, and then we'll unpack it.

MB: Sure. So I suppose in the most dry mathematical language, the four color theorem states that the chromatic number for a graph, for a simple planar graph specifically, is no more than four. And I guess there are some things to unpack there.

KK: Yes. Right.

MB: So a planar graph is a graph that can be constructed, drawn, if you will, such that no two edges cross one another. And a simple graph is one that does not have any self loops. And I guess the other part of it is, what's a chromatic number, right? And so a chromatic number is essentially the smallest k for which the graph is k-colorable. And then that also takes us down the rabbit hole with what is k-colorable? So a k-coloring of a graph is an assignment of at most k colors to the vertices of the graph in such a way so that no two adjacent vertices are the same color.

KK: Right? Okay. So most people might know this in terms of maps.

MB: Yes, that is correct. That actually what got me interested in it. It seems so deceivingly simple, I guess, you know, four colors, all you need are really at most four colors. You could do it in three or two, depending on on the map or graph.

KK: Sure.

MB: But, you know, you can ask people, what do you think? How many colors you think you might need to color this graph under these conditions and constraints? Oh, six, seven? No, you only need four.

KK: Right.

MB: And although I'm no cartographer, you know, I've never really colored a map maybe since pre-K, I think it is just so interesting, and sort of out of the blue, that you only need four colors. And with that, I also think it is really interesting that the proof that the chromatic number is no more than four hasn't been proved by humans, by hand.

KK: Right.

MB: And we've had to rely on computers to facilitate that proof. And being a computer science, or within the computer science field to study, I think that's really, really interesting.

KK: I thought that was part of part of your motivation here. I mean, so although, yeah, recently there, some people announced a proof by hand of the four color theorem. And it's wrong.

MB: Really?

KK: So I mean, anybody who's tried to prove this thing just by hand has come up short. And, you know, so the question of the maps, right? So, for a map, it's like, you know, you don't want two states that share a border to be colored the same, you know, and you can draw examples where you need four, but it's interesting that you can always do it with four, but then it does turn into a graph or a question, because how do you how do you create a graph out of this? Well, you stick a vertex for each state, and you join them if they share a border, and now you've converted it to a graph theory question. So that's how they come along.

MB: Exactly. And actually, I found this, or was presented this theorem in graph theory last spring. And I mean, I really — of course, with a computer science background, I mean, I just, you know, jumped in headfirst. I thought it was the coolest class. You know, it maybe doesn't have the same rigor as, like, really analysis or something.

KK: Oh no, it’s hard.

MB: Well, don't get me wrong. There are some difficult concepts. You know, I guess, with a computer science mindset, you know, I mean, of course, not every topic, you know, had roots in computer science, but maybe with Huffman encoding, or shortest path algorithms, you know, it was just so interesting and fascinating.

KK: Yeah, graph theory stuff is vital in computer science. I mean, it's everywhere, and having good algorithms for that is really important. You know, decision trees and all these kinds of things that you need to know. Cool. All right. So, yeah, but it's true that the first proof was given in, what, 1976?

MB: Around there.

KK: And yeah, and basically, it reduces to some couple hundred special cases that you just check. And then you get a computer to check it. And yeah, so for mathematicians, that's unsatisfying, right? We would just like a nice clean, wordy proof that works instead of relying on computer code. But I mean, I'm okay with it personally.

MB: I am too, but according to the Wikipedia page for this theorem, there are still many doubters. I guess we just have to prove it by hands.

KK: I guess. So that's how it goes. All right. So what pairs well with the four-color theorem?

MB: I really struggled trying to think about something that went along with this, but I landed on something that you could actually, you know, show the four color theorem with, and that would be Skittles.

KK: Okay.

MB: You know, you could lay them out all flat, you know, make it a planar collection of Skittles, basically. you could arrange the Skittles in such a way that no adjacent Skittles share the same color. Of course, there are more than four colors in a Skittle pack.

KK: Right.

MB: I would think it's been a while since I've had Skittles.

KK: Yeah, too sweet. Although, so the five color theorem is really not so hard to prove. Apparently, I've been told. I think I may have even read the five color theorem proof. It's not so bad. But four is tricky.

MB: I looked back at my graph theory notes, and we worked from a chromatic number no greater than six to five. And then we sort of just had a blank, you know, statement. Well, you can prove that the chromatic number is no more than four. But right, the ones for five and six aren't, too, too — I mean, compared to having to do it on a computer.

KK: Right. Right. Compared to people, you know, not necessarily believing the computer proof, right? Yeah. Yeah. We’re convinced about five, so we’re good. All right. Well, Matthew, that was great. Thank you.

MB: Thank you.

KK: All right, up next we have Brandon Isensee. Brandon?

Brandon Isensee: I'm Brandon Isensee. I'm a math major at TCU. I'm a senior. I'll be graduating this semester.

KK: What’s next?

BI: Grad school at Rice University.

KK: In math?

BI: Yeah. Computational and applied mathematics.

KK: Okay, that’s That’s great. It’s a terrific university. You're going to have a great time there. Well, I mean, grad school is what it is. It's fun and work and all those things. But it will really a great experience. All right. So, what’s your favorite theorem?

BI: So my favorite theorem is called Sharkovskii's theorem.

KK: Sharkovskii's theorem?

BI: Yes.

KK: Okay.

BI: Have you heard of it?

KK: I have. But let's tell our listeners.

BI: So kind of the theoretical way of saying it is that the theorem relates to discrete equations that are in, it’s one-dimensional discrete equation, so you only have one variable, and if a certain period exists — if a certain periodic orbit exists — that implies the existence of other periodic orbits. And if there's a three-cycle in particular, that implies the existence of all the other cycles

KK: Yes.

BI: And so that's more of the theoretical way of saying it. But if we put this in more concrete terms, if you have an equation that models a population over time, and it's discrete time, so it's years 0, 1, 2, 3, etc. So this equation tells you the population values for each year, right? And say there's a population growth parameter within this equation that you can vary, so we'll call it K. And this parameter tells you how fast the population is growing. And so say your growth parameter is two. And for this growth parameter, no matter which population value you choose, your population ends up oscillating between three different values across time.

KK: Right.

BI: So that's the long term behavior. So as time goes on, your population oscillates between, say, four individuals, five individuals, six individuals, and it keeps repeating. So it's 4, 5, 6, 4, 5, 6, 4, 5, 6. So if that's the case, then that implies that there is a four-cycle. So maybe it's 8, 9, 10, 11, there's a five-cycle, there's a six-cycle, there’s a 1-million-cycle, all the other cycles exist.

KK: Right.

BI: And so it's quite interesting, because when you look at particular examples of equations, like the discrete logistic map, and you look at where the three-cycle exists, you only see that three-cycle. So you may be wondering why? Why am I only seeing that three-cycle and not the four-cycle or the five-cycle? That's because Sharkovskii's theorem tells you that these cycles exist, but doesn't tell you whether or not they're stable. And so when you see the three cycle, it's stable, because that's where your populations are oscillating between. But when it's unstable, well those unstable cycles repel the population values away from them, and it ends up settling at that three cycle. So it's almost like there's an infinite number of fixed points, if I'm understanding this correctly. It’s like there's an infinite number of fixed points where you see the three-cycle, but all of them are unstable, except for the three-cycle, because that's what you're seeing on the graph.

KK: Right. Yeah. So sometimes this theorem is stated as “period three implies chaos.” Right?

BI: Right. So there, so there was actually like a difference between those two. So Sharkovskii's theorem is the one that's stronger, because that tells you exactly which periods imply the existence of other periods.

KK: Yes, right.

BI: And so period three, that relates to the three-period, right, the three-period implies everything, but Sharkovskii's theorem tells you exactly which periods imply the existence of others.

KK: Yeah. And if I remember right, he puts some weird order on the natural numbers.

BI: Yes. So the order the order is, if you're doing this in rows, the first row is your odd numbers, so 3,5,7…. Your second row is your odd numbers times two. And then the next row is your odd numbers times two to the second power.

KK: Sure. Right.

BI: So it's odd numbers times powers of twos. It’s very interesting. It took me some time to understand the order, but now I get it.

KK: And then, like, one is at the end or something, right?

BI: Yeah. Your two-cycles and such, you know, your periods of twos are all the way at the bottom.

KK: And then this is very important for discrete dynamics, right? It's just kind of the whole story. That's very cool. It's very, very cool. All right, so what do you think pairs well, with Sharkovskii's theorem?

BI: So maybe it's very a superficial connection.

KK: That doesn’t matter.

BI: There's a story with Alan Turing. He's the mathematician — for those that don't know, he solved the Enigma code during World War Two, the Germany Enigma code, and because of that, I forget the exact estimates, but it's like at least a million lives were saved because of that in, like, two years.

KK: It was vitally important.

BI: That shortened the war by two years. And so there's a story with Alan Turing where he rode his bicycle, and after a certain number of revolutions of the bike wheels, the bike chain would fall off. But instead of him just fixing it, he would just count the number of revolutions as he's riding the bike, and right before the bike chain would fall, he’d get off the bike and he would just put the bike chain back on.

KK: I wonder if that’s true.

BI: I suppose so.

KK: It’s a good story either way. Yeah. Okay. That's that's a good pairing. Excellent. Thanks.

BI: Thank you.

KK: All right, and our last willing volunteer today — I think they were all willing — is Julia Goldman.

Julia Goldman: Hi. Yeah, I'm Julia. I'm in my first year of grad school here at TCU.

KK: In math?

JG: Yes.

KK: Okay. How do you like it so far?

JG: I like it so far.

KK: Math’s pretty cool.

JG: I think so.

KK: All right. So what's your favorite theorem?

JG: So my favorite theorem today is Brouwer’s fixed point theorem.

KK: Brouwer! All right, good. Finally a topology theorem. Good.

JG: So last week, I was trying to come up with a theorem talk about, and one of my professors suggested this one. And when I was going online, and reading it and learning about it, I was looking at all these proofs of it that seem fairly technical, and I’d probably want to take a topology class to really get into it.

KK: Sure, right.

JG: But the theorem itself is, I think, very understandable. And I came across so many of these cute little fun real world examples that make the theorem pretty explainable to anyone of any math background, and I really appreciated that aspect of it is approachable.

KK: Okay, so what's the theorem? Let’s remind everyone.

JG: Oh yeah. The theorem is for any continuous function of a convex compact set onto itself, there's going to be at least one fixed point.

KK: Right. One point that doesn't move.

JG: Exactly. I think that’s really fun. There’s a couple examples. Like if I had a map of Fort Worth right now, and I laid it on the floor, there would be at least one point on that map lying directly on top of the point it's supposed to represent.

KK: That’s right.

JG: That’s kind of fun.

KK: Yeah. That's a good example.

JG: Then my other favorite example is a cup of tea. You stir the cup of tea. When you're done stirring, there’s going to be one little bit of your tea that's in the same spot as when you started stirring.

KK: That’s right. Okay. Are you a tea drinker or a coffee drinker?

JG: A little bit of both. Maybe it's because I'm not a topologist, but I read that example, and I just thought, I feel like I could stir my tea enough.

KK: Sure.

JG: But Brouwer says I'm wrong.

KK: That’s right. That's right. Okay. So maybe you don't know enough topology to prove this yet? Did you find a favorite proof that sort of made sense to you?

JG: Not a favorite. I just glanced over it.

KK: Right. So the algebraic topology proof involves, like, something with the homotopy groups, or homology groups and things like that. So that's kind of weird. There are sort of analysis-type proofs. So we just talked about dynamical systems a little bit. So if you just pick any point, and you start iterating the function, right? Just keep pushing it around, then eventually it will converge to a fixed point. Well, some subsequence of it will, because you're in a compact set. So that sort of analysis thing. So that's another way to think about it. But yeah, this is a popular theorem among topologists. You know, we all really dig this theorem a lot. It's probably one of our favorite examples. It is a fan favorite. Absolutely. Yeah. Okay, good. I like the map example, too, because that's really illustrative. Okay. So what do you think pairs well with Brouwer’s fixed point theorem?

JG: Well, I wanted to pair my favorite theorem with my current favorite TV show, which I'm only a little bit embarrassed to say is a reality show called Love Island.

KK: Okay. I've heard of it. I have not watched it.

JG: I think they must have some topologists on set there because I think the show itself is kind of an example of a theorem if you stretch some definitions a little bit.

KK: Okay. Let’s hear it.

JG: So if you're unfamiliar with the show, it’s a dating show. The very first episode, all the participants are put into couples. And then there’s, like, 65 episodes of just, like, fighting and breaking up and getting into other couples, whatever. From the seasons I've seen, at the end of the show, there's always at least one couple that ends up back in their original pairing.

KK: Right.

JG: So thinking of the participants as our convex set, and all the show drama as a function, then there’s your example of the theorem.

KK: Yeah, it's funny how there are always two people who were like, you know, we were we were right all along.

JG: Exactly.

KK: Do they do it randomly? I’m sure the producers don’t actually do it randomly.

JG: I mean, they kind of mix it up. They let the girls choose the first time, or the boys.

KK: All right. Okay. Well, I’ll have to check this out.

JG: Great show. Highly recommended.

KK: Okay. All right. Excellent. Okay, well, thanks, Julia.

JG: Thank you so much.

[outro]

In another Very Special Epsiode of My Favorite Theorem, Kevin had the privilege of asking a group of nine TCU students about their favorite theorems. We loved the variety of theorems and pairings they picked! Below are some links to more information about their favorites.

Aaryan Dehade led off with the fundamental theorem of calculus.

Duc Toan Nguyen's favorite is the mean value theorem, which some would argue is the real fundamental theorem of calculus. It was also a hit with our past guests Amie Wilkinson and Aris Winger.

Maiyu Diaz shared Stirling's formula for approximating factorials.

Hope Sage chose Bell's theorem from physics, which was a response to a paper by Einstein, Podolsky, and Rosen.

Jonah Morgan shared his love for Gödel's incompleteness theorems, which also came up when we talked with math students from CSULA last year.

Anna Long chose the invertible matrix theorem, a behemoth of a theorem that gives scads of ways to show that a matrix is invertible.

Matthew Bolding highlighted the four-color theorem.

Brandon Isensee chose Sharkovskii's theorem, which was also the favorite of past guest Kimberly Ayers.

Julia Goldman finished out the episode with a perennial MFT favorite, the Brouwer fixed point theorem. We have talked about it on past episodes, most recently with Priyam Patel. See if you agree with Julia that it is the mathematics underlying the TV show Love Island!

Episode 83 - Cihan Bahran

16 februari 2023 | 27 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, one of your co-hosts, coming to you from snowy Salt Lake City, Utah, where I feel like I've said that the past few times we've been taping. Which is great, because we really need the water. It is beautiful today, and I am ever so grateful that the life of a freelance writer does not require me to drive in conditions like this, especially as someone who grew up in Texas where conditions like this did not exist, and so I am extremely unconfident in snow and ice. So yeah, coming to you from the opposite side of the weather spectrum is our other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. It's true. It's the opposite end of the spectrum, but hey, you know, I was putting up my Christmas tree the week before last and I was sweating. So this is my reality.

EL: Yeah.

KK: It’s hard to get in the mood, you know, you put on the Christmas music and you you get the tree out of the attic. And then I'm in, like, shorts and a t-shirt and sweating.

EL: You can sympathize with Australians, who have to deal with that every single year.

KK: That’s right. That's right. Yeah. So anyway, we're looking forward to a nice holiday. My son's going to come home after Boxing Day because he has a part time job at a bookstore in Vancouver and his boss said no one gets Boxing Day off.

EL: Yeah, that's that's a thing in some places.

KK: In the Commonwealth. I think it's a big thing. Right? So yeah, he'll be home on the 28th. So we're looking forward to that. But anyway, anyway, this will be after this will be after the holidays when people hear this anyway. So they'll go, gee, I wonder how that went?

EL: Yeah. Waiting with bated breath for updates about your son’s Boxing Day experience.

KK: That’s right. That's right.

Yes. Well, today we are very happy to have on the show Cihan Bahran, coming to us from I don't know what kind of weather. So yeah, could you introduce yourself and tell us about the local conditions?

Cihan Bahran: Yeah. Thanks for having me. I am joining you from Ankara, Turkey, which is the capital of Turkey in the middle. So it's a continental climate, I would say. But it has been rather mild. We haven't had any snow yet or really anything that close to freezing temperature. So yeah, it's chilly, but yeah, I like it.

EL: Yeah. And so what what kind of math are you interested in

CB: Right. So I am interested in representation theory, especially with functorial methods, and I am doing a postdoc here about that at this at this time. So actually, maybe I'm at a little bit of a disadvantage in that the theorem I will share is not necessarily directly from my expertise, so I'm not really, maybe on top of the literature or the methods, but I thought I would pick that because I find it really interesting.

EL: Sometimes, honestly, that could be a little better, because we are also not experts in that.

KK: That’s right. Yeah.

EL: But yeah, and you run a Twitter account, and I meant to look up the exact — is it called some theorems?

CB: Yeah, it's called some some theorems. The username is something like Cihan posts theorems [Editor’s note: It’s @CihanPostsThms] Okay, let me talk about that a bit. So I guess it goes back to maybe 2020 or something, not this account, so that was the pandemic time and for me, maybe psychologically a difficult time that I was seeking out somewhere to connect with the math world. And I found initially a Facebook page called Theorems. And it is it is still running, I guess. But I started posting there. And I had a lot of, like, some bits of knowledge about some interesting theorems that I would, like, share with my friends. And it became like, I was almost daily posting, like the group became dominated by my posts, to the point that people started asking, like, what are you really doing, et cetera. And then maybe since last year, I've been more on Twitter, and I posted some of these on my personal Twitter account. But then for some reasons, I had to make my personal account private. And at some point, I thought I might repost these things that I have had collected, because that group in Facebook was actually a private group, not everyone can see it before joining. And I thought I would post those on Twitter, and I find it, like, when it gets some responses, it's like a dopamine hit for me.

KK: Sure.

CB: And I was actually almost aggressively posting in the summer because I had all this sort of backlog. And at this point in time, I have posted most of the past stuff, and I post much less regularly. When I see something interesting, I post them to the to that account. And I suppose how that's how I am maybe known in math Twitter-verse.

EL: Yeah. And so as a person with with much knowledge and love for theorems, what is your favorite favorite zero?

CB: Okay, so I don't know if it's my favorite, but at least for this episode of My Favorite Theorem, the theorem I would like to share is the so-called — well, so there's this problem, and the theorem says that this is algorithmically undecidable. So what's the problem? The problem is called matrix mortality.

EL: Which is a really an inviting name.

KK: It’s a great name, right? It sounds like a video game or something. Yeah.

CB: Yeah. So, in the most general sense it asks, so the input is a finite list of square matrices of the same size. And the the the decision problem is whether a product of these things in some order, possibly with repetitions, could be ever zero or not. So, if an algorithm would say yes or no to each such collection. And I think at first this was shown to be undecidable for already 3 × 3 matrices in the 70s. And then there were some further developments as to because of course, if I give you one matrix, then matrix mortality becomes is this matrix nilpotent, and you can determine that by the characteristic polynomial, so that is decidable. So how few, how short can the list get and remain undecidable? I think for 3 × 3 matrices, it has been shown in 2014 or so that six 3 × 3 matrices, the problem is undecidable. So like A, B, C, D, E, F, F, that’s six 3 × 3 matrices. So that's like, what, like 54 entries of integers? These are all integer matrices, by the way.

KK: Okay. I was going to ask that.

CB: Snd then the question is, is some product ever zero or not? There can be no algorithm answering that for every possible input. And if you make the, if we allow the matrices to be a bit bigger, there is a version which says that when you make the size 15 × 15, it is undecidable for even two matrices. So just, like, two matrices of size 15, A and B, the decision problem, is ever a sequence of A's and B's equal to the zero matrix? And such an algorithm cannot exist, it's undecidable. It's rather striking.

EL: Yeah, I guess — I'm actually a little more upset about the six, 3 × 3 than the two 15 × 15’s. Because honestly, I just imagined trying to write down the entries of a 15 × 15 matrix, and I give up maybe 30% of the way through, I'll just, okay, whatever.

CB: Well, there is still a gap in knowledge. So let me talk a bit about what's known. For I think, two, 2 × 2 matrices, just two of them, it has been maybe recently shown that that is decidable. So but when the list is, when you have three or more matrices, I believe open. Also, I believe it's still open, whether if you're given, like, five, 3 × 3 or four the lowest boundary we know is six, although from from the development, you might — I would guess that it will remain undecidable for even two 3 × 3 matrices. But that's, I think, unknown at the moment.

KK: So once you show that it's undecidable for a certain, so for six, 3 × 3’s is undecidable, so that means it's undecidable for six of any size larger than 3 × 3, correct?

CB: Yeah.

KK: Because it sort of stabilizes, right? You can put those inside of the next size up by just sticking a one down on the lower corner with a block.

CB: Exactly, you can even you can even pad them by zeros, right?

KK: Sure. Sure.

CB: The mortality problem will not change once you've artificially made your 6 × 6 matrices into 10 × 10 matrices by writing zeros everywhere else.

KK: I see. So the question is for a fixed n, can you what's the minimal number k for which it's undecidable? Right?

CB: Yeah. So there are two parameters, how many matrices and the size of the matrices.

EL: But I guess there's a chance that it's three for 2 × 2 and two for everything else.

CB: Maybe. If I were to bet, I might bet that 2 × 2 is special and would be decidable always, and like the 3 × 3 introduces — but that's just a hunch. I don't really know much about how these things are done, because, like — I mean, I did look a bit to the into the two 2 × 2 matrices, and the algorithm is by computing some some eigenvalues or such, and I and 2 × 2 is so small that I would guess that is enough information somehow, but I don’t know.

KK: Now, I'm just thinking about this, right? This is sort of different from — so I would think of this in terms of the group generated by these matrices, but that's not at all what you're doing, right?

CB: Yeah, it's more like a monoid because it becomes zero. Also, I would like to, for people who know about the word problem, this this reminds people of the word problem for groups. And there is, of course, a relationship, but I would object to the argument that, “Oh, because the word problem is undecidable, that’s not so surprising.” And my objection is that we can always multiply the matrices. A specific instance is always decidable. We can just multiply them and see. For the word problem, there are even specific instances, which remain, like, is deciding whether a word is trivial can be made into a specific presentation and remain undecidable already there. It’s not like because you have maybe relations between the words, you don't know how to change your word into something. But with matrices, we can always, we can multiply like multiplications doable. But when you when we allow, is there ever zero among arbitrarily long multiplication that that is where the problem is. I think the word problem, the problem arises earlier than that.

KK: And that direct analogy with the word problem, you'd be looking for products where you get the identity, right, as opposed to zero. So I guess you don't want any of these matrices to be invertible. It's allowable, I imagine. But but if you have an invertible one, that’s not going to help.

CB: Yeah, I mean, the invertible ones, you can always — I guess, well…

KK: I don't know, though, maybe you need a permutation matrix to make some product work out correctly? I don't know. This is an interesting question. I like this question.

EL: Yeah.

KK: We’re not going to try to solve it on the spot.

CB: I’m not sure. Like, my first thought is that you can probably even, like, throw the invertible ones out. But now I'm not so sure. Maybe they might help in some way of arranging the zeros.

KK: So where did you come across this theorem? This is an interesting result.

CB: Yeah, well, undecidable problems always have fascinated me, and I guess I might have been looking at some of these, maybe it was, I don't know where I came across it. Maybe it was some survey paper of undecidable problems, or maybe a Math Overflow question. I'm not sure. But yeah, somewhere along those lines.

EL: But it's a nice one that's maybe a little more accessible to most people who have taken, you know, a few upper-level math classes than some of the undecidability things, which are just like, Okay, I need to climb this whole mountain to even understand this. You know, we all take linear algebra at some point, you know, if you're a math major or something, and so it's very concrete, you can immediately understand what it is if you’ve seen matrices.

CB: I agree. The description is rather elementary. You don't need to introduce Turing machines and halting arrays or some abstract presentations of groups and such. It's rather — the operations are ones that, as you said, any linear algebra student has seen before, but somehow the problem is already like, not even difficult, it's impossible in some sense.

EL: It is always really interesting to see, like, what are the limits, not just of our knowledge, but of what we can know about our possible knowledge.

KK: Right. Yeah. So the other thing we do on this podcast is we invite our guests to pair their theorem with something. So what pairs with this theorem that doesn't really have a name, but we'll call it the undecidability matrix theorem or something?

CB: Yeah, okay. So I'm not really a food person, so I didn't think of a food. What this — I would say that it pairs well with a decent table tennis service. Because it — there’s some, like, it’s not a killer service but decent, so you can have a decent back and forth, as we have just had, as to like, how small you can make it, how bad is it, that sort of thing. So that's what it reminds me of.

KK: Yeah. All right.

EL: Do you do you play table tennis?

CB: I like table tennis. I don't play as much as I would like to, but occasionally I do play it, and I like playing it.

KK: I’m much better on the Wii than I am in real life. The Wii table tennis is really fun.

CB: Ah, okay.

KK: In real life, not so much. I mean, I like it, but I'm not very good.

CB: I also heard people play on VR online. You can even like see a table.

KK: It’s not quite the same.

EL: I have not played since I was probably in sixth grade or something, when I think I was pretty capable of beating all of my opponents, who were my younger siblings. So, you know, at that age, you’ve kind of got just some advantage by being a little older. And so I tried to take advantage of that whenever I could, as the oldest sibling. You know, I really, we played two-on-one basketball sometimes, and I always kicked their butts at that because, you know, I was way taller. It helped a lot. I couldn't really play but like, against someone four or five years younger…

KK: My problem with all racket sports is that I played a lot of tennis when I was in high school. And so I play all racket sports like tennis. So you know, with big swings, so that doesn't work in table tennis. I's a much faster, yeah, just shorter.

CB: Much more of a wrist play than the whole arm.

KK: That’s right. Yeah. And racquetball is the same way. Like, I want the ball at my waist. I don't want to be reaching down to my ankles. I'm too tall. I can't get there. You know. So all these things were a challenge for me.

EL: Yeah, well, I do really like this pairing, because just like this theorem is sort of this meta- about, not just a specific case of matrices, but like, what we can know in general, given, you know, any set of information, your pairing was not just about the theorem, but was also about our discussion of the theorem.

CB: Yeah.

EL: So I think this was this was very elegant. I appreciate that.

CB: Thank you. I also like that we still have a gap in knowledge. I think that as, I don't know, like, teachers, we introduce — I remember being as a student, that that would really pique my interest, like, when teachers discover, you know, this is not known. And I think it offers a different landscape versus a completely furnished theory.

EL: Yeah, well, I know, when I was in college, I liked my math classes, but I didn't understand that math was still this active area of research. I was very naive about that kind of thing, and even now, you know, you've run into people who don't know what math research means. It's like, well, we know how to add and multiply numbers. We know how to do all of these things. Like, what else can there be to know? and this is something that doesn't take as much — you know, it's one of those examples that you can give.

CB: Right.

EL: You know, it has a lower, or, you know, a more basic way that you can enter this and like, understand, Oh, we're still trying to figure out this kind of thing. And so, I like that.

KK: Yeah.

CB: Also another thing I like, it's a bit upsetting that this is not decidable. I like those sorts of results. Actually, my account in Twitter has been referred to “the account that posts cursed math facts.” A lot of people say that, and that was not my intention, but it kind of fits with that.

EL: Yeah, well, that's very true because yeah, when I first saw it, I was just like, well, how can we not just, you know, just try all the ways to multiply it. At least in theory, you could do that, but not if it's arbitrarily long.

CB: Yeah.

EL: You're allowed to have as many as you like. Okay, if it was just one copy of each one, well, that's trivial. I mean, not trivial to actually do it, but it's trivial to know how to do it. But it's kind of funny that once you allow yourself multiple copies, it's just like, everything goes out the window.

CB: Yeah.

KK: All right. So you've already plugged your your popular Twitter account. We like to give our guests a chance to let us know where we might I find you online or anything else you're you're you're trying to promote or anything like that. It’s okay if you don’t.

CB: There’s my account. It’s called some theorems. So I think I can just put that in Twitter.

KK: Well for now.

CB: Yeah. I think I won’t add more to that.

KK: Okay. Well, Cihan, this has been great.

EL: Thanks so much for joining us.

CB: Yeah. Thank you for having me.

[outro]

On this episode, we were excited to talk with Cihan Bahran about the undecidability of the matrix mortality problem. Here are some related links you might enjoy:
Bahran's website and Twitter account, where he posts "cursed math facts"
The 2014 paper establishing the undecidability of the matrix mortality problem for, among other cases, six 3 × 3 matrices
The word problem in group theory

We recorded this episode before the devastating earthquake in Turkey and Syria. Our hearts go out to all who have been affected. If you would like to contribute to relief efforts, Doctors Without Bordersand Ahbap Derneği are two organizations doing work in the area.

Episode 82 - Juliette Bruce

30 december 2022 | 29 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcasts with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right. I got to take an overnight Amtrak trip last weekend, my first time, so that was pretty fun. Went from Salt Lake to Sacramento and got to see lots of beautiful Nevada and California landscapes on the way.

KK: Yeah, I did an overnight Amtrak once and it was less fun. It was from Jackson, Mississippi to Chicago. And — which, I mean, it's, you know, it's all night, right? So you don't really see anything. And it's remarkable how many times have to pull over for the freight trains, right?

EL: Yeah.

KK: This is how American rail is really different from European rail. You're at the mercy of all the freight, but that's okay. Anyway, yeah.

EL: I guess, today, living on the only portion of Amtrak's corridor for which they actually own the tracks, is our guest, Juliette Bruce. At least I hope I'm correct, that that's where you're living. Otherwise, that was a weird introduction. So please tell us a little bit about yourself.

Juliette Bruce: Thank you so much for the introduction. I'm Juliette Bruce, as you said, and I am a postdoc at Brown University. So in fact, I am in the northeast along the Acela Express corridor. In fact, I've never taken that Amtrak corridor, I've only taken the very slow ones that you were talking about, but I hope to take it soon.

EL: Yes. Find yourself someplace to go between New York, DC, Boston, I guess to Boston, you don't really need the Acela. It's already pretty close.

KK: You can walk to Boston.

EL: If you're really dedicated.

JB: It’s a pretty far walk.

EL: Yes. So I guess this isn't the train cast. This is a math podcast. So, so yeah. What are your mathematical interests at Brown?

JB: Yeah, so my area of math is kind of in the intersection of algebraic geometry and commutative algebra, which is all about studying the interaction between this algebra, coming from kind of the symbolic equations we get when we write down systems of polynomial equations, and the kind of geometry we can look at when we study the zero set of those equations. So we can look at the simultaneous solutions to the system of polynomials, and that's some lovely geometric object. And alternatively, we can look at these symbols we write on our paper, and somehow, in some point in math, we learned that we can do lovely things, like finding the roots of a quadratic polynomial by graphing them on our graphing calculator pictorially, or we learn we can use symbols and write down things like the quadratic formula, and magically they give the same answer. A lot of my research is sometimes a generalization of this fact that there's two different ways to study the solutions to a system of polynomial equations.

EL: Right. I must admit, I'm pretty naive about algebraic geometry, but there is this kind of magic in it, which is — you know, like, in, what, seventh or eighth grade or something, you start learning to graph the zeros of polynomials. Maybe you might not use that exact language for it, but you start to understand that you can intersect two different polynomial equations and find these intersection points and stuff like that. And yet, this is also like cutting edge math, you know, just add a few variables, or bump up the powers of the the numbers that you're using. And suddenly, this is stuff that, you know, people are getting PhDs in. I's kind of kind of cool,

KK: Right? Or work over a finite field, whatever those are. Yeah.

JB: I mean, I always find it fascinating with just how many different areas algebraic geometry has touched in mathematics and in the world. It seems to start from such a lovely and beautiful, simple idea that we learn in, you know, middle school or high school, and just kind of grows exponentially. And it turns out, it's actually a very deep idea that maybe we don't always appreciate when we first see it. I know I certainly did not.

EL: Yeah.

KK: All right.

EL: So then what is your favorite theorem?

JB: So my favorite theorem, or the theorem I want to talk about today, I know it as Petri’s theorem. I know some people know it as the Babbage-Enriques-Noether-Petri theorem. I'm not sure exactly on the correct attribution here, so I'll stick with Petri’s thereom and apologize to Babbage, Noether, and Enriques, who maybe want the appropriate attribution here. And this is a theorem from classical algebraic geometry, which means from the 19th century, and it's about understanding the interaction between thinking about systems of solutions of polynomial equations abstractly, and how we can realize that abstract solution set concretely as solutions to an honest-to-God set of polynomial equations that we could write down and describing what those polynomials might look like.

KK: Okay.

JB: And so the statement of the theorem, I'll state the theorem, and then we'll walk through it, maybe. And you can ask questions, because I know when it’s stated, it's a little bit of a mouthful and a little scary, is that if I have a curve that is non-hyperelliptic, and I embed it via the canonical embedding, then the image of the canonical embedding is cut out by quadratics unless the curve is trigonal, meaning it admits a three-to-one map to the Riemann sphere, or it's a curve in the plane of degree five. So that's the statement of the theorem. That's a mouthful, I know, to get through.

KK: Yeah, sure. Right.

EL: That’s interesting. So, you know, as I already confessed, this is outside of maybe my, my mathematical comfort zone a little bit. And how, how should I think about these exceptions? Like how exceptional are the exceptions? Is it, like, a lot of things? Or just a couple of little things that and otherwise, everything falls under this umbrella?

JB: Yeah, so that's a fabulous question. And so I gave — there are two exceptions to this theorem, right? If a curve admits a three-to-one map to Riemann sphere, so there's a map that goes to the Riemann sphere, that kind of every preimage has three points, it kind of looks like a sheet wrapped up three times around the sphere. Or it's a very specific curve in the plane of degree five. And so these exceptions, there's an infinite number of them. But it turns out if you think about them correctly, it's kind of a small proportion, or it's not most curves that will satisfy this. So this is somehow saying, with these few exceptions aside, we can actually understand the image of what's called the canonical bundle. So maybe I should say, what is actually going on here. It's something a little deep. So kind of the starting point of algebraic geometry is that we want — I said, we want to study the solution sets of polynomial equations. Well, it turns out that that's how the field started. But pretty quickly, people realized, well, this is some geometric space, it's a set of points. And instead of looking at the solution set to a particular set of polynomial equations, we can kind of abstract this away and forget the polynomial equations together and just think about what possible sets of solutions could I have, and think about that kind of abstractly in the ether. There's no polynomial in sight, we can just say, oh, you know, this is a solution set to some system of polynomial equations. We don't know which. And it's a lovely theorem that, you know, if we're talking about curves, it turns out algebraic geometers have this very weird convention that curves would look to people like us, like a two-dimensional surface. This is because I like to work over the complex numbers. So my polynomials have solutions and the complex plane is two-dimensional. So we have this weird terminology. So abstractly, a curve, if it's smooth and has to satisfy some other conditions, just looks like a closed surface, possibly with some holes in it. So we'd have a genus g surface. So if you've seen a doughnut, or a torus, that's just an algebraic curve of genus one. And if you seen a sphere, that's just an abstract algebraic curve of genus zero. And the beauty of these is that somehow, if we take these objects, we can realize them in space, we can put them into some large, complex space, or some large projective space, and once we've done that, you can ask, well, I know there is some set of polynomials that cut the space out, we have this algebraic variety. It's a system where we know it's by definition, a solution set to some polynomials. And you could ask what polynomials actually cut it out under this realization in space. And often, there are many different realizations. So for example, you could look at the parabola, a very simple example. We can look at the parabola, x2−y=0. This gives us the normal parabola going through the origin. It’s realized in space. But we can also abstractly think about just kind of the parabola floating around, no coordinate system at all. And we could also realize that same parabola in space by just, you know, shifting it up or down the y-axis and moving it around, and the polynomials that cut it out when I start moving it around, we learn, are different, right? We learn how to do transforms, we knew somehow, like (x−1)2−y=0 gives a different solution set, but it looks the same in the plane, just moved around. So you could ask, when I put my abstract curves in space, what are the polynomials that actually cut this thing out? And so those are kind of the input to the theorem, is these abstract curves. And we put them into space. And what are they cut out by? So that's kind of the input. And the theorem is answering that question, what are they cut out by? What are they defined by?

KK: Right. So are you assuming you're starting with a plane curve? Or?

JB: No, so this curve doesn't have to be in the plane, although it's kind of just this abstract notion of a curve, so it’s somehow, just in general, a kind of smooth looking surface that's compact and has g holes, so maybe like a 2-holed torus or a 3-holed torus. It's kind of some very weird donut-looking shapes, essentially, is what the curve goes in, what is the input of this theorem?

KK: Right. And you mentioned something called the canonical embedding. So that might require a little terminology.

JB: Exactly. So what do I mean by the canonical embedding? Defining it exactly is complicated. And it's not something I would want to try to do on this podcast.

EL: Especially with audio.

JB: Especially with audio. But instead, let me just kind of give this notion. I said, you know, if we're looking at perhaps standard parabolas in the plane, there's a lot of different ways we could put it in the plane. We could put it through the origin, we could put it so like the vertex is at (1,1) or (2,1), or we could do all these things, and there isn't, doesn't seem to necessarily be a natural best choice for how we put a parabola in the plane.

EL: Right. It feels very arbitrary.

JB: It’s very arbitrary. And when we change our arbitrary choice, we change the set of polynomials that define the parabola in the plane. It turns out that when we're kind of working in a slightly more abstract setting, where instead of looking at parabolas in the plane, but we're looking at these two-dimensional surfaces, which are what algebraic geometers will think of as curves, because we're looking at the complex set of points, there’s an almost canonical way to put them into some kind of space. And that's called the canonical embedding. It kind of arises by looking at ways you can kind of differentiate on your surface. It comes from looking at what are known as differentials on your surface. And I won't say anything more than that, other than to say somehow, it's this beautiful fact that was developed by people in the 19th century that there exists such a thing that allows you to transport these abstract curves into different spaces in a way that has beautiful properties. And somehow, it's a great tool for studying curves.

EL: Not quite sure if this is the right question asked, but you know, you have this input to this theorem, and then it tells you something about like, you know, what polynomials can be your solutions? How specific is it? Like, would it output something that, you know, we would have recognized as a polynomial in seventh grade? Or does it output something that maybe has a little more technical machinery behind it?

JB: This is an absolutely fantastic question. This is a fantastic question. So, right, as you're saying, the input is I input this abstract surface abstract Riemann surface of genus g that satisfies some properties and the output of the theorem and saying if it doesn't satisfy, if it doesn't fall into these two exceptional collections, which are relatively small when it comes to lists of exceptions, then we know that the defining equations are degree two. And you might ask, well, does the proof actually give — like what are the polynomials? Can you actually write them down? And in part, the version of Petri’s theorem I know, in fact, gives you those polynomials in some sense. There are some choices that have to be made, and those choices kind of arise from some technicalities about defining exactly what is the canonical embedding. There are some choices there. But once you've made those choices, Petri’s theorem actually comes down and says we can write down an honest set of degree two polynomials in a lot of variables now. The number of variables is the number of holes on my surface, is the genus of my surface. So it's a lot of variables. But we can write down an honest set of equations that cut this out. And this is this beautiful thing that takes an extremely abstract thing, you know, this curve that's abstract sitting in our mind, and realize it in space, and it outputs a list of polynomial equations.

EL: Okay, wow.

KK: So okay, so now I'm thinking about elliptic curves in particular. So you mentioned there's just one variable. So that's a torus. Right? You can you can actually write this with one variable?

JB: Yes. Yeah. So if you're looking at elliptic curves,

KK: But I always think of elliptic curves as being, like, y2=x3 plus some change, right? That's not degree two, is it?

JB: That’s not degree two. Right. And you're calling me on a on a little technical point that I swept away in the beginning, which is that I said — when I said theorems carefully, I said that if our curve is non-hyperelliptic, right, and it happens, that elliptic curves will kind of not be in a case where the canonical embedding is, in fact, an embedding. Somehow you can talk about what that map might be, and for an elliptic curve, that map would take your elliptic curve and map it all to a single point.

KK: Yeah, that's a bummer.

JB: And that's a bummer. So sadly, elliptic curves, it doesn't quite work. So it is this interesting issue where if we're looking at abstract curves of small genus, so like elliptic curves are doughnuts of genus one, so there's one hole, or if we're looking at curves of genus two, so there's donuts with two holes, this theorem doesn't really apply, because those curves are extra special. And in fact, that's some of the beauty of things like elliptic curves. But once we're looking at more higher genuses, like genus three or four, and so on, you start to see very interesting things. So for example, if you take a genus three curve, and it satisfies this property being non-hyperelliptic, whatever that means, you can realize this curve in the projective plane, which is kind of a three-dimensional object cut out by some polynomials of degree four.

KK: So I mean, is this this love at first sight? Like, did you, you know, as a student read this in, is this in Hartshorne somewhere, or is it somewhere else and just fall in love?

JB: Yeah, so this is a great question. And it's actually as far as I'm aware of, not in Hartshorne, which is kind of a weird thing, because Hartshorne is notoriously quite comprehensive.

KK: Sure.

JB: But all the players are in Hartshorne. And in fact, the lead-up to kind of this theorem is in Hartshorne. And so the build up to this is to get to this theorem, you spend a lot of time in this fairly hard textbook, and you get to the end, and all of a sudden, they say, let's look at curves. And you think, wow, that's pretty simple. I've spent a year and a half, two years of my life learning all this complicated machinery and now you're going to tell me we're doing the simplest case. And you start looking at them and you see these beautiful things where all of a sudden, you built this machinery that lets you compute these equations in specific cases, so say small genus. And later on, you can read this amazing theorem of Petri and see that there's actually a full argument there, of how you can write down these polynomials, and it's kind of this beautiful synergy of all the things you learned coming together in one.

EL: I mean, I guess you've kind of answered this a little already, but what do you think draws you to this theorem so much that makes you love it?

JB: Yeah. So what draws me to this theorem, I think, is a number of different things. So (a) is this beautiful combination, or culmination, of learning so much, so many of these kind of complicated tools that don't seem closely related to the spirit of algebraic geometry, which is again, studying polynomial equations and their solution sets. But also, its has this amazingly surprising thing, which is that somehow if I take this abstract curve, and I take some realization of it in space and I ask for the equations that cut it out, those equations should really depend on how I put it into space. You know, if I put them into space a different way, I'll get different equations. And what this theorem is saying is that since these exceptions to this theorem don't depend on how I put it into space, those things only depend on the actual curve itself in its abstract form, that somehow, there's this beautiful thing that sometimes when you put things into space in the correct way, and you look at their defining equation, that's telling you something very, very special about not just that particular realization of your curve, but kind of the abstract, ethereal curve that lives kind of off in our imagination.

KK: So part two, we like our guests to pair their theorem with something. So what pairs well with Petri’s theorem?

JB: Yeah, so the thing that I thought of when I was thinking what pairs well with Petri’s theorem, is I was thinking about how, when I first started to see the, you know, glimpses of this theorem, it was Chapter Four of Hartshorne, so partway through this huge book that I had spent, you know, multiple years trying to get to that point. And all of a sudden, you get there and you see this beautiful vista of mathematics, these beautiful examples, that use all the tools you’ve got, and there was no easier path there. And it made me think of another one of my favorite hobbies, which is mountain climbing, or mountaineering. And how, you know, oftentimes you have to slog through these very tedious, long, hard, difficult and exhausting things, exhausting work, not always the most enjoyable — sometimes you're tired, sometimes your legs hurt, sometimes you're just kind of looking around and saying, wow, this is kind of a dusty desert, this isn't very pretty. But all of a sudden, if you put in that work, you get to this vista, and you see kind of the beauty of the world around you. And to me you get to this theorem, and you see the beauty of algebraic geometry, and kind of the essence of why you did what you did why you put in that work.

EL: Yeah, so what are some of your favorite vistas or mountains that you've gotten to climb?

JB: Yeah, that's a great question. So I have lived in California with my partner for a while and so a lot of the things I've liked to climb and do are kind of in the Sierra Nevada range or the Cascades, and not all of these ones I've actually fully summited, but I think things like looking off from Mount Shasta in Northern California has a beautiful view where you see the changing from how it's beautiful green forest around the mountain where it's snowy to this dry browner desert as you move off into kind of Northern California. There's some beautiful vistas out near Lake Tahoe and you kind of climb these peaks and get to the top and all of a sudden you can see this absolutely gorgeous lake spread out in this beautiful forest with peaks kind of circling it as a rim.

EL: That does sound amazing. As you can see, but our listeners can I've got my Zoom background is from a hike I did recently where — I must say this hike is basically beautiful the entire time. So there, there wasn't so much slogging, but you know, it was covered in aspens and the evergreens and things and just, you know, you do sometimes come around this curve, and suddenly you can see the Salt Lake valley, below where before it had just been trees, and there is something really special about that.

KK: Yeah, absolutely. I'm an East Coast kid. So Appalachians, which is still of course very beautiful, but different vibes. All right. So we'd like to give our guests a chance to plug anything they're doing where can we find you on on the intertubes?

JB: That’s a great question. I guess I would say I have a professional website. Google my name, you'll find it. Of course I’m also on Twitter for who knows how long.

EL: Limited time offer.

JB: A limited time offer, potentially, also under my name. And you know, otherwise I'm happy to respond to emails or things like that. And I'll just plug as a final thing, you can't reach me this way, but I am the president of Spectra, the association for LGBTQ+ mathematicians. So I'm also very heavily involved and happy to talk, and, you know, look at that sort of work. So if you're interested in LGBTQ+ mathematicians, I'd plug looking up Spectra and the work we've been doing there.

EL: Yeah. And did they just kind of recently, sort of — I got the feeling maybe it was a little more of an amorphous organization, and now it's sort of coalescing into something that has has a little more structure. I've tried to make an algebraic geometry analogy here and it’s just not working. But yeah, this is relatively recent, right? So are you the first president of Spectra?

JB: Yeah, so that's absolutely right. Spectra has kind of a long and amorphous history, coming from kind of a lot of grass roots activism in the ‘90s, through the 2000s. And it's existed in some form, at least with a website for, you know, a number of years. But in the last few years, we've really been trying to grow and formalize and expand our reach and the ability of support we're able to give LGBTQ+ mathematicians. Part of this includes kind of creating a formal board structure, and we did that over a number of years, going to effect this last year, and I was lucky enough to be chosen by the previous board members to be the inaugural president for this year. So I've been lucky to kind of take the reins and guide the organization through its first kind of formal year this year, although building upon all the amazing work a number of extremely dedicated and thoughtful people have done many years previously.

EL: Yeah, and I think it really has been maybe a lifeline, or really as a place that, you know — young LGBTQ mathematicians have maybe sometimes felt isolated where they are, and able to be like, is there anyone else like me? And, like, of course, there are a lot of people like you. And it's been a place that people can find, and I think that's really special.

JB: Yeah, that's exactly the goal. One of the goals we have is trying to make sure people see other visible LGBTQ mathematicians and see people they might be able to aspire to and reach out to or seek advice from or support from. So that's been one of our goals of formalizing and trying to increase our presence.

EL: Well, that's great. Yeah, check out Spectra. And yeah, send Juliette an email, you know, about anything related to that.

KK: Or algebraic geometry, or mountaineering.

JB: Or mountaineering. Yeah.

KK: All right. Well, this has been great fun. Thanks for joining us, Juliette.

JB: Yeah, thank you so much for having me. It's been a pleasure. I've really enjoyed listening to your podcast prior to this. So I really appreciate the opportunity to tell you some hopefully coherent words about my favorite theorem.

EL: Yes, thank you. I just love all the different perspectives we get by talking with so many different people here.

KK: Yeah, that's the best part. All right. Take care.

[outro]

On this episode, we were happy to talk with Juliette Bruce, a mathematician at Brown University, about Petri's theorem. Here are some links you might enjoy as you listen to the episode.
Her website and Twitter profile
The canonical bundle and Petri's theorem on Wikipedia
Robin Hartshorne's (in)famous Algebraic Geometry textbook
Spectra, the association for LGBTQ+ mathematicians

Episode 81 - Christopher Danielson

26 november 2022 | 33 min

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida and I am joined as always by my fabulous other host, co-host? I don't know,

Evelyn Lamb: Co-host. It’s a host but going in the opposite direction.

KK: That’s right. We reverse the arrows. Haha, math joke.

EL: Yes, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. Actually just got back to Salt Lake from a wonderful trip this past week where I got to meet two new additions to my family, ages three months and three weeks. So that was, that was pretty fun, to hold one of the tiniest babies I've ever held. So yeah, very nice little fall trip to take. And now I'm back here and talking about math.

KK: Yeah, well, last Friday night, I drove two hours over to Ponte Vedra, which is sort of near Jacksonville, by myself to a concert. So this is where I am in life. So I went to see Bob Mould, who many people may or may not know, but he was — Yeah, Christopher's shaking his head yeah. He was at Hüsker Dü and then Sugar. He's been doing solo albums forever. And I've been a fan for going on 40 years, which is also weird to say. Had a great time, though. By myself, that's great. This is what one does in his 50s I suppose. Anyway, not as exciting as holding a newborn but, but still pretty good. So anyway, hey, let's talk math. So today, we are pleased to welcome Christopher Danielson to the show. Why don’t you tell us about yourself.

Christopher Danielson: Yeah, I am coming to you from St. Paul, Minnesota.

KK: Nice.

CD: Bob Mould, also a fellow Minnesotan. [Ed. note: Bob Mould is actually from upstate New York.]

KK: He went to McAllister, right. Yeah.

CD: Nice. Right up the street from where I'm standing right now. I work a day job at Desmos Classroom, which is now part of Amplify, designing — working with a number of colleagues to design math curricula. We are currently working on an Algebra I curriculum, about to wrap that up, and moving on to Geometry. And then on the side, I have many projects, some of which will come up in our work today. But I think I understand that you two are familiar with the Talking Math With Your Kids blog that grew into then a large-scale playful annual family math event at the Minnesota State Fair called Math on a Stick.

KK: Cool.

CD: And I also am Executive Director at a small nonprofit that seeks to create playful, informal math experiences for children and families in the same spirit as the work we do at Math on a Stick, but designed for a variety of other sorts of spaces. That nonprofit is called Public Math.

KK: Very cool.

EL: So I'm probably doing that thing where I generalize from a small number of examples. One of my best friends in grad school was from Minnesota, and just loved the State Fair. So I think that Minnesotans just have it this special relationship with the State Fair. And so I did — I am really interested in hearing more about how you do Math on a Stick at the Minnesota State Fair.

CD: Yeah. Should I pick that up right now? Or is there more on the agenda?

EL: Yeah, that would be great!

KK: No, go ahead.

CD: Yeah, so the Minnesota State Fair, it's the second largest state fair in the country behind only of course, Texas.

EL: Where I am from.

CD: Oh, nice. Texas lasts for a month. Ours is 12 days. 12 days of fun ending Labor Day is one of the mottos. The other is the great Minnesota get-together. The location of the fairgrounds is especially convenient for large attendance. The fairgrounds are right, sort of on the border between Minneapolis and St. Paul. And they have been, for probably the past 20 years, have been working on developing some educational and family friendly spaces, out of a perception that it is expensive to go to the fair, which is true, but then once you're in that there isn't much to do besides look at animals and buy a bunch of food.

EL: On a stick.

CD: Yeah, on a stick. So they’ve been working on that. And that led to a lovely literacy space called the alphabet forest that is about 12 years old now. And the first time I sat down there, it was their fifth year and was like the sky, the clouds parted and the angels sang, and I was like, I’ve got to figure out how to build a math version of this. And so together with some organizational support from the Minnesota Council of Teachers of Mathematics and a bunch of expertise from folks that I know through the blog work and through my work in math education, put together a pitch, and after many very boring meetings, it became a thing. So we've got about 15 to 20, different mathematical, playful, creative math activities, everything from a big table full of tiling turtles, to a set of numbered stepping stones that you just see kids jumping up and down happily counting, counting by twos, creating all sorts of fun things to do with. We have a different visiting mathematician or mathematical artist every day, each of the 12 days and they bring whatever sort of hands-on thing they're into. Sometimes that's sort of the standard stuff with, like, Mobius strips and hexaflexagons, and sometimes it is new and new and delightful, creative things that the world has never seen before. So yeah, Math on a Stick, come on out and play with us. 12 days of fun ending Labor Day, always starts on a Thursday, runs through a full week, two weekends and then ends on Monday.

EL: Yeah, that does sound like a neat thing. Sometimes I go to the farmers market here or something like that, and I just think, like, where are there opportunities to kind of create, like you said, these playful, you know, a non-classroom math experience for people?

CD: Yeah, my, one of my Public Math colleagues has a project called Math Anywhere, Molly Daley. She's in Vancouver, Washington, and also does some stuff across the river in Portland, Oregon. And farmers markets are one of the more successful spaces for her. So she'll pay for, for a booth, she has grant money, she'll pay for a booth and just set up a much smaller version of Math on a Stick stuff, as well as some other stuff that she's designed or harvested from other places, but three or four activities, and yeah, delightful times ensue. However, I had a recent experience at the Mall of America, largest shopping complex, also here in the Twin Cities. And it was really interesting, because the way that kids’ families move through the Mall of America is wildly different from how they move through the state fair. So just an invitation to a big STEM/STEAM carnival. And we brought some — one of our favorite things is called a pattern machine or punchy buttons, a nine by nine grid of punchy buttons that you can drop pictures on. And each button is clicky and on a ballpoint pen. So we bought a bunch of those. And then we also had the mega pattern machine, which is just thousands of buttons from all these machines smashed together to make a nice big floor space. But the way that kids come into Math on a Stick is that there’s, like, this long elastic band between parents and children at the fair, not in the super crowded spaces in the fair, but in the less crowded spaces. And so often kids will see those stepping stones that, by the way, start at zero, and then continue on to 23. Yeah, so they'll start on the zero, and they'll lead the way into the space, like we deliberately set up those stepping stones to that the edge of this outdoor space. And by the time kids get to 23, now they're surrounded by eggs that they can put into — little plastic eggs they can put into large egg crates, and tiling turtles and pattern machines and all sorts of fun things to do. And families will sort of follow along behind. At the mall, there’s none of that. There's none of that. Families move in really tight units. There's no, like ,a child leading the family into a space, which is just a really interesting dynamic. And having been out in Portland a couple of weeks ago with Molly when she was at one of the farmers markets, it felt very much more like the fair. A mom or a dad might be much more likely to say, okay, sweetie, you keep playing with these turtles, I'm going to hop over there and buy some apples and I’ll be back in two minutes. They kind of keep their eye on them and everything, but that that elastic band is much longer. Nobody ever says, okay, sweetie, you know, their four year old, I'm going to I'm going to just hop across to, you know, the department store over here, you keep playing with something in the hallway. So Public Math is our project where we're trying to think about how do you design for those kinds of spaces? What would have been a better design than the one we had for something like them our time at the Mall of America?

EL: Yeah. Interesting different kinds of math problems to solve. Different optimization.

KK: That’s right. Yeah. All right. So this podcast does have a name, though. So presumably, you have a favorite theorem. So you want to tell us what it is?

CD: I do! And it is — Yeah, my favorite theorem is, I'll state it simply. And then I guess we get to talk about like, why it’s my favorite and things?

KK: Yeah, sure.

EL: Yeah.

CD: It doesn't have a name. I feel like maybe, maybe it should have — maybe it has a name. Maybe you'll know a name for well,

EL: We’ll brainstorm about it.

CD: But yeah, let me state it simply, which is that the vertices in a polygon are in one to one correspondence with the sides of the polygon. So for example, the three-sided polygon has three vertices. Is there a name for this?

EL: So, yeah, well…

CD: The polygon theorem or something?

EL: I don't know. Yeah, that’s

KK: I mean, a polygon is just a cyclic graph. There must be some graph theory name or something.

EL: I kind of you know, this has a little bit of an interesting linguistic thing, right? Because we call polygons a little bit differently at different sizes, like we call it we talk about triangles not trigons, or trilaterals. When we talk about quadrilaterals, like I think I have heard quadrangle, that must be the tipping point. Then we get to pentagon, so I guess that's not lateral or angle.

KK: That’s just gon. Then it’s gon after five.

CD: But gons are angles. So you are counting —

EL: Okay, is that the Greek word for for angle, and angle is Latin?

CD: So goniometer is the is the thing that you can use to measure your range of motion. I'm gesturing, so that's great on a podcast.

KK: We do it all the time.

CD: Like in your arm or knee? Yeah. So yeah, gon is angle.

EL: Okay. Learn something new.

CD: So it's only the quadrilaterals whose sides you count. Everything else, you count the angles. And, by the way, we also have elided the fact that the vertices and the angles themselves are in one to one correspondence, right? That’s also, maybe a corollary perhaps.

EL: Yeah. Okay. So maybe I'm playing devil's advocate a little bit here. But why is it a theorem that the angles and sides are in one to one correspondence? Why is it not obvious, other than the fact that, like, I've experienced these shapes my entire life and have never experienced one that did not have this property?

CD: Yeah! So I learned that this was a theorem, and its necessity, by working with five-year-olds. So I wrote a book called Which One Doesn't Belong, which was an adaptation, both of the Sesame Street routine, but also playing on some of the routines that I had seen other people playing around with. But for me, the thing that was novel about which one doesn't belong, was that when my children were small, all the shapes books that they had an opportunity to encounter were wildly simplistic. There would be, you know, a triangle page, and then there'd be a square page, and then a rectangle page, and never a square, never a square on the rectangle page. That's confusing for kids. And all of the triangles would be equilateral and oriented on one of their sides, all the hexagons were regular, and again, sitting by their sides, or maybe if they're feeling a little wild, straight up and down balanced on a vertex. But orientation isn't a thing, like, there's all this work that we know is important to come to understand a mathematical idea that just doesn't get doesn't happen in books that get published for young children, even though if you've ever been around four or five or six year old children, they can think about complex relationships, they can think about complex ideas. But somehow we don't understand or value that when we're creating books for kids. So Which One Doesn't Belong was my way of producing, taking ideas that other people had had and condensing them down into what I thought of as a shapes book that was more worthy of children's minds.

EL: I just want to insert that it is a really fun book. I don't remember when or how I obtained a copy. But I have enjoyed going through it myself, and I probably should have asked permission, but I actually used it as an inspiration for one of the pages in this page-a-day calendar I put together a couple of years ago, where I made just one where, you know, it's a bunch of shapes that all have slightly different properties, and you know, you decide which one doesn't belong.

CD: By the way, I’ll give you a little tip before explaining again, why this theorem is important. If you ever try to design a “which one doesn't belong” set, what you want to do is think about whatever your domain is, so say it's shapes, you want to think about four properties of shapes, and then cover up the first one, and design one that has these three, but doesn't have the first one. And then cover up the next one, design one that has those three, but doesn't have this one. And by the time you're done, you'll either realize that your set of four properties is more intertwined than you had originally thought, and now you’ve got to go back and revise, or you'll have a set where you know for sure that there's at least one reason for each not to belong. But then extra, an important key to this is that you have to be open to the possibility that some kid will see a reason for a shape to not belong that wasn't the reason you'd intended. Right?

EL: Yeah.

CD: this isn't a game of “guess which of the four is right.” But it's also not a game of “guess what was in my head when I designed the set.” Instead, we want to offer up something that we know is rich, and then be open to learning from the kids. So I made this book. I was trying to shop it around to get it published, but also needed to, you know, test drive it with children. So I went on what I called my Twin Cities shapes tour. And visited, I think it was three different elementary schools per week for four or five weeks. So I got into just a ton of different situations, worked with kids, kindergarteners, through, like, fourth graders, all in classrooms, like 20 minute bits, and we just had a ball. And frequently, I would hear from kids, like, one kid would say, you know, that shape doesn't belong because it has three sides and the others have four. The opening page of the book is a triangle, and then there are three rhombuses of various types and orientations. So a kid would say that one doesn't belong, because it has the wrong number of sides, right? It has three sides, the others have four. And then somebody else would talk about some other shape. And then another kid would say that one doesn't belong, because it has three corners, and the others have four corners. And in my mind, the first, like, 12 times I heard this from children, I thought to myself, yeah, you're not listening. Some other kid just said that. Didn't say it out loud, kept it to myself. But it was after about the 12th time that I heard it that I said, “Wait a minute. Wait a minute, you heard you heard when this kid over here said said different number of sides?” And they'd be like, “Yeah, and I said different number of angles.” And so it was at that point that I realized that — they’re kindergarteners, right? They haven’t — I know that they haven't seen any good shapes books, right? So they haven't had the opportunity to consider the relationship between the number of sides and the number of angles. And in my adult mind, I had this idea that it was obvious, which is so true of mathematics, like always, right? That if there's something that we ourselves have internalized and experienced for a large number of years, even if it was hard for us to learn at the beginning, we've probably forgotten about that.

KK: Right.

EL: Yeah.

CD: So yes, that's our that's our theorem. And that's why it's important. It's the thing that you actually do have to learn, it isn't obvious when you're first exploring these mathematical objects. I imagine that's true for those who are studying combinatorics. So we were talking about graph theory earlier. Lots of results that feel obvious in retrospect, because you use them all the time, so much that they're sort of internalized, and you don't even think about them anymore. But there is some some point where that thing had to be learned.

KK: So I'm sitting here trying to think of a proof of this theorem. And of course, the dumbest one that just popped in my head is to use the Euler characteristic.

EL: Is that what the five- and six-year-olds do?

KK: I love using sledgehammers to drive nails! Okay, so all right, this is a theorem; it must have a proof. So let's, let's construct one that doesn’t require Euler characteristic.

CD: Yeah, well, I feel like I would start with a line segment that a line segment has two vertices, right? And then every time — so then now I'm going to add another line segment to get what I remember formally being a polygonal curve, right, made up of straight line segments. And when I add another line segment, now I add a segment and a vertex. So I’m always going to have an extra vertex. Until such time that I come back around.

EL: Yeah, and you add a segment and no vertices.

KK: This is exactly the Euler characteristic proof, just in reverse.

EL: Yeah, it's funny, because my mind actually, I think, basically was the dual of what you said, where I swapped out, so instead of that, I was thinking, when you start with an angle, you've got two line segments, and the vertex, and then I was actually kind of thinking, like, the number of angles you have, they each have two segments, but to connect them, you overlap the two. So you divide by two.

KK: Right, so the number of angles is the number of lines.

EL: Yeah, Little, it may be maybe slightly different, but similar sort of idea.

Yeah. Okay. So it's interesting that children see this as two different facts. Children are more literal, right? I mean, in my experience, one of my favorite stories about my son was we were at open house for eighth grade. And he walks in and his soon-to-be math teacher says, “Do you know what eight times seven is?” And he said, “Yes.” Right?

EL: Yeah.

KK: I mean, she was expecting him to say 56. But children will just give you the most literal answer that you can ever imagine. Yeah. So, okay, well, we usually ask if this is a love at first sight sort of theorem. But I don't know. Maybe that's not the right question here. Although maybe it was for you. I don't know.

CD: Well love at first noticing, right?

EL: Yeah.

CD: For me, the noticing that this thing that I had interpreted as being — these two statements that I interpreted as just being equivalent and repetitious of each other, noticing that that was a thing that required learning, and that these kids were absolutely listening to each other. And it gives me an opportunity as a teacher, right? I'm only in there for 20 minutes or so, but it gives me an opportunity to say, “Wait a minute, is that gonna always be true?” The generality is that this one had three sides and three corners? And these all have four and four. Is that always true? Can we imagine a polygon that has some different number of sides and corners?

EL: And what do kids conclude about that? Or do they have, like, ways that they reason about why they have to be the same? Or do they develop pathological shapes that don't have this property?

CD: Yeah, I haven't had time to dig into that in in depth with a group of students. I've had a lot of sort of related experiences. But yeah, I don't know. That would be super fun to to step in. Posed as an offhand question, kids absolutely will both think that it is probably, be willing to believe that it is true, and there will also be kids who will imagine that maybe there is some shape that they just haven't had a chance to meet yet that isn’t. Of course what that investigation with kindergarteners, that's going to get you into a lot of a lot of really interesting kinds of conversations, because they don't have polygon yet as a defined category of mathematical objects. So we're going to have to start to think about whether a circle is a polygon or whether curvy sides count as sides.

EL: Or if you’ve got, like, a square with a handle on it that's just a line segment, what’s that?

KK: Very cool.

CD: But yeah, that kind of, you know, monster creation, from Lakatos’s Proofs and Refutations, that kind of potential counterexample, and then dealing with whether the counterexample is really a counterexample, that kind of stuff goes on at all levels of mathematics, for sure.

EL: All right. I like this. It is not a theorem I have thought about as a theorem ever in my entire life.

KK: Right. Well, I think I see why you love it. Because it actually it's more of a meta-result than the actual theorem. The theorem itself is less important than kind of the questions that it can trigger. And to get kids thinking about things in an interesting way.

CD: But it’s definitely not a Postulate. Like if we're in Euclid, it’s not a postulate, nor an axiom.

KK: No, it isn't. It’s a theorem.

CD: And there are certainly lots of results about triangles in which we know there are three sides, and so there are also three angles, because it was a triangle. Yeah. So if you don't have it, if you get rid of it — like, we can say it's not important, but if you get rid of it, there's a lot of geometry you're not going to be able to do.

KK: Oh, okay. So right. So now instead of non-Euclidean, we might have sort of non-polygonal geometry. So we don't insist that our polygons have the equal numbers of sides and corners.

CD: Yeah, I was just imagining a world in which the theorem is an undecided result, or that we can’t count on. So anything, any place that we assume it, we've got to work around it or prove it again.

EL: Or we can only use theorems about angles.

KK: All right. So the other part of this podcast is we ask our guests to pair their favorite theorem with something. So what pairs well with this?

CD: I have two pairings.

KK: Okay, good. Good.

CD: I don’t know if that counts.

EL: Yes.

CD: Or we need a new word for a pairing.

EL: Yeah. No, that's great.

CD: Yeah. So I'm going to pair it first with a claim and then with an admonition. The admonition is related to what we've already been discussing. But the claim is, it's going to be controversial here, I imagine claim is that a diamond is a shape.

EL: Okay.

KK: A 2-d diamond or a 3-d diamond?

CD: Oh, yeah. So I'm still in plane geometry. Surely there is some corollary for 3-d geometry. But yeah, I got my start in math education teaching seventh and eighth grade. And I used to, when I was a seventh and eighth grade teacher, mid 90s, I was in a camp that is still still very active in which if a child says diamond, I say again, “No, no sweetie, rhombus, you mean rhombus.” Like we call it, we're sophisticated mathematicians, we don't use the word diamond. But again, through working with the kindergarten kids, I came to understand that they don’t — like, diamond and rhombus are absolutely not the same thing to them. So if we treat mathematics as a human construction, right, then the mathematical ideas that a five-year-old has are worth testing and exploring. And one of those ideas that they have is that orientation of the shapes matters, right?

EL: Yeah, I was wondering.

CD: A square standing on its corner is a diamond, a rhombus standing on a vertex is a diamond. But also, if you cut the top off that rhombus, you now have a pentagon. Still a diamond. It's got a vertical line of symmetry, still a diamond.

EL: Right, right.

CD: So not only is there not a correspondence, because rhombus is a thing that doesn't depend on orientation while diamond does, but also that not every diamond has to have four sides in the way that a rhombus does. They don't have to be equal sides. You can you can stretch it. So you've got short sides and long sides.

EL: Yeah, I was wondering if a kite is a diamond.

CD: Yeah, absolutely. Kites are diamonds. And so the thing that I would be very excited about would be a world in which instead of we as math teachers saying, “No, no, sweetie, that's not diamond, you mean rhombus. Diamond isn't the word we use, it doesn't really count.” That it instead be a place where we press on that in all the ways that we press on mathematical ideas and try to get at definitions. Right? So now we're going to make a whole bunch of different examples. Draw me a diamond that looks different from anybody else's diamond. And we create this category. And so I think the best understanding I have of a definition of diamond that would satisfy most kindergarteners, it’s something that has to have a vertical line of symmetry. And it has to be convex. So darts are not diamonds. And somewhere between four and probably, like, eight sides. Triangles are never diamonds. Never, never, never. But four or five.

EL: And it has to have a vertex on the bottom.

CD: Yes, a vertical line symmetry that goes through the vertex at the bottom.

EL: Oh, yeah.

KK: Yep.

CD: Okay, excellent. So that's my claim: a diamond has a shape and therefore worthy of investigation rather than of dismissal.

EL: I’ll buy that.

KK: The admonition?

CD: Sure. The admonition is stop showing children only the special case.

KK: I seem to remember a Twitter like, like you were…

CD: I started yelling at a publisher

KK: You were you were hot about this on Twitter.

CD: Yes. Okay. It’s a really interesting — I think the thing you're remembering was actually almost the reverse, which is something I alluded to earlier, the thing that there's never a square on the rectangle page. So I went to a public library, doing some research on children's books for some work that I'm doing and happened — of course, was in the shape section and happened to see this book about rectangles. Like literally its title is Rectangles. This is a book all about rectangles, it has no other purpose. And I pick it up and just, like, want there to be a witness to this — but of course, there wasn’t — of my predicting, there's not going to be a single square in this in this rectangle book. And I flip through the pages and of course there isn't. So it's just one of these small sort of regional publishers that publishes educational titles for libraries and school libraries and whatnot.

KK: Right.

CD: But I DM them on Twitter to say, hey, maybe we could liven this up a little bit. And they said, Well, no, according to state standards, you know, we're responding to state standards, blah, blah, blah.

EL: Oh no.

CD: I was like, Oh, that's really interesting. I'd love to see the standard that says that you can't say a square is a rectangle. What they came back with was a Texas standard at kindergarten that says at kindergarten, you are supposed to be studying special examples of shapes such as squares being special rectangles. And this publisher was publishing a book for four-year-olds. And so because it was a pre-K title, they couldn't put the kindergarten standard in. It wouldn’t be well-aligned.

KK: Don’t let them get ahead.

EL: Yeah, it would be too advanced to know that a square is a rectangle.

CD: And we have this idea that we can't provide, again, we can't provide complex ideas. We can't give kids interesting things to think about, or conundrums or puzzles. So yeah, admonition isn't quite that, right? My admonition is stop showing them only the special case, but also please, let's show them the special case and help them integrate the special case with the general one. But yeah, all the shapes books with the triangles that are on their bases. And yeah, you know, it's like if we were teaching kids about even numbers and the only even number we showed them was 2, end of story. It seems like maybe we need a little more.

EL: I’m kind of wondering, you know, if, like, guerrilla math person with like square stickers, like going into all the shapes books, putting them in the rectangle pages…

CD: That would be a fabulous public math project.

KK: It really would. That's good. All right. So we like to give our guests a chance to plug themselves and things they're doing. Where can we find you on the line? Where can we purchase your wares? You have excellent wares for sale.

CD: Yeah, thank you. So Talking Math With Your Kids is the blog and also the online store where tiling turtles and pentagons, hexagon puzzles for small children that have widely varying examples of hexagons, are all available there. The Twitter feed is trianglemancsd. Unfortunately, triangleman was already taken by the time I got to Twitter like 12 years ago, and so I had to tack my initials CSD Christopher Scott Danielson.

KK: But not by They Might Be Giants. So who took triangleman?

CD: Yeah, I don't know, some guy who never uses it. I think he lives in Florida. Never tweets.

KK: Sure.

CD: And yeah, by all rights, it should have been turned over to me long ago. But yes, the Twitter handle is in honor of both They Might Be Giants and my love of shapes and geometry. Okay. So that's the Twitter feed. Yeah, and public-math.org for some of the projects, we're up to over there, but you can get to it all through the through the Twitter.

KK: Okay.

EL: Yeah, thanks.

KK: Excellent. Thanks for joining us and for making us think about the fact that it's a theorem. That's, yeah, that's useful.

CD: Truly a pleasure. Thanks for having me on.

[outro]

On this episode, we had the pleasure of talking with Christopher Danielson, who works for Desmos and is involved with several programs to help kids have rich, creative mathematical experiences. Here are a few links you might find useful after you listen.
Danielson's Twitter account
Talking Math With Your Kids
Math on a Stick
Public Math
Math Anywhere
Evelyn's Page-a-Day math calendar, which takes inspiration for August 8's page from Danielson's book Which One Doesn't Belong?

Episode 80 - Kimberly Ayers

21 oktober 2022 | 34 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. My name is Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah, where fall is just gorgeous and everyone who's on this recording, which means no one listening to it, gets to see this cute zoom background I have from this fall hike I did recently with this mountain goat, like, posing for me in the back. It kind of looks like a bodybuilder, honestly, like really beefy. But yeah, super cute mountain goat. So yeah, that really helpful for everyone at home. Here is our other host.

Kevin Knudson: I’m Kevin Knudson, a professor of mathematics at the University of Florida on the internet, they would call him an absolute unit, right?

EL: Definitely. At least they would have five years ago. Who knows these days?

KK: Shows how out of touch I am. That's right. Yeah, here we are. Yeah, it's actually lovely in Gainesville. Like I've got short sleeves on, but it's like 75 and sunny and just everything you want it to be.

EL: Perfect.

KK: And tomorrow, tomorrow's homecoming at the university, which means that it's closed — this is bizarre — for a parade. But as it happens, tomorrow is also my birthday.

EL: Wow!

KK: So I get the day off, and it’s unclear what I'm going to do yet.

EL: Well, just having a day off to lie in bed as long as he want, you know, drink your your coffee at a leisurely pace.

KK: Absolutely.

EL: It’ll be great. Yeah, and we are recording this shortly after Hurricane Ian. And so you're here, so you made it through okay. I actually don't know my Florida geography well enough to remember where Gainesville is.

KK: Gainesville is north central. And weirdly, this cold front sort of pushed just south of town right before. It was about 65 degrees for three or four days, which is freakish. The hurricane, of course, took its very destructive path entering around Fort Myers, went across over Orlando, then to the Atlantic side. We got about a half inch of rain. It was — I mean, we were expecting, like, 10 inches, and then that that weird path happened. Of course, a lot of a lot of our students, you know, their homes have just been devastated. It's a rough time, but you know, the governor and the President are at least putting aside their differences temporarily and making some good progress. Well, we'll see. It's gonna be a long rebuild down there.

EL: Yeah.

KK: And it's a beautiful part of the state, and I feel bad for everyone down there.

EL: Definitely.

KK: But yeah, it's not the first time, you know?

EL: Yeah. Well, yeah, I hope it it continues to progress on the cleanup and everything. And today, shifting gears entirely, we are very excited to have Kimberly Ayers on the show. Welcome, Kimberly, would you like to tell us a little bit about yourself?

Kimberly Ayers: Hi, thank you. Yeah, I'm super excited to be here. And happy early birthday, Kevin.

KK: Thanks.

KA: So I am an assistant professor in the math department at California State University San Marcos, which is about half an hour north of San Diego, so for those of you who are less familiar with California geography, and my research is in dynamical systems and ergodic theory.

KK: Cool.

EL: Nice. And I said “shifting gears” because I know you're also a biking enthusiast like I am.

KA: I am. Yes. I love to get out on my bike. And California weather is — San Diego weather, it’s hard to be outside. So I’m a big bike fan.

EL: Yeah, I — the other day, someone on a local social media thread was posting like, you know, “We shouldn't have good bike infrastructure in Salt Lake because, you know, we're not San Diego, so there's so little time that you can bike here.” And I was like, well, first of all, that's just not true. But you do live that dream of the, like, San Diego biking weather all year.

KA: Yeah, it’s — I can't complain about it.

KK: Sure. Well, it doesn't really snow that much in Salt Lake right. I mean, it hits the mountains. But yeah, I mean, so when I was a postdoc in Chicago, I cycled a lot, but come November I was finished, right?

EL: Yeah, because the roads just never get all the way clear, but here it's dry enough that they do get cleared. And so — you know, I am not an especially hardy person. But, you know, if you’ve got some layers on and the ice is off the road, it's actually doable.

KK: It’s not a problem.

EL: I discovered. I mean, this was a pandemic discovery because I grew up in Texas, and I would just put my bike away in, like, November here, but decided I mental health-wise that I really needed that during especially the height of that covid winter — our first covid winter. Anyway, lovely to have, I guess three people who enjoy biking on this show, but we are not here to talk about biking. We are here to talk about Kimberly's favorite theorem. So yeah, what is that?

KA: So my favorite theorem is a theorem called Sharkovskii's theorem, which is a pretty famous theorem in dynamics. To back up a little bit, when I talk about dynamics, right now I'm talking about discrete dynamical systems, where the idea is if you have a function that has the same domain and codomain, you can think about compositions of that function with itself, right? You can take successive compositions over and over again. And so as a dynamicist, I'm interested in looking at these sequences, like if I start with the point x in my domain, and then I apply f, so I get f(x), and then apply f again, get f(f(x)), and then f(f(f(x))), and so on and so forth, right? This is a sequence. And so we can ask questions about sequences, right? We can ask, like, do they converge? Or maybe if they don't converge, do they have a convergent subsequence? Do they ever repeat themselves, and that repeating themselves is actually what Sharkovskii’s theorem is all about. So Sharkovskii's theorem is about continuous functions on the real line. So it's important to say that there is at the moment, no, like higher dimensional-analog to Sharkovskii’s theorem. This only applies to one-dimensional functions. But Sharkovskii’s theorem says, okay, so we have to take a really weird ordering on the natural numbers. So we're going to start with all of the odd numbers except for 1. So starting at three, we'll take all the odd numbers in a row, so 3, 5, 7, 9, 11, so on and so forth. And then, once you're “done” with all the odd numbers, yes, then you'll consider 2×3, 2×5, 2×7. And again, all of those, right, and then 22×3, 22×5, taking higher powers of to multiplied by 3, 5, 7,.… And then again, once you're “done” with all of those, the only numbers that you haven't included are the powers of 2, so then you take all of the powers of 2 in descending order, until you get back to 1. So it's not a well-ordering, right? You have to kind of wrap your mind around this fact of like, once you're “done” with the odd numbers, then…

EL: Yeah, right.

KA: But it is a total ordering.

EL: Yeah, so you can take any two numbers, you can tell, like, which one is before the other one.

KA: Exactly.

KK: Right.

EL: But you can't say this one is 17th. Well, you can actually say which one is 17th in the series, but, like, a number that isn't an odd number, you can’t say what position it has.

KA: Exactly, exactly. If I were to count, like, the natural numbers in their usual ordering, you know that eventually, you're going to get— like, if you asked me are you eventually going to hit 571? Yes, I will. Right. But if I were to try to do this with the with Sharkovskii ordering, that’s not going to happen, right? So it's kind of weird that there's a minimal and a maximal element in the Sharkovskii ordering. So I haven't told you the punchline yet.

EL: Yeah, we’re just wrapping our head around.

KA: But yeah, exactly, right. So we start by taking this really strange ordering on the natural numbers. And then what Sharkovskii’s theorem says is if you have a period — so I guess I didn't quite define which way which the ordering goes, but let's say that 3 is the maximal element and 1 is the minimal element. So Sharkovskii’s theorem says that if you have a period N orbit for some discrete mapping on the real numbers, then for every M that's less than N, you also have a periodic orbit of that period. So, for instance, if you have a period 2 orbit, you have to have a period 1 orbit, which is what we just call a fixed point, right? If you have period 64, then you also have a period 32, 16, 8, 4, 2, 1, etcetera, and probably most excitingly, is that if you have a period 3 orbit, then you are guaranteed to have periodic orbits of any other period.

KK: Right.

KA: So sometimes people talk about Sharkovskii’s theorem, and what they say is period 3 implies chaos.

EL: Yeah.

KA: Now, I haven't told you what chaos is. And I kind of joke, actually, that chaos in the dynamics community is sort of a bit of a chaotic concept in and of itself. Because there is no really one universally accepted definition of chaos. There's several different types of chaos. There's what we call like Devaney chaos or Li–Yorke chaos. But this definition of chaos, which I believe is Li–Yorke, says that in order to have what we call chaos, you need periodic orbits of all periods. So there's that period 3 gives you that condition, you also need an orbit that is going to be dense in your space. So an orbit that kind of fills up your entire space. And then you need this other thing, which is probably the most famous aspect of chaos theory, which is the sensitive dependence on initial conditions, otherwise sometimes termed the butterfly effect, which basically says that if you have two points, no matter how close together, your initial points are, if you apply f enough times, their sequences, their orbits eventually grow some distance apart from each other, no matter how close together, they start. So there's essentially like no room for error if you have a chaotic system.

KK: Right, right.

KA: Sothat's why once you have period 3, you're guaranteed at least one of the requirements for a chaotic system, right? So my students asked me the other day, they were like, what's your favorite number? And I was like, Oh, that's a really hard question to answer.

KK: Three!

KA: But I think it has to be three because of this, like, you see a period 3 orbit, you kind of automatically get excited, because those are, like, pretty rare. So that is, yeah, I guess I have to say that three is my favorite number

EL: So we get a favorite number for free on this episode.

KA: A favorite number and a favorite theorem!

EL: Okay, so I want — I'm dragging a little bit this morning. And so having a little trouble with, you know, putting these things together. So what I mean, so we have this weird order on the natural numbers. Like sure, I'll let you do that. Can't really stop you. So what kind of — what is our f? What? What kinds of dynamical systems are we talking about here?

KA: Yeah, so let's talk about a couple of famous examples. So oftentimes, I'm talking, I've been talking about functions on the real line, but I guess really dynamicists really like studying functions on compact sets. Because if you have compactness, you're guaranteed things have convergent subsequences. So so we get what we call this limit behavior. So a lot of dynamicists study functions on, like, the closed unit interval. And so some examples are what's called the logistic map, which is a quadratic function. So it's some parameter R times x times 1−x. So it looks like an upside down parabola, right, it intersects the x-axis at 0 and 1. And then in order to make sure that you map back to things in between the 0 and 1 interval, we're going to require that R be between 0 and 4, right?

KK: Right.

KA: If R is bigger than 4, then the top of that parabola bumps up above 1, and, and there are actually cool things that you can study with that as well. But I'll talk about that another day.

KK: Oh, the bifurcation diagrams. That's what you want to talk about. Right?

KA: Yeah, once you vary that, so that's the cool thing, is like as you vary this parameter R, if you start at 0, and then kind of think about what happens is you increase R a little bit, you see these periodic orbits appear in exactly the order that Sharkovskii tells you, they're going to happen. So you start with a fixed point, so that's your period 1 orbit, right? And then I believe it as once R gets above 3, that's when that period 2 orbit shows up, right? And then if you bump up R a little bit more, then you see a period 4 orbit appear, and then a period 8, and we call this a period doubling cascade. And then, of course, as you can imagine, there's only finite room for R, right, we only consider up to 4, but there's infinitely many kinds of periodic orbits that have to appear, so they have to start coming at you faster and faster and faster. And then once R gets above — I want to say it's roughly 3.87, and it's only a number that we've sort of gotten that numerically, unfortunately — then you enter what we call the chaotic regime, which means that period 3 orbit shows up. And then we know we have out there somewhere periodic orbits of all periods. So people might be familiar with this bifurcation diagram, which shows kind of like, where the periodic orbits appear. And it starts off, like on the left hand side, looking pretty simple and smooth. But as you go to the right hand side, it gets more, like, fractally looking. And that's because, once again, those periodic orbits have to show up more and more and more quickly, right, as R increases in value. It’s very cool.

EL: You are just blowing my mind, that there isn't just, like, what I thought was going to happen here was like, we would only get these powers of 2 ones. And then you're going to tell me a different dynamical system later that would have — but like, because it makes sense, it somehow makes sense to be like, at 3, we get this, at, you know, 3 1/2, we get this, at 3 3/4, we get that and, like, the fact that, that like, we some at some point do shift to this period 3 place?

KA: Right!

EL: And we can't even know what the number is? I’m sorry!

KK: And these functions are so simple, right? It's just a quadratic. It's a tent map, basically. Right?

KA: I know! It blows my mind every time I think about it. Yeah, so okay, so you're seeing the beauty in this theorem now. Like, just how cool it is. And I guess I'm gonna, you know, shift gears, pun completely intended there, once again, because not only is the statement of the theorem really cool, but the proof of the theorem is really incredibly beautiful. And so I'm going to do my best to explain it without being able to draw anything. But let's talk specifically about the period 3 implies periodic orbits of every other period, that kind of statement right there. So let's suppose that you have a period 3 orbit, so you've got three points on the real line, call them a, b, and c, such that f(a)=b, f(b)=c, f(c)=a, so they go in this cycle, right? You can label — so that's going to partition, at least between those, into two different intervals, right? So let's say that you have a, b, and c arranged from left to right. And I'm going to, for purposes, it's going to be a little bit weird. But there's a reason I'm going to call I1, the interval between b and c, and I2 the interval between a and b. So what you see happening, then, is that because of continuity, that interval I1, when you when you think of what it maps to under f, it has to cover that entire interval of I1 and I2, right? And also I2 maps to something that at least contains I1. Right? And so you've got this, I'm gesturing a lot with my hands, which I realize nobody can see.

KK: We do this all the time.

EL: Just to slow it down a little.

KA: Yeah.

EL: So the I2 has to cover I1, it’s kind of obvious-ish if you've drawn this like I have.

KA: Right, because you think about where the endpoints go.

EL: So yeah. Can we slow down on this I1 thing?

KA: Yes. So remember, b goes to c and c goes back around to a. v EL: C goes to a. Okay.

KA: Right? So if you think about what I1 does, is it kind of flips upside down, and then maybe kind of stretches and comes back around, right? And so once again, I'm assuming that f is a continuous function, so you're guaranteed fixed points when you map an interval to itself, right if you have a continuous function. And so since I kind of have this structure, what I'm going to do is I'm going to draw a graph, a directed graph, where my two vertices are going to represent I1 and I2. And then the way that you can kind of visualize this is that I1 has a loop that maps, because you think about I1 kind of covers itself. Right? There's an edge that goes from I1 to I2, because we said that I1 covers I2, and then there's also an edge that goes from I2 to I1. Okay?

KK: So you’ve got a directed loop in this graph.

EL: Yeah, a lollipop.

KA: Exactly. And now the really cool thing is you can create closed paths of any length in that graph, right, just by starting at I2, going up to I1, and going around, I1 as many times as you need and then coming back to I2. Right? And that — traversing along that graph is essentially what tells you about what the periodic orbits are doing. There's some analysis here, when you think about intervals mapping to themselves and guaranteeing fixed points. And so we say a periodic orbit sort of follows this loop if it kind of travels between these intervals, and then comes back to itself.

EL: Yeah, so you've got like some other theorem kind of sitting in the background saying, like, some interval along this has to — or sorry, some point in this interval, has to do this precise thing.

KA: Exactly. And you can prove that via, I think it's a result of the intermediate value theorem, because your function is continuous.

KK: Brouwer fixed point theorem, essentially.

KA: Right, exactly. You're essentially applying that and then you're just walking along this graph, and all you need to do is make sure that you start and end at the same point, in order to get that periodic-ness. And so then the punch line is that once you have this graph, now I can create, you know, a path, a closed path, of any length that I want, just from the existence of that period 3 orbit.

KK: Yeah. Okay.

KA: And so I really love that,— you know, I'm not a graph theorist. For a long time, I was calling closed paths loops, and my graph theorist friends got really mad at me, told me I had to stop doing that. So now I'm like, okay, closed paths. But I love that it builds — you look at the structure of the graph, and you kind of just read it off from there. And so the proof of the entire Sharkovskii’s theorem basically constructs these graphs with these intervals, and looks at which intervals map to each other, and then you draw the edges as they are needed. And then you, again, you just read off, like, what length closed paths can you get from this. And it’s so cool.

EL: Yeah, there's something so appealing about this, because, you know, you see this, and you're like, oh, man, I'm going to have to find a fixed point. Okay. Is it going to be, like, 3/4 of b plus? Like five? But no, you just have to lose all of this.

KA: You don't need to find it. You just know it's there yet?

EL: You can almost forget the whole dynamical system!

KA: Exactly.

EL: The whole actually nitty gritty details of what's happening with this dynamical system and just say, like, oh, look, I made my little lollipop.

KA: You just take all that information, and you put it in the graph, and then you can kind of almost forget about the dynamical system and just look at the graph.

KK: Well, until you want to actually find this the orbit, right? I mean…

KA: Yeah, well, sure. And that is a much more challenging question, right? Because these orbits are often what we call unstable, which means that other orbits don't converge to them. And so in order to find them, you kind of have to start, like, exactly on top of them, right? And that's very hard to do.

KK: Right, because because of the chaos business?

KA: Exactly, exactly. There’s zero room for error, literally.

EL: And so was this a love at first sight kind of theorem for you?

KA: Oh, absolutely. Once I learned it in grad school, I was just really taken away with — I started by being really captivated by this ordering on the naturals, because I never thought to, like, rearrange the natural numbers — you know, the usual ordering seemed good enough to me. But like, oh, we can rearrange in this ordering, like, maybe is — sorry, my dog is groaning behind me. I hope that that's coming through. We can take this different ordering, and it still like, makes sense. I can still compare any two natural numbers. But it just completely like violates my intuition. And then I saw the proof and I was like, oh, that's just beautiful. Taking all of this information and reducing it down to this directed graph and then just reading off the graph is just so cool to me.

EL: Yeah, so I've kind of maybe opening Pandora's box here, but, so we just did the period 3 implies the all other periods kind of thing. So I — so is the proof, if you wanted to show that, like, period 7 implies period 9 or something, do you do the same kind of thing?

KA: It’s the same idea. Yeah, you draw again, you, you think about a period seven orbit, you label the intervals in a very specific way, and you have to be a bit careful about the way that you label the intervals to make sure which intervals are going to cover which ones. And then again, you draw the graph, and then you can just read things off the graph.

EL: Okay.

KA: So that's exactly how it works. And so there is kind of a generalized graph that has structure that's a little bit too difficult for me to explain right now. But, you know, we'll say, like, take an arbitrary natural number k, we can generally describe the structure of what that graph is going to look like, and then we can describe what the lengths of the closed paths are going to be.

KK: Right, right.

EL: Yeah. Okay. So cool. So, like I said earlier, well, before we started recording, I've heard this “period three applies chaos,” it’s such a great tagline and stuff. And I always nod, like, yeah, yeah, it totally implies chaos. You know, I have looked at this a little bit, but I'm really happy to get to know a little more about what this theorem of actually means so I don't have to just pretend I actually understand what’s going on anymore.

KA: Yeah, it is a really cool, like you said tagline to this theorem. It's very punchy, and succinct.

EL: Yes.

KK: Yeah. Very cool. So okay, now, I’ve got to know. So the other part of this podcast is, we ask our guests to pair their theorem with something. So what what pairs well with, with Sharkovskii’s theorem.

KA: Okay, so I gave this a lot of thought. And I think I came up with a really great pairing. I don't know if you have all ever watched taffy pulling videos?

KK: Absolutely.

EL: Yeah, I actually wrote an article.

KK: Right. I remember this.

EL: Yeah. Dynamical systems in taffy pulling at one point.

KA: Yeah, so I will say I don't personally particularly enjoy taffy. I think it's sweet and sticky and makes my teeth feel kind of weird. But I could watch videos of taffy being pulled for hours. So the idea, and I you know, I don't make candy. So I don't necessarily know what I'm talking about. But my understanding is, they have this big mass of sugar that they've boiled. And they need to basically aerate it, they need to introduce air into it somehow. And so you can do this either on a machine or by hand. But basically, the taffy has to be stretched and then folded back on itself and stretched and folded back. And they just do this over and over and over again. And I think about like, well, that's exactly what we're talking about doing with the unit interval when we talk about a continuous function on the unit interval. We're kind of deforming the unit interval somehow and then putting it back on top of itself. And then just doing that over and over and over and over again.

KK: Right? Yeah, so.

KA: Go ahead, Kevin.

KK: No, I was just going to say, this reminds me, so Evelyn, I think had this diagram of the taffy pulling machine in your article, right? It is really fascinating. That's how these sort of it's 3 things. Yeah. 3 shows up everywhere, right?

KA: Yeah, and now again, I don't know anything about this, but I wonder if that's intentional. And what's also really cool is sometimes they'll put, if they’re going to dye the Taffy a certain color, they put a little bit of coloring just somewhere on the taffy, and then as the taffy gets pulled, that color works its way through the entire thing. And I like to think of that as an analogue to the dense orbit: you start in a very concentrated little area, but slowly this dye works its way throughout the entire the entire mass of candy.

KK: No, I think that's actually an excellent analogy. That's exactly what's happening. And so I wonder if — the inventor didn't know this theorem, of course. But somehow taffy pullers intuitively knew that 3 would do the trick.

KA: Three is the one that would work.

KK: Two’s not enough to just going to do what we want, but just sort of flip it over itself, but three will really you know, yeah, braid it at least.

EL: Yeah, it's cool. It was so long ago at this point that I wrote that that I can't remember the punchline of my article. I might need to go find find my article and read it again. Be like, oh, that person really wrote so well! But yeah, that's fun, and actually as a kid I loved taffy. But as an adult with, you know, various tooth and jaw problems, it's not the very friendliest candy.

KA: No, it's not. But at least you can watch videos of taffy being pulled without having to actually eat it.

EL: That really is the fun part.

KA: And actually, there’s one more thing that I should tell you about Sharkovskii, that is very cool about it. There's this thing that's called the — maybe this is not quite correct what they call it, but they call it the converse Sharkovskii — which says that, so we have this ordering on the natural numbers. There is, for every tail of that ordering, there is going to be a continuous function that has exactly periods of those exact — periodic orbits of those exact periods.

EL: And, like, none before it.

KA: And none before it, every single one. So it's kind of like this sharp, you know, it goes both ways.

KK: So for every natural number, there is a dynamical system that has that many of that order, but then everything less than that in the Sharkovskii ordering, but nothing before.

KA: Exactly, but nothing that came before. So it goes both ways, in a way, right.

KK: So there's something with an orbit of order 57, but not 55.

KA: Exactly. Exactly.

KK: All right. Interesting.

KA: So I think that's also a very cool result.

KK: Dynamics is hard!

KA: Yeah, it is. And I feel like I'm constantly having to, like, I feel like my spatial sense is never very good. But maybe it's gotten better over the course of me studying dynamics more. I’m, like, constantly having to turn things around in my brain and, like, fold things over. And yeah, I love it a lot.

KK: So we also like to give our guests a chance to plug anything. Where can we find you on the worldwide intertubes?

KA: So I am on Twitter. My handle is @kimdayers. Some other cool projects going on is I've gotten I've been doing a fair amount with LGBTQ people in STEM. And so I did an interview over the summer talking to this organization called LGBT Tech, and you can find that on YouTube, just talking about my experience as a mathematician and my identity as a queer woman. And so that's something that I'm very passionate about. And you know, if anybody ever wants to talk more about that, they're welcome to reach out to me on Twitter.

EL: Nice.

KK: Excellent.

EL: Yeah, we'll include a link to that in the show notes. Check those out. Yeah. Thanks for joining us.

KK: Yeah, this has been great. Really great.

KA: Thank you so much. This was a lot of fun.

KK: Good. Take care.

[outro]

On this episode, we were happy to have Kimberly Ayers of California State University San Marcos on the podcast to talk about Sharkovskii's theorem. Here are some links you might enjoy perusing after you listen to the episode.
Ayers' website and Twitter account
Her interview with LGBT Tech
Tien-Yien Li and James A. Yorke's article Period Three Implies Chaos
Our "flash favorite theorem" episode, where Michelle Manes also professed her love of Sharkovskii's theorem
Evelyn's Smithsonian article about the mathematics of taffy pullers

Episode 79 - Philip Ording

15 september 2022 | 35 min

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to rehydrate myself after taking a long bike ride yesterday in the Utah desert in July, basically.

KK: That’s not cool. Okay, so here's my here's my story. So the last 10 days of June, my wife and I were out in Vancouver visiting our son. And it was lovely. It was, you know, 65 degrees every day, and we took a side trip to Banff. And which, if you've never been, I cannot recommend enough expect ACULA really beautiful. We had a wonderful time. We took the redeye Wednesday night back from Seattle to Orlando. Thursday morning. I had a sore throat for a couple of days.

EL: Uh oh.

KK: Yeah. I took a COVID test and it was negative. Okay, but I still don't feel great. Thursday I didn’t feel good. Friday morning I'm feeling worse. Take another COVID test. Guess what?

EL: It got you?

KK: It got me. I had a good run. It was two and a half years. But anyway, so

EL: This is a podcast from quarantine, although it's exactly the same as all our other podcasts because we’re always on Zoom anyway.

KK: Yeah, so anyway, here I am. So if I sound a little froggy, that’s why. I'm feeling a lot better and so is Ellen, but yeah, it's been a rough few days in the Knudson house. And it's 100 degrees here and miserable.

EL: Right? Yeah. A little less conducive to fun.

KK: Yeah, yeah. But enough of my petty problem problems, which — look, you know, everybody, if you're not vaccinated, get vaccinated, right? I'm of an age where I can be double boosted. And you know, I just, I got a bad cold. That's it. So get your shots, people. There, enough political — anyway. It shouldn't be political, but somehow it is. So today, we are pleased to welcome Philip Ording to the show. Phillip, why don't you introduce yourself?

Philip Ording: Hi, thank you, Kevin. Thanks, Evelyn, for having me on the show. Yeah, I am a mathematician and a writer and I teach at Sarah Lawrence College in Westchester, New York state. It's not as hot here right now. It was over the weekend.

EL: You’ve got your moments up there, I’m sure. It can be very oppressive.

KK: Yep.

PO: And if you hear some some child background noise, that's because COVID got the summer camp up here. My son came back from upstate Catskills camp, because they had to shut it down after a week.

KK: That’s a bummer.

EL: Oh, man. Yeah, that's rough. Well, we have invited you on the show to talk about your favorite theorem. But first, I wanted to sort of digress to a theorem that you have written about quite extensively.

KK: A lot!

EL: That apparently is not your favorite theorem. But I wanted to invite you on here because of this amazing book 99 Variations on a Proof that came out a few years ago, and I read it, you know, last year, or something, and I kept thinking, “Oh, I should invite him on here.” And it's 99 proofs of a theorem — maybe we might not even call it a theorem, a statement.

PO: It’s generous to call it a theorem.

EL: Yeah, that about the roots of a cubic polynomial, one particular cubic polynomial, and you just talk about it, you have 99 different ways to prove that the roots of this polynomial are 1 or 4. Sorry, a little minor spoiler for this book. I think you’ll still be able to enjoy it. So yeah, can you talk about that? Like the how you had the idea to write this book and kind of maybe tell us about some of the styles of proof or styles of presentation that you've included in here?

PO: Yeah, sure. Thanks for bringing that up. And the, the book, yeah, it’s not my favorite theorem. I chose it almost at random. And the book is really about everything around it. So I was interested in whether or not you could fill a book by thinking about the expressive material of mathematics outside of the content, or almost parallel to the content. I had a friend in grad school who said that he — he said this, I think, over drinks, but with gravitas — that he thought that the thing that mathematics had over other subjects was that it has so much content; you know, if you make one statement in mathematics, it's the kind of thing that not only is very condensed, and is probably the result of a long, long track to study, but it's also something you can return to a lot. So I was interested whether or not even a very humble equation and solution, something that anybody who has been exposed to math would recognize as mathematics, would be able to support that kind of an investigation of something — mathematicians don't talk about style that much. I think philosophers maybe are starting to talk about it more recently — and just carry it through the things that I like about math, the things I don't like, the history, and some of the folklore as well. So the titles are kind of the style for each of those chapters. And they range from things like “Psychedelic” to “Medieval” to “A proof that's found in a book.” And everything in between. So there are proofs from school, from graduate school, from college. There are person different languages. There are proofs that are linguistic, I guess, you could say, that draw attention to the particular notations, or the sound, that the proof reads as. Yeah. And it was a lot of fun. It kind of was a project that once it started, it took over and had a life of its own, which was probably what what got me to the very end of it, even though it was a long project.

EL: Yeah, a long time to be thinking about one cubic equation. I was flipping through today, and I did you know, towards the back, you have a mondegreen, which is one of those kind of misheard lyrics sort of things. “Their omelet: eggs, beer, eel” is the the first line of it. So you know, “their omelet” instead of “theorem: Let”

KK: Yeah. Yeah.

EL: And I read through it. This is one, you just have to concentrate so hard to read it and try to figure out the math version of it, but yeah, so you got, yeah, so many different ways to roll over this equation. So yeah, I hope people will check that out. It's a lot of fun.

PO: That was a that particular proof was a lot of fun to work on. I had some students helping me in the summer, and we just turned over the language of one of the simplest proofs in the book from a mathematical point of view. I think it comes from a kind of sleight of hand that could easily be misunderstood if somebody wasn't paying attention. And I remember when I was in college, I had a friend who said that she liked math, and she'd taken some courses but gave up after a calculus course in which she couldn't understand what the professor was saying. And all she remembers is this professor would get very excited and say “Knees the baby, knees the baby.” And she didn't have any idea what that meant, but she knew it was important. And so I tried to think of things that sounded very similar.

KK: Yeah.

PO: And I think it's an experience that everybody has, at some point, you're sitting in a talk and you kind of are reading the person's emotions as much as you're reading, you're listening, to that particular details of the techniques that are used, and there's often things that are lost in that channel. So it was fun to make fun of that phenomenon that I think most people who have studied mathematics at a certain level have experienced.

KK: What’s knees the baby? I can't figure it out.

PO: I still don't know. If anyone figures it out.

EL: Listener submissions.

PO: Multivariable calculus

KK: Okay, well, anyway, that’s, okay. We’ll try to figure it out offline.

EL: We’ll try not try to be thinking about that the whole time we’re recording this.

KK: That’s right. Okay, so you've told us what isn't your favorite theorem. You do have an actual favorite theorem. Why don’t you tell us about it?

PO: I do. And I love this question, because it’s, to me, it's very appealing. It's also very challenging. It's not the first time, actually, somebody asked me for my favorite theorem. The first time it wasn't for a podcast, it was for bathroom. I had a friend, some family friends, that had remodeled their apartment and they thought that this bathroom they had designed, it was like black paint or wallpaper inside. And they thought it would be fun to have their mathematician friend make a theorem or some kind of statement of gravitas in the bathroom. Or maybe they just thought it would go well with the marble sink or something. I'm not sure. But I thought about it for a long time. And I thought, okay, you know, is this going to be like something that is, I think, the most important piece of mathematics? Or is it going to be something really personally meaningful? Or maybe, like, I was in a bathroom at a bar once downtown and some, I think it was a grad student at NYU maybe, had done, like, the de Rham cohomology sequences, and I thought that looked cool. Maybe it should just be graffiti. But yeah, I sort of never got around to it, because I felt like I didn't really attach that much meaning to particular theorems. But anyway, what I came up with is something that's instead of a theorem, it's an idea. So it's called the Erlangen program.

KK: Okay.

PO: And it's credited to Felix Klein, German mathematician, 19th century. And a program — yes, so it's kind of a project or an assessment of the state of mathematics at the time, but also a direction forward. So to say what it is, it's, I reread the Erlangen program, which is a lecture that he prepared, actually. It’s named after the university that he was going to be teaching at, a professor. And actually, there are no theorems in it. The thing that's maybe closest is a statement that says that if you want to learn about geometry, you can find everything that you want to know by studying the motions of geometric objects in that space. So what does that mean, to give some example — Have you heard of this, by the way? I don’t —

EL: I definitely knew the name. I could not have told you. The first thing of what it actually was.

KK: I think it's actually been a remarkably influential idea for the last 150 years, right? I mean, I think it's driven a lot of of what happened in the 20th century.

PO: I think so. My background is in geometry and topology. So I might, you know, I might be biased.

KK: Yeah, me too.

PO: Yeah, so, I mean, to give an example, if you wanted to understand, say, points in the plane, the idea is that you can understand points just as well as anything that you might do with, say, intersections of lines, or coordinates, or quadrants, or distance, by just studying, say, rotations of the plane that fix that point. Or collections of rotations that fix a set of points. Or if you wanted to study line geometry in the plane, you could study, well, I don't know if this qualifies as a motion, but the transformation takes every point on one side of the line to the point reflected across the line. So just studying reflections in the plane that fix axes, you can really express everything you'd want to know about lines in the plane. And just to give some sense of, you know, why is this interesting, and not just a complication, so if you wanted to have — say you had two reflections, and you compose them, so you reflect across your first line, and then you reflect across the second line — and that's two operations, you can you can combine them, you're going to get something back — the result is going to be a rotation about — if there's a point where those lines meet, when lines typically do — you're going to get a rotation about that point. And when I first sort of started to get this idea, and use it, it's sort of a yoga, you get used to it after a while, of going back and forth between the world of geometric objects and the world of the structure of motions, or the group of motions. I loved it, and it was very useful, and it seemed like it joined together areas of my brain that were were divided before. So I think that's why it’s my favorite, but I could say more than that too.

EL: Yeah, so where did you first encounter it?

PO: I think it was in my senior year in undergraduate, I was given a project by my senior thesis advisor, wonderful professor and Troels Jørgensen, and he cut his teeth studying hyperbolic geometry. So one of the things that I think is really amazing — and this was maybe Klein's motivation for introducing the Erlangen program — is that if you have many different geometries, so if you've been introduced to the idea that there isn't an absolute singular geometry out there, what we call now Euclidean geometry, that used to be just geometry, and now there are non Euclidean geometries or even wilder things, like topology that we don't call a geometry, exactly, then you might want to know, how are you going to do anything in those weird spaces? And if you have the Erlangen program, it's telling you, as long as you can understand the structure of the transformations, which we think of as generalization, or restriction of congruence. So congruence is the word we usually use for those motions of the plane that that fix them, that preserve them. Okay, so what I had to do is understand something about the hyperbolic, the non-Euclidean analog of a pyramid, tetrahedron is the term. And so tetrahedra are kind of, like, the dumbest of the platonic solids, I mean, maybe it's got four sides, four corners, it's a good shape for a die if you want to have just four options, because it's so symmetric. But when you go into the this world of negatively curved space, you can study tetrahedra that are formed by points that are at infinity, meaning that you don't see, actually, a finite object in front of you, you just see these sets of lines that are going off into space. But it turns out, they bound a finite volume, which is very, very bizarre. And if you want to understand anything about them, you're kind of left scratching your head, if you're just going to be limited to the tools of Euclidean geometry, measuring things like area and volume in the traditional way. It turns out that if you take that idea of, okay, I'm not going to think about the tetrahedron as made up of lines, I'm going to think of it as constituted by rotations in that space, it turns out, you can write down those rotations, that whole set, quite easily using once you've gotten used to using some matrix algebra, so kind of higher dimensional generalization of the regular algebra on the real numbers. And that — so combining those kinds of representatives of lines, you can just go to town computing things, and you can compute intersections, just like I said, with these compositions of reflections in planes, instead of say, instead of Euclidean planes, now hyperbolic planes. So that's a long answer to where I first encountered it. And yeah, I wouldn't have known really how to approach the problem I was assigned if I hadn't had those tools.

KK: Sure. Yeah. I mean, trying to think about the actual geometry of 3d hyperbolic space is sort of weird, right? I mean, like you say, you can't see it. You can, but you can't, and you might draw it, but you have to remember the metric is different and the distances don't look — things that look finite aren’t. And, yeah, it's a very bizarre feeling to try to move into that space.

EL: I love those representations of hyperbolic space. I mean, they're stunning. And they produce some of the most interesting kinds of ornaments out there. But it is hard to know where to start when you're just looking at these dazzling representations or models. And I guess, the other thing that I was made to understand was that this Erlangen program is a little bit like a Rosetta Stone, because it's not only telling you how you can work within any given geometry, by studying its associated group of transformations, but if you know that a geometry has among its transformations a subset or a subgroup that has this kind of coherence, then that becomes sort of a sub-geometry. And you can relate them. And I think this was going back to Klein, when he was up to — they had all these great methods in projective geometry, one of the kind of early alternatives to Euclid, and they were able to use those to study and relate geometries one to another in this kind of zoo that exploded in the 19th century.

EL: Yeah, and that's, you know, thinking about, I guess, when most of us go to grad school in math, you know, one of the powerful things that we do is see this relationship between the algebra, you know, group actions, and geometric objects in some way. And so, this is building from that connection, I guess, from the Erlangen program. Is that somewhat right?

PO: Exactly. Yeah. I mean, there are other connections between geometry and algebra, right? I mean, we learn with Descartes, and once we start plugging in coordinates for points, and then writing out lowly cubic equations for expressing pictures of curves. So, you know, I think that when — you know, the process of learning math is usually, even though math seems like a very strict discipline, it has its own subfields. And those subfields don't always work together in obvious ways. So we tend to teach them in by these isolated textbooks, you know, algebra, or group theory, and analysis and geometry, and so forth, or calculus. And I think Klein was very much a synthesizer. And I like this idea. As was William Thurston, who was the person setting out programs for geometry when I was a grad student, and I think we're still untying some of the things that that he that he set out.

EL: Definitely.

KK: Yep. All right. So another thing we do on this podcast is we invite our guests to pair their theorem, or in this case program, with something. What do you think pairs well with the Erlangen program?

PO: Oh, yeah. Okay. So, yeah, this is something I thought about. And along the lines of the question of your favorite theorem, I went to a kind of personal, like, trying to think about, taking this question very sincerely. Because I like the idea that there might be a connection between our personal taste and the things that we do. I mean, it's a high bar for mathematics, I guess, because we're working with abstract things. But there's a piece of sculpture that I would pair with this theorem that's in the Museum of Modern Art. It's by a sculptor, it’s a postmodern piece by Richard Serra. He’s an American sculptor. From 1967, I think. And it doesn’t — it sits on the floor, it doesn't have a pedestal. And it's not much to — you might you might step on it by accident, if you didn't, if there wasn't a cord around it or something, but it's a rectangular piece of rubber, like black vulcanized rubber. And it looks a little bit like, it has a graceful form, it rises in the middle and then descends to the floor. It looks like maybe like a cowl over a monk at vespers or something, I don't know. Or like the hills on the screen, the green screen behind Evelyn of Utah. And the name of the piece kind of says it all. It's called “To Lift.” And it's part of a series that that he made. I think he was inspired first by dancers that he was seeing, choreographers at the time, in downtown Manhattan. But the idea, he made this this kind of Erlangen program for himself that was called the verb list. And it starts out, like, to crease, to fold, to roll, to twist, to torque. And he made pieces for many of these that kind of instantiate this verb by applying it very simply to material. So it's not — a lot of them are I think were rubber, but others are lead or steel. Later on, he got into more. And yeah, so the thing about the Erlangen program, besides its connective properties between different disciplines, I really like the way that it takes things that I've thought of as more solid, geometric objects, concrete things, and then trades them with verbs, with actions or motions or transformations. So it makes geometry much more dynamic, anyway in my way previously of thinking of it as this static world that you kind of enter and measure things. Instead, it's this world where you have all of these permutations of the space that you're looking at. And you kind of play. I think of it as more playful. But that's the sense I got from, I had a chance to work as a grad student in Richard Serra’s studio. And it's very serious business. I don't want to pretend like it's kindergarten, but sometimes it has that feeling of like, this is a proposition, can we make a form that embodies this movement, or this daily kind of task or transformation? And he made videos around that same time, “Hand Catching Lead,” and they're very simple, but they just act on you in a way because you start imagining your own participation with the material world around you in this way. So that would be my pairing.

EL: Yeah, well, your description I haven't, you know, I'll look it up later, to see if I can find a picture of the sculpture. But your description, you know, it sounds like you had a sheet of rubber on the ground and just lifted it up. And it does sound like the simplest thing. I'm sure it wasn't the simplest thing to actually make something that that gives you that feeling. But yeah, I guess that there's that — when you see it, you immediately understand what the aim was. And it's funny, when you're listing that these other verbs that that he did, so many of them also have mathematical things that you can almost imagine a textbook that's telling you, like, Okay, this is what a Dehn twist is, and shows you a simple example of that. This is what something with torsion is or this, I don't remember exactly, all the words, but even “lift” has a mathematical meaning as well. And so these choreographic and artistic things, also connecting to the these mathematical ideas we have.

PO: I love those those suggestive verbs and those little diagrams in kind of combinatorial, or cut and paste topology. Yeah, definitely.

KK: So you mentioned that you worked in the sculptor’s studio as a graduate student. How did that come about?

PO: Oh, yeah, that's right. So it was kind of one of those “only in New York” moments. I think his studio manager reached out to the tutoring email at Columbia University where I was a graduate student. And I think they were kind of stuck with the communication between the studio and the engineers that make the large-scale sculpture that that Serra's known for most widely today. So there was some kind of communication breakdown there. And they thought, you know, we're not understanding what the engineers are telling us about what is not possible, and what is possible. And it had gone, it had left the converse the bounds of what was engineering-ly, possible, like, it was actually what was formally possible. So at that point, you know, this was in the early 2000s, he was already working at a very sophisticated level, in the sense that, to make some of the large-scale forms, they were using cutting-edge architectural design and engineering tools. And once they had a design, they could send it out, and people would sort of develop it to a point where it would be stable, rigorous, and so forth, pass the test. So yeah, when I saw that, I jumped on it because I've kind of always been interested in places where mathematics might speak to the arts. And had at that point already, I think I was tutoring a professor in the architecture department. And his name is Peter Macapia. And he was, he became a very good friend, but he also kind of, once he knew I was going to go try to meet Serra, he gave me a bit of a crash course on why Serra might be interested to talk to a geometer or mathematician. And so I don't know if that's the reason that I was asked and invited to work in the studio, because I had done a little homework? It might have just been that — sometimes I would joke that I was a math therapist, a geometric therapist. I would listen to the things that they were trying to do and ask them why they talked about them the way they did. And often I think they came up with their own solutions. But certainly it was a formative experience for me. And part of the reason I wrote the variations was to see if there was a way to take seriously reversing the direction from instead of applying math to art, to see if I could borrow some of the ideas of art-making to do math.

EL: Yeah, that is such a great opportunity you had as a grad student, even, to get to do that kind of thing. And yeah, it’s, I think both of us, a lot of people love that murky boundary between math and art and the ways that, you know, we can apply the very — I guess, I think of math often as a theoretical art, we're doing a similar kind of thing. I mean, I know this isn't original, but it's the similar kinds of thing. We've got aesthetics, we've got rules of our discipline, and like, we apply it to abstract objects, and using that to apply it to concrete objects after that is very cool.

KK: Yeah.

EL: Thanks. I'm looking forward to looking up that sculpture. Yeah, after we get off the call.

KK: All right. So we've talked about your book, we always like to give our guests a chance to plug anything else, or where we might find you on the worldwide web or anything like that.

PO: Oh, sure. Yeah. The book came out in paperback last fall, so you can find it even more affordably priced, I would say, from Princeton University Press. I want to plug my friend's book. I don't know if you've had Jessica Wynne on.

KK: No.

PO: She’s a photographer that made a beautiful book called Do Not Erase.

KK: I have this book.

PO: Yeah. Okay. Yeah. So she was going to ask me to make a board and for her to photograph and I wrote — I used the maximum word count, I think — to write a little bit about my senior thesis advisor Troels. So if you want to page through that, but the book is amazing, and there are many celebrated mathematicians boards in there. So that's fun. I think your listeners would enjoy that if they haven't seen it already.

EL: Yeah. We'll put a link to that in the show notes. Yeah, thank you for joining us. I really enjoyed getting to talk to you and think more about that connection between math and art.

KK: Yep.

PO: Thank you for having me. It's been really fun to talk to you.

KK: Yeah. Thanks, Philip, it’s been great.

[outro]

In this episode, we talked with Philip Ording, a mathematician at Sarah Lawrence College, about the Erlangen program. Attached are some related resources you might enjoy.
Ording's website
The website for his book, 99 Variations on a Proof
John Baez's links related to the Erlangen program, including Klein's original paper on the topic
Royce Nelson's page about 3-dimensional hyperbolic geometry
Jessica Wynne's book Do Not Erase about mathematicians and their chalkboards

Episode 78 - Daina Taimina

11 augusti 2022 | 28 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I like how we're always besmirching other math podcasts, which as far as I know, also don't have quizzes at the end. I am your host Evelyn lamb. I am a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. Okay, full confession. I don't listen to other podcasts, so I don't know if they have quizzes at the end or not.

EL: Shame. They probably don't. I mean, how would you even administer that?

KK: That’s right. That's right. Yeah. Yeah. We don't need to find this out.

EL: Yeah. Well, we are, we should, we should just get right.

KK: Let’s do it. Yeah.

EL: We are very happy today to invite Daina Taimina onto the show. So Daina, please introduce yourself, tell us a little bit about yourself, and we'll get started.

Daina Taimina: Hello. Thank you for inviting me. And, actually, I didn't prepare much to tell about myself. Because usually, I tell my students, you know, just search for me on the internet. That knows more about me than myself.

EL: Maybe even some of it’s true.

DT: Maybe, yes. Sometimes it's true. Yeah. So well, okay. Well, I was teaching about 20 years, I was teaching mathematics at the University of Latvia. And at about the same time, I was teaching in Cornell, where I, now I have stopped teaching. And I officially count as a retired, so it means I have free time to do whatever I want.

KK: Nice.

DT: Yes. And then sometimes, well, it has been now for 25 years, I have crocheted hyperbolic planes. And I guess that's what people know most. Because sometimes I am introduced on people. “Oh, yeah.” And then this person who is introducing me says, “You know, she's the one with hyperbolic planes.” “Oh, yeah! Yes, yes, I know that!” Okay, I guess that's my other name.

EL: Yeah. And our in your multitalented our listeners can't see it, but when we were saying hi we saw that you have some of your very own paintings in the background on Zoom, where I'm seeing right now, and they look very lovely. So you do art in addition to crochet and math?

DT: Actually, that was before, and I and the reason why I was doing I was doing art, is I signed up for a watercolor lessons because I knew that I'm very bad at art. Because when I was in school, I was told I can do anything but that. And at that time in Cornell, I was teaching students who were very afraid of maths, and most most of them were actually architecture art students or music students, and I really wanted to experience how it is to take some subject where you are told, and you believe all your life, that you are bad and you can’t do it. So I did it. And so yes, that was interesting experience.

EL: Well, and it looks like with some practice, you gained skills. Amazing how that works.

KK: I did that to actually about a year ago. I cannot draw. I'm terrible. So maybe we have the same issues here.

DT: Yes, yes. Yes, exactly.

KK: And I took it I took like a just a two hour drawing workshop online where we draw birds, and I actually drew something that looked like a bird at the end. So you know, it can be done.

DT: Yes, because this is what you what you learn, is that — and then I was explaining to my students, too —I brought in one of my paintings, and I said, it's actually, what I realize is that it was things which I knew. I knew, well, perspective. I knew how to do composition from photography, you know, just like doing some photography. And all I needed was, you know, I did need some technical skills, and that is the same in math. You do need to learn some technical skills, and then you can then you can get on, so it's not that different.

EL: Yeah, that's a great, a great lesson to learn and to help share with your students like, “Hey, we're all learning various things. I have this background, you have this background, but we can all improve in various areas of our lives.”

KK: Yeah.

EL: Well, the name of this podcast is My Favorite Theorem. So, what is your favorite theorem?

DT: Well, as I told you, my favorite theorem is Desargues’ theorem. Yes, and then, well, actually it started with some more ancient theorem, which was Pappus’s theorem. And it was somewhere, I think I was in middle schoo, and I was reading something in a math history, and I read that there is this ancient theorem, where if you are having two lines, and then you choose three points on each of them and those lines are non-parallel. Well, if they parallel than they are, that’s a very simple case. But if you have like two lines at some kind of angle, and then you choose, and then you then you connect in pairs points from these lines, and you always have three points which are on the same line. And I know like I was just like, that's — okay, I'm so old that at that time, there was no Geometry Sketchpad or any of these programs, there was no computer. So I just kept drawing these lines and finding those points, then it was just amazing. And then of course, I was like, “Oh, what else is there?” And that's when I discovered this, this Desargues’ theorem, which said, okay, if two triangles are situated so that three lines joining their corresponding vertices all meet at a single point, then the points of intersection of the two triangles’ corresponding sides, if those intersection points exist, all lie on one line. I couldn't — I read it, and I couldn't believe that. So again, I took a pencil and took a straightedge and I started to draw, like, various ways, and it was really finding these points and having them and, and then later learning that the converse of this Desargues’ theorem is also true, and then that’s a converse, theorem, that's also a dual theorem. So it was just so fascinating. So that was something, like, different from the geometry, you know, like, exactly the geometry we were having going to school, and so that's kind of led to perspective. And yes, I was just like, really it was fascinating.

EL: Yeah, this is one that has come up a few times for me in things I've read, or people I've talked to in the past few years. But yeah, I loved geometry for a long time, and this is not a theorem that I was exposed to, in most of my geometry education.

KK: Yeah.

DT: And it's very interesting that you can you can prove this theorem using another ancient theorem, Menelaus’s theorem — okay, I'm not going to talk about that — but that sounds very algebraic, because that uses uses proportions, and it's totally in Euclidean geometry. But I like the way how Desargues himself saw, and he actually was thinking about it in three dimensions, and then it's simple. When you are cutting, like, a triangular pyramid with two planes, and then it's just totally obvious.

EL: Okay, I'll have to sit down and try to visualize that a little better.

KK: Right, isn't the simplest proof, don't you use three, you have to go into 3d and then it sort of, like you say, sort of becomes obvious?

DT: Exactly, yes, yes. Yes. That's one of those. Yeah, that's one of those cases, and that was so great! You know, you just jump out, and then it's obvious. And then it's also, the other thing is, if you are having — so you know, like you can imagine that you are having a book, or though now you're having a point, and then you are projecting a triangle, and then all you do is, you imagine that those lines, that it stretches, and then all you do is you open it up in one plane. And there's the theorem.

KK: There’s the proof. Yep, I wish our listeners could have just seen that.

DT: I don't know how to describe.

KK: So this actually came up for you in school in Latvia? Like, your instructor actually taught?

DT: No, I believe I was reading something from Martin Gardner, or something outside, but I did have a wonderful geometry teacher.

EL: So you were interested in math very early in your education?

DT: It was just one of my easiest subjects. I was interested more in literature and languages. That's what you are saying, you know, like an art. Math, it’s just something that comes by nothing. It’s just simple, just seeing things.

KK: I mean, well, so Evelyn, maybe you had this experience. I mean, I became a mathematician because I was always good at math, right? It was the thing that I could easily do. And so it's sort of interesting that it was sort of the easy thing you could do, but you liked something else more?

DT: Yes, that’s true. Yeah.

EL: Yeah. I I had sort of a similar experience to you Daina, I think, where I was, you know,“good at math” — good at arithmetic, basically — in elementary school. And I liked the proofs in geometry, although I didn't understand that those were “real math” also. I thought it was just a diversion.

KK: The two-column proofs?

EL: Yeah, I liked the logic part of it. You're working it through, but I thought arithmetic was real math. And so yeah, I wasn't as interested in that. I was more interested in — I really liked science, but I did a lot of music also and stuff. But eventually, it wasn't until college, that I really kind of fell in love with it, and decided to devote my life to it in at least some form.

KK: Including podcasts at this point. So yeah.

DT: Well, you've been successful.

KK: Sure. So, have you used this at all? I mean, Is this a theorem that you use in your own in your own mathematical work? Or is it just something that you just love?

DT: No, I this is something which I love. Yes. Because, ya know, it’s — well, using it in teaching, you know, and sometimes I have used it talking in schools, you know, you go meet students and show them something, like, here are some fun and some beautiful things. But no, also I was teaching history of mathematics for many years, too. And I loved that Desargues himself, he never published his theorem. It was it was published by his student, Abraham Bosse. And I think he mentions exactly that Desargues had this three-dimensional proof. But also there was an interesting thing: When Desargues — he belonged to Mersenne’s circle circle, and that is the same circle where there were also Descartes and Pascal. And there was all this mathematical writing and just that exchanging of ideas, so that's fun. I was trying to find out, I know there was a some, in discrete mathematics, there is this 10-line configuration, and then that can be used to solve some problem. And I remember there was some sports problem, but I didn't remember like, you know, just precisely what. It’s interesting.

And then the other thing which I like, about this, not only this perspective, but that you can exchange points with lines and lines with points.

EL: Yeah.

DT: Yeah, so in some ways, maybe this is why I was getting more interested in mathematics, like what's beyond what we learned in school. This was this was like, I guess a very first example that I found that is something more about math, you know, more fun than you'll learn in school — which I wouldn't say that I was bored, but still, that there was something more.

EL: Yeah, well, and you mentioned exchanging points for lines, which, I always feel like I've sort of pulled one over on someone if you could do that kind of duality thing. And you know, move intersections of lines to a different line, and then the points intersect at where the lines were, something like that, and so yeah, that's always very satisfying, I think. So part of this podcast is that we ask our guests to pair their theorem with something the way you might pair food and wine or food and your favorite jazz CD or something like that. I’m not appealing to the youths if I say CDs. I don't know if Gen Z knows what that is.

DT: Yeah, it's like floppy disks.

KK: You have to say vinyl now.

EL: Or Spotify. But anyway, what have you chosen to pair with this theorem?

DT: With travel.

KK: Okay, yeah.

DT: So for me like it is exactly because you can change these points and lines. And then if we go back to that, what I was talking about ,projecting the triangle through that one point. And that's what I was imagining if I was that point, and then I'm looking on someplace on this first triangle. but when I travel, it really expands what I'm seeing when I get to that second triangle and see it in reality, and then I can go back. And that would be like, exchanging, so that would be this duality. From that place, now I can see myself differently.

EL: Oh, I love that.

KK: This is a very thoughtful pairing.

EL: Yeah. And where are some of your favorite places that you've traveled?

DT: Well, there are places like, I always like to travel to Sicily. So this is a very, very significant place for me, because that's where I met my husband, and then we had returned back there, and it's never too much to go back there. So I liked that, well, we managed to travel, and I liked my travel to South Africa. And actually, what I like in my travels is to meet people. And I think that's what those those big triangles when you project is, you meet the people and you talk with people, and I don't travel like without some purpose. And mostly it can be like really meeting people and doing some workshop or talk, and then of course learning about the place where I am.

EL: Yeah. And I always love how you learn both that the way you're used to doing things isn't the only way to do things, and that there are similarities across cultures, that we're all kind of the same in some way. So they're kind of those two contrasting ideas, but maybe they're dualistic in some way as well.

DT: Yeah, it’s like one of the things, since I was in Europe, and I have been knitting a lot. And I come here, and I see that there is another way of knitting, and I'm just staring and it’s just so totally different. But at the end, well, we get we get to same thing.

EL: Yeah, well, and since you bring that up, I did want to mention, as you said, you might be best known in the math community for your hyperbolic crocheting. And I know that Desargues’s theorem is from Euclidean geometry, not hyperbolic. But I would love to talk a little bit about kind of where hyperbolic crochet came from and how you got the idea to do that. Everyone loves it.

DT: No, I'm glad. It's just because I had to teach hyperbolic geometry, and, well, it happened. It's like this. So it’s summer, so in a month, it's 25 years since I crocheted my very first plane. And I'm really surprised people are still interested in that. And then it's kind of now a usual thing. It’s also like with Desargues’s theorem, you go into 3-d and then the hyperbolic plane, you can see only in 3d, because if you if you project then those are maps, like on in 2-d, but in 3-d and then particularly, you can just touch it. And you can do a tactile exploration. My main purpose was exactly for my students, so that they could touch and explore it. Because I saw a paper model, which was done, but one thing is to glue a paper model, which I did, but then once you fold it, for the next class, you have to do it again. And it was like, okay, no, I need something more durable. Yeah, so it was really for teaching. And then it's interesting that this hyperbolic plane started to teach me. When I got suddenly, unexpectedly for me, invited for the first art show, you know, now I had to learn, okay, what does that mean? So finding colors and ways to express, so and it has been going on and on.

KK: So, my wife actually crocheted me one once. So she, she can crochet really well, and it is remarkable to hold that thing in your hand. You can sort of begin to really understand how distances work in this weird, floppy, hyperbolic plane. It's really beautiful.

DT: Yeah, because it's another way to get our knowledge, because we need to feel it. It's not like when you are reading — well, okay, if you are an experienced cook, you can read a cookbook and taste a recipe, you know, like field tasting. But actually the best is that you try to cook it, and then you taste and then, you know, like, that's when you really learned about it.

EL: Yeah, and I think in my experience making hyperbolic crocheted planes, one of the the lessons, the math lessons I learned the most from it is just exponential growth is even more than you think. Because I think the first one I made, I started with something like 10 stitches, and I did it, maybe it was a five to four increase ratio. And, you know, five or six rows down, I was like, am I ever going to finish this row? Even as, you know, five to four to me, it doesn't seem like a big number. Like it's just barely above one. But if you, you know, multiply it by itself a few times, it takes a while.

DT: My hope is when the pandemic started, and then people were told that this virus is spreading exponentially, I hope that at least those who had crocheted hyperbolic plane instantly knew what it is what it means.

EL: Yeah.

KK: And then, of course, this also spawned this whole, like hyperbolic crochet reefs that they would do. So our librarians here actually organized an exhibition, they put it in the this display window in the front of the library. They got people to crochet coral, but it's still the same basic thing that you you came up with.

DT: well, it just it came like that, it all came up. So yes, it's actually that came up when — that was from my very first lecture to the general audience. And I was just thinking, Okay, well, one thing is to show the mathematical object, but then I need something, you know, in real life, too. And I remember that before that, first off, that's when I really was finding, like, lettuce leaves. And it was finding like, some curly things, and that gave an idea to see, you know, like, okay, here we go. It’s in a nature and then that just span us by not just the spin off. Idea. Yeah.

EL: Yeah. And I have to put in an advertisement for your book. Is it Crocheting Adventures in the hyperbolic plane? Is that the title?

DT: With.

EL: With hyperbolic planes.

DT: Crocheting Adventures with Hyperbolic Planes. Yes. And then there is another one, Experiencing Geometry, 4thedition, is open source. And there are a lot of hyperbolic planes in that one, too. It's on Project Euclid. And yeah, that’s open source. That was my husband's wish, that it would be open source. So that's — when I finished this fourth edition, that’s it.

EL: Yeah, and I just can't recommend — I haven't looked, I think I've looked at Experiencing Geometry, but not spent as much time with it as the crocheting one because around the time I, you know, a friend gave me a little crochet kit (not because of math at all, just because like, Oh, you like crafty things, you might like this), I happened to see your book. So it’s, like, this confluence, and your book has more than just making a plane. There are a lot of other interesting math ideas that you put in there. So I just can't can't recommend it enough. And it won an award for weirdest title, didn't it?

DT: Yeah, yes. Well, of course, that that's when — it was the strangest book title, the Diagram Prize — that went around the world. Of course, two years later, when I got an Euler Prize, which is much more serious, from the Mathematical Association of America, that press wasn't interested at all.

EL: Yeah, you know, the strangest title, I guess is a little little more headline-grabbing or something like that. But yeah, yeah. Anyway, I know we sort of went on a diversion from triangles, but I'm glad we got to talk about the hyperbolic crochet a bit because really, I think, so many people have had positive experiences with math and with experiencing geometry in a different way than just looking at a flat on a piece of paper thanks to you.

DT: Thank you.

EL: So, we've plugged your book, but we do like to give our guests a chance to share anything else that they'd like to share, other resources or websites or anything like that for people to, to look at?

DT: Well, when the book came out, I started to write a blog which is called hyperbolic-crochet.blogspot.com. But I'm not very good at keeping it up, so I’m not sure how many people are reading.

KK: Is anybody?

DT: But there is still lots of material and trying to answer a set of questions. And then of course, as I said, on the Project Euclid forum, it's on Project Euclid, look for Experiencing Geometry. So that is my newest book, which I’ve finished. And yeah, so as I said, a lot of interesting geometric things, too. If you are interested in geometry, that would be a good thing to look up, particularly for teachers, because I added one of the appendices, suggestions from various geometry projects, which you can do in class. Because those were questions people were sending me, like, okay, so what can we do in class and what can be some fun things we can do? So yes, I was trying to help. I hope it's helpful.

EL: Yeah. Wonderful.

KK: All right.

EL: Well, thank you so much for joining us. I was really glad to get to talk with you.

DT: Okay, thank you.

[outro]

On this episode of My Favorite Theorem, we were pleased to talk with Daina Taimina, recently retired from Cornell University, about Desargues's theorem. Here are some links you might find interesting after you listen.

Her website, blog, and Twitter account
Desargues's theorem on Wikipedia
Our episode with Annalisa Crannell, who also loves Desargues's theorem
Taimina's book Crocheting Adventures with Hyperbolic Planes, which won the Diagram Prize for oddest book title and the Euler Prize from the Mathematical Association of America
Experiencing Geometry by Taimina and David Henderson on Project Euclid

Episode 77 - Tien Chih

13 juli 2022 | 26 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Evelyn lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, currently enjoying very beautiful spring mountains, which my guest and my cohost can see behind me in my Zoom background. And this is my co host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. My wife and I are off to California this weekend. So, you know, she's she's a book artist, and there's a there's a big biannual, every two years is biannual, right? Or is that semiannual?

EL: Maybe?

KK: Who knows?

EL: Biennial is also a word. Maybe they’re the same word? [Editor’s note: dictionary.com says biannual can mean the same thing as biennial (every two years) or semiannual (twice a year). The Oxford English Dictionary, on the other hand, says biannual means twice a year and biennial means every two years.]

KK: Every two years. Except that this two years is this two years is three years, because two years ago was, well, anyway.

EL: Right.

KK: So yeah, so it's called CODEX, and she is exhibiting there, and I am tagging along because I like Berkeley.

EL: Fun!

KK: And we're going to do stupid things like spend too much for a meal at Chez Panisse.

EL: That sounds great.

KK: Yeah, we're really looking forward to it. So anyway, but let's talk math.

EL: Yes. And we are excited today to be talking with Tien Chih. Tien, would you like to introduce yourself?

Tien Chih: Hi, my name is Tien Chih, and I'm currently an assistant professor of mathematics at Montana State University Billings, which is a comprehensive teaching school in the middle of Montana. But later on this fall, I will be joining the faculty of Oxford College at Emory University. Oxford is college separate from the main campus of Emory that's a small liberal arts spin off, so students can do a small liberal arts experience with the first two years of their undergraduate degree at Emory and then go to the main campus to finish, so I'm really excited for that.

KK: Yeah. And that's that that's a little bit outside of town, right? So Emory itself is in Decatur, correct? And the Oxford College is where? I know it's not right there.

TC: Oxford, Georgia, about a half hour or 45 minutes or so east, I think.

KK: Okay. Nice.

EL: Yeah. Well, I am excited that we got you while you're still in Montana, because I just love having guests who are also in the Mountain Time Zone because that they don't think I'm, you know, a layabout because I never want to do anything before 11am.

KK: This is a common problem we have.

EL: Yeah, I am excited, I'll be going to Glacier, or I'm working on planning a trip to Glacier National Park for this summer, which I know isn't actually that close to Billings, because Montana is enormous. But I think it will be very beautiful. Some of the pictures that you post sometimes are really beautiful with the scenery up there in Montana. I'm sure, maybe you're a little behind us on the spring timeline, but similar mountain beauty.

TC: Yeah, I went to graduate school, actually at the University of Montana in Missoula, which is much closer to Glacier. So while I was a grad student I managed to go up there a couple of times, and you're in for a really good time this summer.

EL: Yeah, I'm excited.

KK: Excellent.

EL: Yeah, well, let's dive into the math. What math would you like to talk about today?

TC: Okay, so my favorite theorem is not exactly a theorem, or is a theorem, depending on your point of view. But my favorite math concept that I'm going to talk about today is mathematical induction. There are a couple of reasons why I chose this. One is that I am a combinatorialist/graph theorist. In our field, we don't have a lot of big foundational theorems or theories. In our line of research, we tend not to build skyscrapers, we tend to sprawl. And so because of that there aren't like these big foundation things, so it's hard to point at, like, one theorem and say this is a key theorem in our discipline, but the idea of induction is always present in our work, and especially in my work. Another reason I like induction is because I do a lot of math outreach kind of things. I'm involved in the Math Circle community quite heavily. I run a student circle here at MSU Billings. And mathematical induction is one of those things that almost all students, even children, intuitively understand. The actual mechanisms and formal logic, of course, is not something most people are familiar with. But this idea of if you have something that's true, and you know, you can do this thing, and it's still true, that this means that it just keeps being true, that’s something that students inherently just grasp right away, especially if you compare it with visuals. So I think there's a lot of concepts in math that are like this, that are technically kind of difficult results to articulate, but intuitively, everyone already understands them to some extent.

KK: Right.

EL: Yeah. And you mentioned visuals. So what are some of the visuals you might use to describe induction?

TC: So one of the classic induction proofs that you would give as an example or an exercise in an intro to proofs class is the proof that the sum of the first n odds is n squared. And very often with as presented, that's done with the usual algebra, and the flip, and then the 2k+1 and all of that. But you can easily show that if you take a square, and you take a two by two square, you have to draw a one by one by one extra L around the original square to get the two by two square. And then you get the three by three square by drawing a two by two by one L around the two by two square. And then you just say, Okay, you keep doing that, right. And most students or people without formal mathematical training will recognize, oh yeah, that just keeps happening. But the “keeps happening” is induction, right? That's what we don't say out loud, but that's inherently in this reasoning.

KK: That’s a good visual, because as you say, the algebra — I mean, so when we teach our sort of intro to proofs course, this is where students really get their first taste of induction. And I think it's kind of cold, right? So you say, okay, yeah, so it works for the base case: check. You know, so 1=1. All right, we understand that. And then you say, assume it's true for k and then show it's true for k+1. And I think students learn this mechanically, but I'm never sure that they really grasp what's going on.

EL: Yeah, well, and you if you've learned it by manipulating, you know, like k+1, and, you know, multiply that out to square it or something, that sometimes does remove you from actually thinking about the concepts. I mean, it's an important way to be able to work, but if you have pennies that you're arranging on a desk, or cards, or something in a square, that can be a lot more like, yeah, look what's happening. Here's the next odd number, here we go. Yeah. So just maybe to back up for a moment. Maybe we should actually, state, you know, what induction is? Because it is I mean, it as you said, it's an idea that a lot of people will intuitively feel is correct, but might not have have actually seen as you know, this like packaged, you know, with a little definition bow on the top kind of thing.

TC: Right. So the idea of induction is you need two prerequisites. One is a statement, at least one typically integer value for which that statement is true. So you have a statement, let's say capital P, at an integer k, and we verify that P(k) is true. And then we couple that with an argument that shows that anytime something is true for, say, an n, it must also be true for n+1. So P(n) implies P(n+1), then starting at k, the statement is true for all n greater than or equal to k, so for k and then k+1, and then the implication gives you k+2, and so on. And then you keep going.

KK: And so the visual that I sometimes use with students is you know, it's imagine you have an infinite line of dominoes. If you knock down the first one, they're all going to fall down, right? Which isn't exactly correct, but it's a reasonable visual.

EL: There’s this TV show — not to derail totally, but I'm about to — I think, I don't remember if it's called Domino Wars, or Domino Masters, or something like that it [Editor’s note: It is called Domino Masters, and mathematician Danica McKellar is one of the hosts]. We were in a hotel room flipping through channels, and we saw this and they make these domino things, and in fact, sometimes not all the dominoes fall down when you push the first domino because there's some sort of problem in the the domino line of implications there. But in mathematics, the dominoes are all set up at you know, just the right distance or angle that they do fall down.

KK: Ideal dominoes, right?

EL: Yes.

TC: Although sometimes the dominoes, you can find arguments that that end up skipping a step in some of the dominoes. And so I don't know if you've ever seen these induction, like non-proofs. But I give this one as a challenge to my discrete math students: All cows have the same color.

KK: Yes.

TC: Yes.

EL: I don't think I — so this sounds familiar, but I don't remember what it is.

KK: Well, I’ll let Tien tell us, but I have a story about this. So when I was an undergrad at Virginia Tech and MAA members almost universally heard of Bud Brown, who was very well-known among the MAA community and really entertaining and won the Polya award several times for the things he wrote. And this was his standard joke about induction. He used horses instead of cows. But yeah, let us let us hear the theorem that all cows are the same color.

TC: So let's prove via induction that all cows are the same color. So we start with a base case, n=1, and we say, all right, if we have one cow it’s the same color as itself, ergo for a size of set n=1, all cows are the same color. And then, all right, we assume that it’s true for a set of size k and say, all right, so let's take a set of k+1 cows. Well, by induction, we know the first k cows are the same color. And by induction, we also know the last k cows have the same color. And so ergo the k−1 cows in the overlap, they're also the same color. And more importantly, the first cow has to be the same color as the k−1. The last cow must also be the same color as the k−1. So all k+1 cows are the same color, right?

EL: Yeah. Convinced me.

KK: Sure.

EL: We only see one color of cow.

KK: Right? But you know, for Bud’s joke, it was somehow he made this work where it's like, well, that's a horse of a different color. And now the trick, of course, is to get your students to understand why that proof doesn't work. Because it seems convincing. Right?

TC: Yes.

KK: It feels pretty good. So why don't you explain to our readers why it doesn't work.

TC: So it is a true statement at a certain point that this first cow — in a sense, a true statement that the first call has to be the same color as the middle k−1, as does the last cow. But of course, if k is 1 as in the base case, then that’s zero cows, so it's vacuously true. So the first and the second cow are not not the same color as the middle k−1. But that’s not useful information. So this is what what pops into my — visually, how I interpret this, is as that missing domino, right? We actually took away the second domino, right, the n=2 case, and so because that domino is missing, that induction obviously does not carry through and we have more than one color of cow on this planet.

KK: Yeah. And what's fun about induction is you never stop using it. So you know, you're a combinatorialist, you're clearly going to use it all the time. I'm teaching graduate algebraic topology this term and just maybe two weeks ago, I used induction. I was computing the cohomology of the Eilenberg-MacLane spaces of type K(Z,n). And it's a double induction, because you know case 1 and so you pull yourself from something odd to something even. And then it's a different argument for something even to something odd. So, it just never stops.

EL: So I guess I'm going to betray my naivety about like what combinatorialists do, but as a combinatorialist are you kind of mostly using like the regular induction, not transfinite or you know, some spicier flavor of induction. I don't know, do you have to induction — I’m even forgetting, there are a few different ones that you know go for, like different types of sets. So rather than the integers, you can do induction on real numbers, right, if you set things up right. But are you basically just doing it on the integers mostly? That's the only place I would want to do it.

TC: Yeah. Don't feel bad if you don't know what combinatorialists are doing. Most of us don't know what each other are doing. Right? Again, the sprawl that we talked about. Yes. So in my case, I try not to even deal with infinite things that aren't countable. I have to for some of my constructions, but I would prefer not to have to do anything uncountable. And so yeah, I'm only doing, me in particular, I'm only doing induction on integers. I'm certain if you define graphs on uncountable sets, or this sort of thing, then it would make sense possibly to invoke Zorn's lemma or those other kinds of induction-ish things, but that's not what I do.

EL: So well, the base version is powerful enough, I mean, I don't want to besmirch the fine name of induction. So you know, of course, this is important in your work. My big experience, really learning about induction in my first heavy-duty proofs class in college was a transformative moment, for me, in really being able to work with axioms and stuff. So it's also really important to me, maybe from a pedagogical — I mean, I was the one being pedagog’d at — but, you know, from a pedagogical point of view, I think it is also really important. Do you find that too? Is this a really important moment for your students?

TC: I think so. I think — again, like I said, the idea of induction is something that’s inherent in us, I think, but it's like this idea that you're able to keep doing something. And so when I cover induction, I kind of also use it as maybe one of the first examples where you try to you try to formalize what your intuition told you to do anyway. And there is often a tendency I notice when teaching entrance to proofs type classes that because the idea of writing proofs is such a new thing, that somehow, it's totally like — when they write these proofs, they want to use a lot of jargon, and symbols and theorems maybe that they learn. And it sometimes can get away from their actual understanding of what it is they're trying to prove. So when we start with induction, it’s like, look, you tell me why this is true without trying to prove it, just tell me why it's true. And our goal is to turn that into math, and then also clean up any loose bits or cases or whatever, along the way. But that's really what we're trying to do. So if we're trying to improve that, you know, again, the sum of the first n odds is n squared. It's like, all right, why is that? Tell me why I keep adding these rows, and it still keeps being a square. Okay, now what we need to do is just make that concrete. And that's such an important skill throughout, right, because you can't reasonably prove anything unless you have some idea of why this thing might be true. And then all we're trying to do with the formal writing is just say that. Say exactly what you're thinking, but very, very, very, very carefully.

KK: Yeah. So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with induction?

TC: Let’s see if there's a couple of different pairings I could go with.

KK: You’re allowed more than one.

TC: Okay, sure.

EL: And if you have one, and then you have n of them, you're allowed to have n+1 of them.

TC: I don't think I want to go through the gamut of naturals. So when I first learned induction, I was an undergraduate, and I think I was drinking a lot of whiskey at the time.

KK: That sounds illegal, but we’ll allow it.

TC: And so that's one possible pairing. In the math outreach stuff that I do, back when we used to have it in person, I would often buy little granola bars and little single-serving bags of chips for the students to snack on while we're working on this stuff. And so that's a pairing I would give for working with induction with middle school students, which is the target audience for our circle here. And then, in my current work now as a grownup mathematician, I am making a lot of homemade noodles recently. And I’ve been really enjoying them. And so maybe that's what I would pair for for the induction that I'm doing right now in my own stuff.

EL: I like that. I mean, I would like to try your homemade noodles also. That’s always a lot of work, I think, to to make those at home. But yeah, I like the potato chips idea a lot too. Because, of course, there's the Lay's slogan. I don't know if it's still in their branding, “You can't eat just one.”

KK: Yep.

EL: Which works really well. Plus, for me personally, every potato chip tastes like one more potato chip.

KK: That’s right. Yeah, it truly is. You have wide and then you have to have another one. Forever.

EL: Yeah, but noodles. Noodles, I think are good too. Because, like, it’s hard to count the number of noodles there are. Plus delicious. Is there a noodle dish that isn’t delicious?

KK: What do you do with your noodles? How do you prepare them?

TC: So far I've stir-fried them and I've also made a beef noodle soup with them. Which are both excellent. And they're actually much easier to make than I would have thought.

KK: Do you have one of those pasta maker things where you crank them out? Or?

TC: No, I just roll them into a rectangular-ish sheet. And then I just dust it and fold it up and then

KK: You just cut it?

TC: Cut it with a knife.

KK: Okay, excellent.

TC: Yeah. So they come out a little thick, but I do prefer it that way anyway, so that works for me.

KK: All right. So I think Evelyn I already looking up flights to Billings.

EL: I’ll just drive. Yeah, I could be there—

KK: You’ll “just” drive? I mean, look.

EL: I'll be there in eight hours. I don't know how long it is from where I am.

KK: It’s got to be more than that.

TC: I’m closer to one of you now, but in a few months, I will be closer to the other one.

KK: That’s right. It’s like five hour to Atlanta.

EL: Yeah, well, this has been great. I really glad that we've we got to talk about induction which really has deserved an episode of My Favorite Theorem for quite a while. Would you like to let people know how to find you online or any anything you're involved in that you'd like to plug to make sure people know about it, anything like that?

TC: You can find me on Twitter at @TienChihMath.

EL: We’ll include a link to that.

TC: As you both know, I am a co-organizer for the online seminar Talk Math With Your Friends, which has collaborated with this podcast previously in the past for a live taping. And I don't have too much going on right now. Again, I'm in that very odd phase where I'm in between things, but I am very involved, like I said, in math outreach, in teaching, undergraduate research, that sort of thing, a very teaching-focused kind of career. And so I'm certain that I will have things along those lines happening soon in the future, once I get settled at my new position, so I'm really looking forward to all that.

KK: Alright, sounds great. Well, Tien, thanks for joining us.

TC: Thank you for having me. It was a blast.

[outro]

In this episode, we talked with Tien Chih, who will soon be starting a position at Emory University's Oxford College, about mathematical induction. Here are some links you might enjoy with the episode.
Chih's website and Twitter profile
Talk Math With Your Friends, the online math colloquium series he co-organizes (and with which My Favorite Theorem has collaborated!)
A wikipedia page dedicated to the proof by induction of the statement that all horses are the same color
Domino Masters, a TV show about dominoes

Episode 76 - Math Students of CSULA

9 juni 2022 | 59 min

Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined by your other co-host person.

Evelyn Lamb: Hi, I am Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And I was actually thinking we should have a quiz at the end of this one.

KK: We really should.

EL: It’s just so jam-packed. There's gonna be so many different things floating around. So, like, be prepared…actually don't because we haven't prepared a quiz for you, so we don’t want you to be disappointed.

KK: I’ll start writing the quiz now. Yeah, today we have an interesting new experiment that we're going to try. So Mike Krebs from Cal State University in Los Angeles reached out to us with an idea. Mike, why don't you just introduce yourself and explain?

Mike Krebs: Hi, my name is Mike Krebs. I'm a professor of mathematics at California State University Los Angeles. Graduation is tomorrow, and I think our students have had enough of quizzes, so thank you for passing on the quiz. Yeah, I listen to a lot of podcasts, and my origin story of finding your podcast is sometimes to find a new one, I will go to Wikipedia and click the “random article” button, and then whatever comes up, search to find a podcast on that.

KK: Okay.

MK: I found various things that way like the story of Sylvia Weiner, an octogenarian marathon runner, and so on and so forth. And then one time, I clicked “random article,” and up came a page on differential geometry of surfaces.

KK: Okay.

MK: And one Google Search later, I started screaming at my laptop, “There’s a podcast called My Favorite Theorem!” So, yeah, I discovered that at the time I was teaching, this past semester, a capstone course for our math majors, in which students select a topic and then have to write about it and present about it. And I said, “Oh, I wonder if the good folks at my favorite theorem would be interested in doing something like that with students.” So I recruited some students from that class, as well as a bunch of other students from our university. And here we are now.

KK: All right.

EL: That’s amazing. And so you're mostly graduating seniors about to graduate and you're spending the morning before your graduation with us? I feel so honored.

KK: I really do. This is something else. Yeah.

EL: Well, let's get to it.

KK: We have nine students. And so as Mike pointed out, there are nine factorial or 362,880 possibilities here. And we have chosen one of those orders.

EL: Yes. You know, if you so choose, you can always divide this into tracks and listen to them in every possible order and then get back to us and tell us what the optimal order would have been. But for now, it's the order in which they appeared on my Zoom screen. So our first guest today is Pablo Martinez Gutierrez. Great to have you. Would you like to say a little bit about yourself and let us know your favorite theorem?

Pablo Martinez Gutierrez: Hi, thank you for having me on the show. Yes, I'm Pablo. I'm currently a math undergraduate at Cal State LA, hoping to complete my Bachelor's, not this semester, but hopefully next fall next semester. And my favorite theorem that I'm covering for today is Euler’s formula and Euler’s identity. It's something that I got exposed to back in Professor Krebs’s class when I took his class for differential equations. He was teaching us about second order linear homogeneous differential equations. And in one class session, he introduced the topic of Euler’s formula and identity as a side gem. And I was like, “Oh my goodness, this thing is so incredibly beautiful.” The way that I learned it in his class was he introduced the mathematical expression ex as a Taylor series, and he expanded it out as a series. And then when plugging in eix, then you spat out that series and because of the i, something interesting happens where it starts to be, you could split it up into two individual, or two smaller series, so to speak, of cosine, and i sine x. So you would have the expression eix equal to cosine x plus i sine x. And that to me just seemed that for me, it was like I was gobsmacked. It was just baffling. It was incredible.

EL: Yeah, everything just falls out after that, right?

PMG: Yeah, you're seeing all these terms that come from math, you have e, that comes from compounded interest back when you're learning about it in algebra, you have sine and cosine, that are coming in from the unit circle and trig. And then you have i from complex numbers. So all those just coming in together is is like mind-boggling, right? And then if that wasn't amazing in and of itself, something interesting and amazing, even more amazing, happens when you plug in π for x, right? So you have eiπ is equal to cosine π plus i sine π. And so the cosine π just becomes negative one. And the i sine π becomes zero, which just goes away. So then you have eiπ equal to negative one. And then if you add one to both sides, you get eiπ+1 is equal to zero. And that's just — when I saw it, I was in awe. And I was just like, how do these things align and assemble so beautifully and neatly and concisely? It doesn't seem like, it seems crazy that it would happen that way.

EL: I have an unpopular, or possibly controversial opinion about this, which is, it's cooler to leave it with the minus one on the other side, instead of doing the plus one equals zero. Don't cancel me for my controversial Euler formula takes, but I’ve just got to put that out there.

KK: My favorite part about complex exponentials like this is that you can forget all of those sum formulas, right? Like, if you want to know the cosine of three theta, you just use the complex exponential. It makes your life so much simpler. So that's my fun thing. Okay, this is a really beautiful fact. So what have you chosen to pair with this fact?

PMG: So my pairing for this formula and identity is this. I don't know if anyone's seen the Stephen Hawking movie Theory of Everything. The ending scene of that movie has this musical score that I like to listen to, that evokes a similar feeling of elegance and beauty, and awe about the universe, which is the same feeling I get from this identity and formula. It's called the Arrival of the Birds by the Cinematic Orchestra and the London Metropolitan Orchestra. You can give it a listen on YouTube. And any you as you listen to it, it elicits that feeling of awe.

EL: Yeah, listen to it while you do some complex integrals, maybe. I like it. Yeah. Thank you.

KK: All right.

PMG: Thank you.

EL: Yes, well, the bar has been set high. But yeah, we will see — no, I won’t pit anyone against each other. Our next guest is Holly Kim. So yes, Holly, if you'd like to tell us about yourself, and tell us your favorite theorem.

Holly Kim: Hi. So my name is Holly. And I'm currently a grad student at Cal State Los Angeles. And I'm not graduating this semester, so I still have about, like a year or year and a half before I graduate. But I'm happy to be here. So thank you for having me on the show as well.

KK: Absolutely.

HK: My favorite theorem is currently Ore’s theorem from graph theory, which states that for a given graph that’s simple and finite, and for two vertices that are distinct and non-adjacent, if the sum of the degrees of those two vertices is greater than or equal to the total number of vertices of your graph, then the graph is Hamiltonian, meaning you can find a Hamiltonian cycle, meaning you can find a spanning cycle that reaches every vertex once it's in the graph. So that one is my favorite. And it's interesting because I was not a math person when I got my bachelor's. So when I took my first proof-based course, the professor quickly mentioned Hamiltonian graphs. And I had not seen graph theory, I think in that form, at least, ever. So it was really interesting at that time. And he had made a joke about like, “It's not the Hamilton that they made the musical about.” And around that time, I thought that was so funny because I was also listening to Hamilton the musical, or had started listening to it, even though it had been out for a while by the time I'd taken that course. But it just sort of stuck with me, and I thought Hamiltonian graphs, and Hamilton the musical, they’re just sort of like, every time I thought about it, I thought, oh, how fun and how interesting and how funny Hamiltonian graphs are. And then what makes them even more interesting is that unlike Eulerian graphs, where you can tell a graph is Eulerian quickly by looking at the — you know, it's if and only if every vertex has an even degree. So then you know that graph is definitely learning. The Hamiltonian graphs don't have sort of a defining characteristic, like Eulerian graphs. So Hamiltonian graphs are sort of elusive, like there are some theorems that will work for certain families, or types of graphs, but nothing that quite, I think, captures, yes, for sure every graph — or this graph is Hamiltonian if and only if these conditions are satisfied. So it's not been discovered or found out yet. So that's my current favorite theorem.

KK: I don't know this theorem. So this is sort of interesting, right? So it basically says that if you have two vertices in the graph that have enough edges out of them, basically, you're guaranteed a Hamiltonian cycle. That's just, that's pretty remarkable, actually.

HK: Yeah, and the proof has like a funny — it's like a proof by contradiction, but there are certain edges, like you cannot have, as you construct this proof, otherwise, you will end up having a Hamiltonian cycle. So it's like, you’ve got to have just enough, but not too many, or where you actually end up with another, like a Hamiltonian cycle kind of embedded in, in your graph. And so it's very fun. The converse is of course not true. You can see, the other direction would not.

EL: I'm sitting here trying to doodle myself a graph and see, but I think I think I need to do it a little bigger graph, because there weren't enough vertices in this one. And I can't doodle and talk at the same time.

KK: I can’t either. What’s that about?

EL: So yeah, no multitasking for me.

KK: So what pairs well, with this theorem?

HK: Well, it might be on the nose, but I'm going to do it anyway. I paired it with Hamilton the musical. And I've mentioned it before, but beyond just, it being a good soundtrack to listen to with just about everything, I thought, well, certainly, there must be a deeper connection I can draw between the musical and Hamiltonian graphs. And if you listen to the musical, a motif of it is that oh, Alexander Hamilton is just like never satisfied in terms of his goals and ambitions always wants to do more. And he's never at a point where he's like, Oh, I'm, I'm good. And I don't need to keep going. Just based on the musical. And I kind of thought that Hamiltonian graphs, they aren’t personifications of Alexander Hamilton, but that you know, there is nothing that quite satisfies them at this time, or at least as a whole, like Hamiltonian graphs as a whole. So, there is nothing that that would satisfy like, oh yeah, for sure, I am a Hamiltonian graph if these conditions are met. And so that was my connection. And if you want to go further, that symbol, like the iconic logo of Hamilton, or the Hamiltonian graph of Hamilton the musical, there is a star, which is isomorphic to a C5, a five-cycle, so my pairing was Hamilton the musical.

EL: I like that. Well, just in case you have not seen it, there is an excellent parody of it. (singing) William Rowan Hamilton (end singing) of the the Alexander Hamilton song, done by, I'm forgetting the name of the YouTube channel [Editor’s note: it’s A Cappella Science]. I think it's it's run by a guy called Tim Blais, B-L-A-I-S. So, yeah, check that out. I was actually, I was trying to write a parody, I just would always get in my head that (singing) William Rowan Hamilton (end singing).

HK: That’s so funny.

EL: Then I discovered some other person did it already. But they've got a bunch of people coming in. They've got, like, someone playing Emmy Noether, and a few other math contemporaries. So yes, definitely check that out. We'll include a link to that in the show notes. Yes, excellent. We're doing great here.

KK: Yeah. Who’s next?

EL: So thank you, Holly. Bryce Van Ross, welcome to the podcast!

BVR: Hi, thanks for having me. Excited to be here. I guess I'll introduce myself.

EL: Yes, please do.

BVR: My name is Bryce. Come tomorrow, I'll be a master's graduate in math from Cal State LA. So I'm pretty excited about that.

KK: Congratulations!

BVR: Thank you. Yeah, so the theorem today, I like learning a lot and learning new things. So in the process of choosing a theorem was like, find something new. I found the Hales–Jewett theorem, I believe that's how it's pronounced. The general idea is it's a combinatorial theorem, usually applied to game theory, and you start off with two positive integers, N and C. N would correspond to, for example, like a grid, right, like how many columns or how many rows you have in a grid, and C would correspond to the amount of players alternating in some game. So, the Hales–Jewett Theorem states that for any two positive integers N and C, there will exist some H-dimensional cube corresponding to that N and C. So by cube I mean, like an N by N by N by N H-many times, grid. Now, more than that, that theorem says that corresponding to that cube, there has to exist some, at least one, row, column, or diagonal that is all of the same color, which is pretty impressive. Yeah. And a great way to conceptualize that is, for example, like generalizing the notion of Tic-Tac-Toe, where you have a bunch of like Tic-Tac-Toe, grids as faces of some very big cube. This theorem tells us that, at some point, you're going to find a very long line that's either a row, a column or a diagonal where someone wins, guaranteeing that also someone loses. So that's the theorem I picked.

KK: Wow, okay.

EL: Yeah, so what what kind of — so you gave the example of Tic-Tac-Toe. I must admit I'm fairly ignorant in game theory. It's like, I sort of get, I like the part where you're like, Oh, you've got an N-dimensional, or H-dimensional cube. It’s the part where it's actually the games, that’s why I don't know that much about game theory. Because like, I'm a little bored by my that kind of thing. So what other kinds of games can show up in this?

BVR: Yeah, I'm also unfamiliar with game theory, never even like read a book on game theory or taking a course, but from my understanding, you could extend it towards notions are familiar games like Connect 4, any turn-based game such that it requires somebody satisfying a line of something of the same color.

EL: I know Game Theorists always have these super tricksy ways of like, oh, yeah, a chess game you could just make into this or, you know, Nim you can represent by this or, you know, any game you want, whether it's like a real game, like chess, or a math game like Nim. Yeah. Sorry, hashtag not all math games. But yeah, that's interesting. So you just kind of stumbled on this theorem in a some sort of curiosity rabbit hole?

BVR: Lots of googling at 2am. And Wiki-ing at 2 am.

KK: That's what happens. This sort of reminds me of Sperner’s lemma, right, where you try to, if you're coloring the vertices, you start coloring vertices, then you have to get a triangle with all the same color vertices. I wonder if it's sort of related to that in some way. Or maybe — I'm a topologist, so I’m always trying to think of ways to just turn this into a topology question. It feels like it should be one. But anyway, yeah, all right, so this is a good theorem. What pairs with it well?

BVR: Yeah, so I was trying to think of something where audiences can very much relate to. I'm a big fan of games and media. And I think several people in the world last year, watched the show Squid Game. And season two is upcoming. So Squid Game that's my pairing. The reason why it's my pairing is not just because game is in the title there, but also because for those familiar having seen season one, at certain points in the game, I think it was with marbles or something, they were like, oh, make your own game using these marbles. And I was like, if I were to like change up and impose my own ideas of season two, what if a player gets to choose a game, any game and make it and compete against any number of players. I think that the Hales–Jewett theorem is ideal because everyone would be intimidated by a giant cube of Tic-Tac-Toe. But if you know the theorem, then technically you have an advantage because you know, hey, someone's got to win and ideally, it's you.

KK: Okay.

EL: I must admit, I am too much of a weenie to watch Squid Game.

KK: Yeah, I never watched it either.

EL: From just — from even the SNL sketch that was based on Squid Game, I was like, Nope, not gonna watch that one. It looks too scary for me.

KK: Yeah, my wife really doesn't like anything violent. So we're watching a couple of things right now that there is some violence and she has to you look away. It's just really pretty rough. But I know it’s very popular.

EL: Our very brave listeners, I think, will enjoy that pairing.

KK: Okay, excellent.

EL: Well thank you. Oh, and Bryce, I know that you wanted to share an exhibit that you've got up right now.

KK: That’s right.

BVR: Yeah. Yeah. So I really value, for example, like the themes of this podcast, to give an opportunity for people of any background to get a different perspective on math. And I like that for a variety of topics within STEM. So I work as a library archivist at Cal State LA, and I'm developing a STEM exhibit for all faculty, students, etc, to just visit and change their minds. So there are going to be interesting math related artifacts, as well as just unconventional things. You're going to see, like, dinosaur bones and whatnot. And it will be debuting in the fall. So just wanted to give a hype for anyone interested, they could reach out to Cal State LA Special Collections and Archives to find out more, and it'll be open for everyone.

EL: Awesome. Yeah. I hope our LA-based listeners will check that out.

KK: Sounds very cool.

EL: Okay, thank you, Alvin. Alvin Lew? Would you like to join us? Sorry, I don't want to introduce people differently. Yes, please introduce yourself and let us know about your favorite theorem.

Alvin Lew: Hello, everyone. I'm very glad to be here. I'm Alvin I'm currently a third-year computer science major and math minor. Not directly just in math, but I've been doing machine learning research and kind of being at the intersection of two different subjects there. And so today, I'd like to share not so much a theorem, but I guess the actual theorem is that the cardinality of the real numbers is greater than the cardinality of natural numbers. And specifically, I really, really enjoyed the proof of Cantor's diagonalization.

KK: Class.

AL: And so I don't know, has anyone played the game in elementary school where each person is trying to come up with a bigger number?

KK: Sure.

EL: Yeah.

AL: And then someone would play the trump card of infinity, you know.

EL: And then infinity plus one.

AL: But I think what people don't realize is that there are actually different levels of infinity. And I remember watching a TED Talk video back in middle school, and my mind was so blown by the fact that there could be an infinity greater than a different one. And so having taken a more formal class last semester, talking about Cantor's diagonalization, I think it's such a beautiful proof, because it's so easy for even non-math majors to understand. And it just goes something like: suppose you can list everything out so that you can match each natural number to a real number. And what you'll find is that the real number you can write out as an infinite decimal, so a number with infinite decimal places. And so you just write all of them out, you pair each of them up. And what you ended up doing is you take one of the digits from every single real number, and then you create a new number based on that by changing up all the digits, and you'll find a contradiction that you actually didn't list out that number. And so just that very simple idea there leads to actually quite a few paradoxes. I think there's like a Hilbert hotel paradox if anyone's heard of that, and some other ones, but something so simple like that, just one idea, and it opens the kind of, you know, mathematics world where there's different levels of infinity. And for a normal person who's not too involved in math, I think it's actually very interesting that it's accessible. It's a simple idea that anyone can go ahead and check out.

EL: Yeah, yeah, that such a great one. And I remember that is the one when I was in undergrad, when I saw that and finally understood it, it was it was one of those like, mind blown kind of moments for me, a very special place in my heart.

KK: Yeah, yeah. And then you want all these other weird things like, you know, like the cantor set is also uncountable. It's kind of the same argument. Yeah, it definitely blows blows everyone's mind the first time. And but the second time, sometimes too.

EL: I’ll just think about it every once in a while be like, is that really true? Do I just need to add that number to the list? And then I'll fix everything.

KK: Well, yeah, it doesn't. You can you can prove that too. But you know, I saw some Twitter thread the other day, that was always trying to argue that there's some models out there where the or the reals are countable, you know, like, you can change your rules and get different answers. But I don't like that. I like my real numbers to be what I think they are. Right? Although, who knows what they are really. Anyway.

EL: So what what is your pairing for?

AL: So I was really actually thinking quite hard about how to match some concept of infinity with real life. And so I think some people might have heard the analogy of, like, you put a box in a box in a box or whatever. And so to me, that's actually very hard to actually even imagine, right? Because in real life, you could never actually make an infinitely small box. It's not practical. Theoretically, it's possible. So what I was thinking is I pulled the old mathematician trick and pair it with alphabets. Because mathematicians, every time we struggle to match numbers to something, we just slap a variable. We take the Greek alphabet, we take everything. And in terms of uncountable and countable infinities, we actually take from the Hebrew alphabet, right, the Aleph and the Bet characters to represent the two different sets. And so I decided, you know, by pairing it with different languages, I figure, if we ever do come up with infinitely, or more more discoveries like that, we can just slap another letter on it and hope it pairs up. And we'll have a new method of explaining something without without the numbers with the more general letter from from a random alphabet.

EL: I like that. Yeah, I've seen, you know, the Latin alphabet, the Greek, Hebrew, and Cyrillic. But yeah, we need to just the next time when one of you comes up with with some new concept that just needs you know, Greek letters are not sufficient for the amazement of this, we need to start using other alphabet.

KK: Oh, yeah. How about like those Southeast Asian ones? Like the like the Thai alphabet? That’s really beautiful and completely different. Right?

EL: Yeah, we’ve got a lot of options.

KK: We do.

EL: New goal. I like it. Thank you, Alvin. Very good. Okay, next up, we have Judith Landau. Judith, please tell us about yourself and share your favorite theorem.

Judith Landau: Well, thank you for having me. My name is Judith Landau, thank you for introducing me. I'm also graduating as an undergrad in math tomorrow.

KK: Congratulations!

JL: Thank you. And I'll be moving on to an interdisciplinary biology program, as a Ph. D. program. And I've been doing some research modeling biological systems. So my favorite theorem is actually the fundamental theorem of Markov chains, which is from the field of probability theory and statistics. And I'm very interested in Markov chains, because they are a way of modeling systems based off of probability instead of deterministically. So instead of saying we know what's going to happen next, we're going to say it's based off of probability. Markov chains can be represented by a directed graph. So a set of vertices with edges that are directed pointing to a specific vertex in one direction or the other. The outgoing edges of each vertex are actually, the that's the probability — sorry, the random variable associated with that vertex. So each vertex has its own random variable, its own probability distribution that tells you how likely you are to go to any other vertex, and so in a very simple weather model for down here in Southern California where we could only might only consider sunny, rainy and cloudy days, no snowy days, those could be our three vertices or our three states. And we could talk about the probability of moving between them from day to day. And so the fundamental theorem of Markov chains can be stated in many different ways. There are long ways and short ways to state theorem. But the simple way to state it is that an aperiodic irreducible Markov chain has a stationary distribution. So an aperiodic Markov chain for our sunny rainy cloudy model would basically mean that the chain is set up so that there is no regular period for any of those states. So there is no regular period between the sunny days, cloudy days, or rainy days. And the irreducibility just means that there is some path, directed path, obviously, between every vertex in that directed graph for this Markov chain, so you're able to reach every vertex from any other in some number of steps. And the stationary distribution: basically, for the Markov chain, if I were to give you a longer version of the theorem, the longer version of a theorem says that the stationary distribution for the Markov chain is actually the long-term probabilities of ending up on any given state. So in my example, sunny, rainy and cloudy. My pairing, if I can move on to that, is actually in biology. It's kind of a pun, because my pairing is proteins. Proteins are actually chains themselves, they’re chains of amino acids. And so since there's this dependency going on in Markov chains, where the next day's weather is dependent on today's weather, and that's a really characteristic idea in Markov chains is that there's this dependency from one state to the next, proteins, the order of those amino acids, is dependent on a gene, because genes are what code for proteins.

KK: Okay.

EL: I love that pairing. That's very nice. It it kind of makes me wonder, like if — I do not know anything about biology, the last biology class I took was in, probably you weren’t born yet.

KK: Ninth grade.

EL: But it makes me wonder like, are there things where like, it's a little more likely that you'll have a tryptophan after a lysine, than something else?

JL: Yes, we actually study that in bioinformatics heavily, what amino acids are more likely to be next to each other and things?

EL: Oh, cool! I’d never thought about that. But it's kind of like, of course there's going to be some sort of tendency, because it would be extremely improbable that it would all be exactly equal all the time.

KK: Right.

JL: And according to our theories, in biology, it's all related to the structure of that protein and how that structure will affect the function of the protein. And that's why proteins evolve as they do.

EL: That is so cool. Well, thank you for sharing that theorem and that biology connection, and I'm so glad that you are going on to study this more.

KK: Yeah, that's very cool. Yeah. Good luck.

JL: You too.

EL: All right. Next up, we've got Kevin Alfaro. So yes, welcome. Please let us know about yourself. And I see that you've got the Golden Gate Bridge behind you in your Zoom background. Are you from the Bay Area?

Kevin Alfaro: No, it’s just my favorite bridge.

EL: It’s a good bridge.

KK: I mean, I've stood in that spot and taken that picture. I think I think a lot of us have, right?

KA: Yeah. Thank you for having me. It's a pleasure to be here. I'm Kevin and I'm majoring in math at Cal State Los Angeles. And for my theorem, it's actually Archimedes’ theorem on his approximation of pi. So I'm going to be taking us back a couple thousand years.

EL: Yeah, I love it. I love the variety.

KA: Yeah, I'm a big daydreamer. So this is all something I could daydream about all day. And so what he was trying to do, he was trying to approximate pi. And what he did was since it’s included using a circle, he placed the hexagon inside the circle. And this hexagon had already included the radius of the circle as one of its sides. And this is because the hexagon is made of six equilateral triangles. So each side ends up being the same, and each side also corresponds to the radius. So he uses this to calculate the circumference of the hexagon. And since it’s inside the circle, he knows it has to be smaller than the circumference of the circle. So he, he knows that the circumference of the hexagon is 6R since it's six radiuses each, each part of the triangles. So already has a bound on that. He knows that it has to be greater than 6R. So now what he does is a series of calculations to try to get a bound. He tries to squeeze pi in two hexagons, a hexagon outside of the circle and a hexagon inside of the circle. So he knows that that circle has to be between these two hexagons. But the way he does it is so so laborious. Yeah, so for the for each of these. So he starts with the first hexagon, right? And it's a six-piece hexagon. So he has to build that up each time, he has to create a longer hexagon that corresponds to the actual circle. So he has to cut up each arc in half. And he does that. So he cuts every one in half, and he creates a midpoint. So he has to do the Pythagorean theorem for each one. And yeah, so he's just doing it like every day. So if you guys have like, a couple of days of free time, you guys could do this yourself.

KK: And you didn't have algebra either, right? I mean, he didn't actually have algebra. So he was — and these numbers didn't even exist to him like square root of two or whatever.

EL: Yeah, it’s amazing. I’m trying to remember, is the last level, was it, like, a 96-gon, or a polygon with 96 sides or 100-something sides?

KK: I think that's how far he got. Yeah, yeah.

EL: And it's just like, I immediately give up on hearing that. And I've got a little calculator in my phone that I carry in my pocket all the time. And like, I've just like, I've noped right out of doing that. But yeah, it's amazing. Yeah. So is that called the method of exhaustion, which I think is perfectly named?

KA: Yeah.

EL: Yeah, it’s so amazing to think about 1000s of years ago, the people like like Archimedes, and other people like having, having the wherewithal, the persistence, to go through and be like, Well, I don't have a calculator, and I don't know what pi is yet, so I guess I'm just going to, you know, figure out the length of this 96-sided polygon. I don't have anything to do for the next week. So did you first encounter this theorem? Or this idea?

KA: I actually got this from a really great book. It's called Infinite Powers by Steven Strogatz. So I recommend reading that book to anyone listening. It is good. Yeah.

EL: And so yes, what is your pairing with Archimedes exhaustive proof of — or exhaustive approximation of pi.

KA: My pairing is actually more of an abstract pairing. And it's more of an idea, because I think around the context of math around this time, too, and the context is always changing. And around this time, it was, it was more like a spiritual level of math, like people were more into it. And it was more spiritual. So then if we connect it to now, and how advanced math is, and we know what Archimedes was doing, since he could never find out the numerical value of pi, since it's an irrational number. And he couldn't do that back then. But he still knew that it was between two fractions, between 3+10/71, and 3+10/72, I believe. And that's all he needed to know. So he knew that it was between two numbers, but he can never really find out what exactly this number was. And I think that really speaks a lot about what math is, or what people do, in terms of knowing what we don't know. So in this time, he knew that he didn't know, which is a lot what, what's happening on today. And I think that's just a cycle of what we do for maths or for anything, really. We try to get to the point where we know that we don't know, and then we're done.

EL: That can be such a difficult part of basically anything in life. This whole pandemic thing we've been living through, like knowing what we do and don't know about what's happening with that has been such a challenge. It's like, oh, that really affects my life in a way that maybe knowing the exact value of pi doesn’t.

KK: Right.

EL: But yeah, it's so hard. And sometimes you think you know what, you know, what you don't know, and you don't realize what you don't know. So yeah, I like that kind of, from the very concrete polygons to this sort of abstract. Thank you, Kevin.

KA: Thank you.

KK: That’s a good one.

EL: All right. Next up, welcome to the show, Francisco Leon. Would you like to tell us a little bit about yourself and share your favorite theorem? Yeah,

FL: Thank you for the welcome. Tomorrow, along with Bryce, I'm graduating from Cal State LA through the master's program, so I'm really excited about that.

KK: Congratulations.

FL: Yeah, thank you. Yeah. Today, I want to mention, when I was asked, what's my favorite theorem, at the time, I really had one on mine from my topology class. I’ll state the theorem now. It says a topological space is regular if and only if for every element in the set and open set containing the element, that there exists another open set that also contains elements whose closure is in the first mentioned set. So maybe to label some of this, to get a better visualization, so we have some element, say P. And then say, some open set that contains it, U. Now in between P and U, this alternate characterization of regularity says there's going to have to exist another open set V, that also contains P inside of U whose closure is also inside of U. So it goes, maybe nested: P, V-closure, V, and then U.

KK: Yup.

FL: And that's equivalent for regularity. And you know, at first, this was a homework assignment from the class. And one of the key tools to do this problem is to understand that if you illustrate an element in open set, so you draw it ,right, you kind of draw a little point P, and a dashed circle around an open set, that's going to be equivalent to having P outside that complement. So you can draw P outside of a box and a box, just kind of know that, since of the complement of open sets are closed, that's why it's represented as a box, to be able to comfortably go between those two perspectives of the same situation, I found difficult to get at the time. And so, you know, I fostered an appreciation for this theorem because it made me resolve that difficulty. So it led me to some trains of thoughts that were really interesting for me at the time. And it's been a while since a theorem held my attention for so long. So just to share some lines of thoughts I was having when working through this exercise: okay, I was like, how do I go from being an element in this open set to being outside its complement? I started imagining myself inside of a room, just as the element would be inside the open set. And I'm like, how do I push the walls of this room so that suddenly the outside is contained inside these walls, right? Because you want to go from being inside the open set to being outside the complement. So how can I move this boundary, so that the outside is contained? I felt like I couldn't do it. I look outside, and I know the universe is huge. How can I push the walls of this room to contain everything else outside and suddenly be the one outside? So I had great difficulty. And then as I would think about this problem and look outside, I would see the window. And then I realized that the illustration of the element inside the open set, you know, when you have that point P and those dashed lines around it, those dashed lines aren't necessarily representative of a border, it's really representative of a window, because what you do at the illustration, is you look at what's within those dashed lines to see the elements. You are really looking into the set to see what's inside of it. So what I really needed was a window in this room for me to be able to understand the outside. So now if I reimagine the situation, I'm in a room, which is representative of the open set, and there's some element in here. What I need is a window so that I can see what's on the outside. And then if I change my perspective, if I pass through the window and then turn back around, then I see through the window, the element inside the room. And now I have the outside perspective that I want it. So now I see the element inside the room. So now I'm seeing the complement of the open set. And that really helped me change that perspective from being inside the open set to being outside the complement, which is a closed set. And then I was able to do that exercise pretty straightforwardly, but I was really struggling before that. And I never really think about theorems as visually as I just described, so that's why I really appreciate this theorem and that's why it's one of my current favorites.

KK: You know, when I was an undergrad, I was always going to be a math major. That was always what I was going to do. But that class, point-set topology, is when I really fell in love. I mean, I was already planning to go to graduate school, but my professor for that, Peter Fletcher, who passed away a couple of years ago, really put me on this path to being a topologist. I didn't know what I was going to do until I actually hit that. And of course, you know, that flavor of topology is pretty well settled and has been for quite a while. But it's still really beautiful to think about these basic properties of topological spaces. And I love your visual description of this sort of thing, like regular spaces. You know, I haven't thought about them in a long time, but I think you really nailed it right there. So nice. Yeah.

EL: And I like, I like seeing that little glimpse into how you were thinking this through, even if that isn't what you wrote down in your proof in the end. But it seeing — so often, when we read math, we don't, we don't see those little glimpses of how the person actually thought about it. We see the cleaned up version that, you know, is presentable for professional company or something, but I really liked seeing that. So what what is your pairing for this theorem?

FL: Oh, yeah, so I'm thinking maybe a hot apple pie, just because we typically do place it on a windowsill to cool down and I just figured that this theorem is cooler with a window.

KK: That’s good.

EL: And yeah, who can argue with a nice warm slice of apple? It's delicious. Yeah. Excellent. Well, thank you so much.

FL: Thank you guys.

EL: All right. Next up, we have Marlene Enriquez. Marlene, would you like to introduce yourself and tell us about your favorite theorem?

Marlene Enriquez: Yes. So hi, my name is Marlene, and my favorite theorem has to be the ham sandwich theorem.

KK: Yes!

EL: You've got company there with Kevin.

KK: So in episode zero, Evelyn and I had our favorite theorems. That was mine. Here we go. All right.

EL: So please tell us what you like about it.

ME: What I like about it is just, it's the first time that I encountered a theorem that really didn't have an exact answer to how to solve it. So the idea of basically having a line cut something and split it into equal volumes, per se. So when I first read it, I was like, oh, that sounds cool. When I read it some more. I was like, oh, no, this is really interesting. And it's actually the the theorem that I used to write the my research paper for my senior seminar class with Dr. Krebs. So the theorem, when I read it, I read it like this is like, you have a sandwich made out of bread, ham, and another piece of bread and like, they don't have to be exactly aligned. Or even touching each other. But there exists, a line, or a cut, a straight cut that will split the bread, ham, and the other piece of bread in equal volumes. And then when I saw that, like, wait, you don't really have to be next to each other, or like on top of each other, or even in the same room as each other? And you can still be cut and split into equal volumes? I was like, Oh, wow. I can read more about this, it’s pretty remarkable.

EL: Yeah, it can be the sloppiest made ham sandwich ever, and it’ll still work.

KK: Like you can have one slice of bread in Los Angeles. I have one here in Gainesville and a piece of ham could be in Salt Lake with Evelyn, and there is — it will take a big knife — there is a big knife that can cut them in half. That's right.

EL: Where did you first see this theorem?

ME: I was actually searching for a topic for my paper and I was online Google searching, every time at 2am, it always happens. And I kind of stumbled upon it. And I think I saw a YouTube video and I was like, Oh, this is interesting. Then I clicked to another one. I'm like, okay, this is very interesting. And then I just started searching and searching and reading and yeah

EL: Excellent. Yeah, I guess this podcast brought to you by the time 2am: the perfect time to be just finding weird math stuff to learn about.

KK: So what's what pairs with a ham sandwich there?

ME: Well, it's not on the nose, not a ham sandwich. But actually, when I was typing my paper, I was in my living room, and I have siblings. And they were fighting over a chocolate bar. They were saying, oh, like, let's split it equally. And they're like, No, you got the bigger piece or you got the bigger piece. And just the idea of sharing. Even when you go out with friends, you split the bill and stuff like that. That kind of just kind of brought it to, like, that's kind of like this ham sandwich. But like, in like, money, or a chocolate bar, who got the biggest piece, right?

KK: That’s a hard problem, actually, the whole equal division problem is extremely difficult. As I'm sure you found out with your siblings already over a chocolate bar.

EL: The answer is probably just giving up all earthly desires.

KK: Yeah. Right. The Buddha, the Buddhists understand.

EL: Well, thank you so much Marlene. And our final guest is Daniel Argueta. Thank you for joining us. And please tell us a little bit about yourself and your favorite theorem.

Daniel Argueta: Well, thank you, Kevin, and Evelyn, for having me and Evelyn, don't worry, you said my last name correctly. So I'm also graduating tomorrow. I'm getting my bachelor's in math.

KK: Congratulations.

DA: It’s a big thing. I’m first gen. So when Dr. Krebs approached me about this, I was like, What am I going to talk about? What's my favorite theorem? And when I was doing research for our capstone class, I actually stumbled across you guys’ podcast. And the one, I think it was episode 17 with Dr. Naomi Joshi or something like that.

EL: Nalini.

KK: Nalini Joshi.

DA: And it was the Mittag Leffler theorem. And that theorem was just way too complicated for me. Dr. Krebs helped me bring it down a little bit. And I stumbled across the infinite product convergence theorem. And I was like, okay, maybe that’s what I’m going to talk about. I wrote a whole paper on that. And then I thought about it and I was like, you know what, that's not my favorite theorem. So I started doing some googling, and I came across the headline “The theorem shook math to its core.” So maybe that might give you guys some ideas as to where I'm going to. And I'm actually doing I'm going to talk about Kurt Gödel’s, incompleteness theorems.

KK: Oh, yeah.

EL: Okay, yes, is a great way. I’d just like to say, of all the however many thousands of possibilities, I like that, at least that this one was at the end, to sort of be like, okay, we talked about all this math that we can do. Now let's finish it off with: Oh, yeah, we can't do math. That's probably not the best summary of the Incompleteness Theorems. So why don't you tell us about the incompleteness theorem?

DA: Well, what Gödel pretty much set out to prove, everybody at the time was trying to prove that math is perfect. And they were trying to say, hey, math is perfect. They were trying to find like this theory of everything, and Gödel’s all like, hmm, I don't know about that. And so pretty much what his proof did, and not to get super technical or anything, was he showed that in any formal system, there are certain truth statements about the system that can not be proven by said system. So a way to think of this is our classic logical problems with the knights and the knaves. In this case, you only need the knight, where you have a knight in shining armor who can only tell you the truth. So so you ask yourself, what is a sentence that this knight cannot say if he can only tell the truth? And one sentence is: I cannot say that sentence. You know, because if he can say the sentence, then it's true. But if he can't say the sentence, then you know, it's not gonna work. So I just really like this theorem, because to me, you know, we usually tend to think of math as perfect. It's, a lot of people call it the universal language, because math across everywhere is perfect. But to me, it's super interesting that the fact that we don't know everything about it, and we will never know everything about it. So that's why I really like this theorem. And it actually brings me to my pairing, which is — I know usually you guys do food on the on the podcast, but in this case, my pairing is our Final Frontier, which is space. I think I think it super coincides with it. Because, you know, we're going to always keep discovering more and more about space as our technology advances. But at the same time, our universe is ever-expanding. So will we ever completely know everything about space? There's so much to learn and discover, and the same can be said for math. And that's my favorite theorem.

EL: Excellent. I love that. Yeah, it's almost like okay, yeah, math is like, perfect, and you can do everything. Just don't look too hard. This incompleteness theorem stuff is just saying, oh, yeah, you think you could do calculus? Just don't look too hard.

KK: Right, right. Yeah. Well that’s everybody.

EL: This has been so fun. I’m so optimistic about the future of math right now. And biology. And whatever else you all do.

KK: And machine learning, and whatever it is yes.

EL: Yeah. Good luck to all of you in your next steps. It’s so appropriate that we got to do this right before many of you are graduating and, you know, taking those next steps. So thank you so much for sharing part of your, Monday morning with us.

KK: And thanks to your professor, Mike, for reaching out to us and making this happen.

MK: Well thank you for hosting everybody. This is great.

EL: Yes. I don't know if you all want to unmute it and do a goodbye. I don't know. How do you end a podcast with 12 people? (I just did multiplication there.)

KK: Is there a CSULA kind of cheer? Like I'm at the University of Florida so it'd be like a Go Gators kind of thing. They’re looking at me like, no, we don’t do that.

EL: Math majors might not be the best choice for for you know, knowing all the sports things, not to invoke any stereotypes.

KK: I know every word of the Virginia Tech fight song.

EL: Well, it may not be universal.

KK: Maybe not. All right. Well, you’re all unmuted. Maybe say goodbye.

(General chaos of 12 unmuted people on Zoom.)

[outro]

In this Very Special Episode of My Favorite Theorem, we were excited to welcome nine students from California State University Los Angeles, along with their professor Mike Krebs. Each student shared their favorite theorem and a pairing with us. Below are some links to more information about their favorites.

Pablo Martinez Gutierrez talked about Euler's formula and identity, which connect trigonometric functions with the complex exponential.

Holly Kim told us about Ore's theorem about Hamiltonians in graph theory. Check out A Capella Science's Hamilton parody video!
Bryce Van Ross shared the Hales–Jewett theorem from game theory.

Alvin Lew chose Cantor's diagonalization proof of the uncountability of the reals, which was also a favorite of our previous guest Adriana Salerno.

Judith Landau shared the fundamental theorem of Markov chains, which relates to her work in bioinformatics.
Kevin Alfaro talked about Archimedes' approximation of pi.

Francisco Leon chose a theorem from point-set topology about regular spaces.
Marlene Enriquez chose Kevin's favorite, the ham sandwich theorem.
Daniel Argueta finished off the episode with Gödel's incompleteness theorems.

Episode 75 - Dave Kung

17 mars 2022 | 31 min

Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, a professor of mathematics at the University of Florida. And I'm joined by my other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we just got a lovely few inches of snow last night. So I've developed a theory that podcasts cause snow here. Although it could be the other way. Maybe snow causes podcasts.

KK: Maybe.

EL: It’s hard to tell.

KK: I don't know. It's 85 degrees for today. Sorry.

EL: Yeah. I meant to say don't tell me what the weather is in Florida.

KK: It’s very nice.

EL: It’s too painful.

KK: It’s very nice. Yes, speaking of painful, we were having our pre-banter about I had a little hand surgery yesterday, and I have this ridiculous wrap on my right hand, and it's making me kind of useless today. So I have to do anything left handed, and to control everything on the computer left handed. But you know, it'll resolve my issue on my finger. That'll be good. So anyway, today, we are pleased to welcome Dave Kung. Dave, why don't you introduce yourself?

Dave Kung: Hi, there. I'm Dave Kung. I'm a mathematician by training. I spent 21 years at St. Mary's College of Maryland, and I've recently moved on from there, and I work at the Dana Center, the Charles A. Dana Center, at the University of Texas at Austin. I work with Uri Treisman down there on math ed policy.

KK: It’s very cool. So you got a really serious taco upgrade.

DK: Definitely. I'm still living in Maryland, but I get to visit to Texas every few months.

KK: Are you gonna relocate there? Or are you just gonna stay in Maryland?

DK: I’ll be here for now. Yeah.

KK: I mean, it seems like the sort of work that you could do remotely, it’s policy work mostly, right?

DK: A lot of policy work. There's going to be a fair amount of travel once that's more of a thing, but not all of it will be to Texas. Some of it will actually be in DC, in which case, I'm pretty close.

EL: Yeah, you’re right there.

KK: Yeah. Yeah. Well, they do really great work at the Dana center. I've been involved a little bit with the math pathways business, and it is really vital stuff. And of course, Uri is, like, well, he's the pied piper or something else. I don't know. But when you hear him talk about it, he's an evangelist, you really you can't help but like him.

DK: Yeah. And making sure that we know that students have the right math at the right time with the right supports, we're far from that goal right now, but we can get closer.

EL: Yeah, very important work for all mathematicians to care about.

KK: Yeah, yeah. Okay, but I think we're going to talk a little bit higher-level than math pathways today. So we asked you on to have a favorite theorem. What is it?

DK: My favorite theorem is the Banach-Tarski theorem, which is usually labeled the Banach-Tarski paradox.

KK: Yes. Yeah. So what is let's hear it. Well, let's let our listeners know.

DK: So the Banach-Tarski paradox says the following thing: that you can take a ball, think of a sort of a solid ball, and you can split it up into a finite number of pieces — we’ll come back to that word pieces in a bit — but you can split it up into a finite number of pieces, and then just move those pieces and end up with two balls the same size and the same shape as the original. It is incredibly paradoxical. And I remember hearing this theorem a long time ago, and it just sort of blew my mind.

EL: Yeah, it was one — I think I was an undergraduate, I don't think I'd even taken, like, a real analysis class. But I heard about this and read this book, there's this book about it that I think it's called The Pea and the Sun. Because another statement of — I mean, once you can make two of the same size things out of one, you can make kind of anything out of anything.

DK: Just repeat that process. That's the other statement, you take a pea and you do it enough times. And if you do it, well, you can reassemble them to form a sun. Absurd.

EL: Yeah, I just, I mean, reading it, there was a lot that went over my head at the time, because of what my mathematical background was, but at first, I was like, Okay, this means that math is irrevocably broken. And then after actually reading it, it’s like, okay, it doesn't mean math is broken. So maybe, maybe you should talk a little more about why it doesn't mean math is broken. If you have that perspective. Maybe you do think math is broken.

DK: It certainly feels like. I mean, I think my first reaction was: Cool. Let's do that with gold, right? We'll just take a small piece of gold and split it up and keep doing that. I think there's a lot in this theorem, right? And so I mean, it's one thing to understand it, sort of at a deep sense why it's not absurd. And I think it helps me to think about just sort of, you know, once you know a little bit about infinity and the fact that there are different sizes of infinity. And once you know that somehow the even numbers, the even integers, have the same size as all the integers, which is already sort of weird, this feels a little bit like that. I mean, you could sort of somehow take the odd integers and the even integers, and each one of those is the same size as the full integers. But some of the integers themselves, it's weird to be able to split the integers up into two things, which are the same size as itself.

EL: Right.

DK: And so fundamentally, this is about infinity. And the reason this is a little bit more than that, well, first of all, obviously, the ball here is not countable, right? We're not dealing with a countable number of points. This is uncountable. So now we're talking about the continuum in terms of the cardinality of the points. But I think then the surprising thing is that it works out geometrically. So it's not just about cardinality, but you can do this geometrically. And so you can actually define these sets. And, you know, the word pieces, when we say you can split it up into a number of finite number of pieces. I think the record is somewhere under 10 pieces.

EL: It might be just like five or something. It’s been a while.

DK: But the word pieces is doing a whole lot of work in that statement.

EL: Yes.

DK: And these pieces are not something you could ever do with like a knife and fork or, you know, even define easily. It requires the axiom of choice to define these pieces.

KK: Uh-oh!

DK: So it's really high-level mathematics, to understand how to do these pieces, but the fact that you could do these pieces, and then geometrically it works to just reassemble them, to just rotate and translate these pieces, and get back two balls the size of the origina, that’s just astonishing.

KK: Yeah, that's why I've always had a problem with this. I mean, I, I can read it and understand it and go, yes, you can follow every logical step. But you're right, it doesn't work visually, if you think about it. So can you describe these pieces at all?

DK: They are screwed up. So the analogy I like to make is, you know, if you're working on the interval from zero to one, you know, so first of all, the Banach Tarski paradox does not work in in one dimension. But in terms of these pieces, if you're thinking about the interval from zero to one, you could think of the rational numbers as a piece of that, right? And so it's a piece in the sense that it's part of the whole, it's not a piece in the sense that you could cut it out with a butter knife, or you could model this with a stick of butter or something like that, right? You have to hit 1/2, 1/3, 2/3, you have to hit, you know, 97/101 in there, right? All of those are rational numbers, but you can certainly think of all of the rational points in the interval from zero to one as a single piece. You can talk about them, you can define them, and then you can talk about moving them. And so you can think about it that way, right? So you have all those rational numbers, and you can think of that as one piece. And it certainly is a lot more complicated than that. You know, when you think about Banach, Tarski, one of the things I love to do is with students is to go back and think about what it would mean for a set to be non-measurable. So we can measure things like intervals, but it doesn't take too much mathematics to sort of dive into the fact that there have to be sets — if you have a sense of measure, which works on intervals and things like that, and you want it to have other properties, like when you take two disjoint sets, the measure of the two disjoint sets together should be the sum of the two measures, like really basic properties, it doesn't take too much to be able to prove that there are sets that are non measurable. And once you can prove that, like Oh, then then the world gets really screwed up. Because in the world, in our everyday living, everything seems measurable, in some sense. Like even if you have some screwed up sculpture, you could measure its volume. You dunk it in water and see how the water level rises, right? I mean, it is, it has a measure. And the idea that there is no way to measure something is just incredibly counterintuitive. But once you get that, then it's it's it's a little bit more of a leap, but to understand at a fundamental level that you can define pieces that are so screwed up that you could just rearrange them and get two copies of the original, it’s fantastic.

EL: It’s interesting that you, when you first introduced this, you said it's something about infinity and I remember what — so now I'm thinking I might have taken a real analysis class before I had read this book, because I remember when I read the book, thinking, Oh, this is telling me something about non-measurability and, like, really giving me a concept of what non-measurability does to things. And that's that's how I viewed it. It's a statement about how important measurability is.

DK: And just to be really clear, if these pieces were measurable, right, then we would just have some volume or something like that. And there's no way to double the volume. Right? So you can't you just can't do that. So clearly these pieces, at least some of them have to be non measurable.

KK: Right? So much for your alchemy idea, right?

DK: No more doing this with gold. Okay.

EL: It won’t work on atoms.

DK: There’s this fundamental idea that's permeates mathematics that we can continue to divide things, right? You can get things as small as you want. You see this — this is basically what calculus is all based on, infinitesimally small things. And it's just a reminder that the real world does not work like that. You take a gold atoms, you could keep splitting something up. But eventually, you’ve got one gold atom in each piece and you can't go any smaller than that.

KK: Well, you could, but then

DK: It wouldn't be gold anymore.

KK: Right. All right. So you sort of hinted at this, but why do you love this theorem so much?

DK: I love that it makes you question so many things. I mean, I love paradoxes in general, right? paradoxes are these moments when there's, there's so much cognitive conflict going on and cognitive dissonance, that it forces you, in order to resolve that cognitive dissonance, you have really have to question some other fundamental aspect of the world. So it's something that you were thinking before, is not true, or this paradox is like totally crap, right? So something like that. But in this case, the theorem is true, right? The Banach-Tarski paradox is true. And so it forces you to just go back and question some fundamental ideas about the world. And I love that there are statements like that, that can force you to go back and question so many things. I think we as humans need to do better at this. There are so many things that we just accept, as, as we take for granted, right, and we take them for granted as if they are true with a capital T. And in this case, it's about, like, you can measure all things. But of course, in our in our everyday lives, it's things about the world, it's things about people, it's things about politics, it's things about, you know, topical issues. And we grow up, and it's so normal that we think these things are just part of the world with a capital T on truth. And I love those moments that force us to go back and question those those fundamental “truths,” which all of a sudden turn out to be assumptions, some of which may not be right, or some of which we might want to reassess.

KK: So maybe you're arguing that we should study more mathematics to make the world better.

DK: We certainly should do that, Kevin.

KK: To build a better citizenry, right?

DK: Certainly if we all understood mathematics better, we would have been better off during this pandemic. That's certainly true.

EL: I have a question. So as you mentioned, this is called a paradox often. Do you think it is a paradox?

DK: Yeah, that's a great question. You know, paradox has a couple of different meanings. One is sort of this deep philosophical meaning, like, is it really something which is somehow both true and false simultaneously? And, and this is not in that sense, a paradox. It is a paradox in a sort of weaker sort of more everyday sense of that word where it really throws us into into some cognitive dissonance that forces us to question other things. We can't hold both true that like everything can be measurable and things have volume and you can take a ball, split it into six pieces, rearrange them and get two balls. Those are fundamentally in conflict, and one of them has to go.

KK: It does rely on choice, though, right? So there's something to argue about there. You know, there are those people who deny the axiom of choice.

DK: They're few and far between in the math community, but they're out there. Yeah, they're not measure zero, so to speak. So yeah, so it does, it does use the axiom of choice. So this idea that you can, you know, you can make infinitely many choices. And you can see in the proof where you have to do that. So you end up with uncountably many sets and you choose one point from each one of those uncountably many sets, and that's part of the way you get one of the sets that creates the the Banach-Tarski paradox.

KK: Yeah.

EL: I guess I'm a little surprised that the Banach-Tarski paradox hasn't made more people reject the axiom of choice, to say like, Okay, well, clearly, you can't do this. So, therefore, what is this relying on? Well, it's relying on the axiom of choice.

KK: Well, there's a lot of things about like that, like the Tychonoff product theorem. So that's equivalent to the axiom of choice. So do you want your product of compact spaces to be compact or not, you know, I mean, I don't know.

DK: Yes, I think Evelyn, you're hitting upon this fundamental thing: at a very deep level, math gets weird. Yeah. Right. And you can have things. I mean, you see that in, you know, in Gödel’s work, you see like, well, is that true? Well, you know, it can either be true or false. What's your pleasure today? Right? To the axioms or we can reject it. And, and fundamentally, it kind of doesn't matter. But you know, go ahead like, is there an infinity between the integers and the real line? Like, oh, you know, take your Take your pick?

EL: Yeah, what seems more more useful to you right now? Or what sounds like more fun to play with?

KK: Sure.

DK: And in some sense, that that is kind of the beauty of math, because so much of what mathematicians do is based on wherever you want to start. It’s theoretical, like, oh, well, if we start here, this is where we get. If we start at a different place, we get this other thing. And so you can hold those both in your head, despite the fact that maybe you can't have the hold them simultaneously. Like, either the axiom of choice, you take it to be true or you take it to be false, but you can't take it to really be both. Then things break down. But you can you can do a lot of mathematics either way.

KK: Yeah. So another thing we like to do on this podcast is invite our guests to pair their theorem with something. So I'm curious, what have you chosen to pair with this paradox?

DK: So there's this piece that I played, I think I first played this when I was in high school. I'm a violinist and Evelyn and I share this as string players, but it's called the Enigma Variations. It's by Edward Elgar. And it's a really interesting piece with sort of a fascinating story behind it. And the story tells, or explains the name, enigma. So the idea is that Edward Elgar was sitting there, and he played this little melody, and he was sitting at the keyboard — this is in, like, 1898, late 19th century. And he started riffing on this on this melody, right? And he's like, oh, okay, like, we could play it this way, or this way. And then he started putting names on it, like, Oh, my friend would play it this way. Or, you know, my wife, she would play it in this way. And then he sort of just kept taking that further and further, and he ended up writing a bunch of different movements based on how different people in his life, he imagined would play this particular melody. And one of the it's sort of ingenious if you want something to go viral, I guess. But he never, he never lays out what the what the melody was. And so that's the enigma part of it. Somehow out there somewhere is this melody, and all you're hearing is different people's take on it. He does tell you who the people are, and so some of these movements have initials telling you who some of them are. Some are explicit with names. But all you have is that, right? And it's beautiful music and when you listen to it, some of the some of the movements are really fast loud and really exciting. And others are just incredibly slow and languishing. And it's hard to imagine that all of that could sort of in some sense be based on a single melody, and somehow it is. That's what I'd like to pair it with, the Enigma Variations.

KK: Do you have a favorite so that we can insert a clip?

DK: Um, let's see. I do. So probably the most famous of these is called Nimrod. It's an incredibly slow movement. I've played it a bunch of times. It's the sort of thing that, you know, if you want to get people to cry in a movie you could play a little bit of Nimrod. I don't think I've ever successfully been on stage playing this without crying. It is that emotional come and it's also, for me it's more emotional if you play it slower, so it's incredibly evocative. [Clip of the Nimrod variation]

KK: Do you still play?

DK: I do. I do. Well, in theory. We’re in this pandemic where I haven’t been able to play much. But yeah, I play in a local community orchestra and play with my daughter and yeah, I've had some had some fun over during the pandemic, by myself playing, picking up some old pieces and playing them.

KK: Very cool.

EL: Yeah, I was thinking probably everyone has heard Nimrod without realizing it, because it's in the background. If it's not Adagio, for strings, it's Nimrod in the background of that, you know, swelling, emotional scene, your farewell or someone’s dying or whatever. You know, it’s there.

DK: And when you hear it, you can hear little hints of Pomp and Circumstance, which we hear all the time during graduations. And so you can see like, yeah, those are, those are similar sort of in structure, in the melodies and the harmonies and how they fit together,

KK: Right, plus it’s public domain by this point, so you can just throw it into movies pretty easily. Right? That's true, too.

EL: Yeah, I guess you're the practical one here.

DK: Yeah, that's important when you're doing things like podcasts. Yeah, yeah.

KK: Yeah. All right. So um, that's a really good pairing. I like that a lot. So we also like to give our guests a chance to promote anything they want to promote. So So we've talked about the Dana Center a little bit. Anything you want to pitch? Where can we find you online?

DK: You can find me at the at the date of the Charles Dana center, it's UT Dana Center. dot.i.com. I think maybe .edu I'm not sure actually sure. [Editor’s note: it’s neither! It’s actually https://www.utdanacenter.org/], but it’s a pretty easy web search to find us. You know, we're trying to make sure that mathematicians, that everybody has an opportunity to see themselves as a mathematician, and that everybody has access to the right math for them at the right time with the right supports. And so much of the math community, so much of our curriculum is grounded in things that are really old. T algebra, geometry, algebra II sequence was decided in 1892 by a group of 10 white men in the northeast who decided that that would be the right thing. And this focus on calculus comes out of the Cold War, it comes out of the need to produce a small number of engineers who are going to work pencil and paper, and then with massive computers that they could program, to win the Cold War, to beat the Soviets into space and to do all of that. That's no longer the world we live in. And we need to expand what we think of as mathematics. And it's not that calculus isn't important, it still is, but maybe not sending everybody in that direction would be better. And so some students would be much better off if they took statistics. Other students, Quantitative Literacy would be great. We're swimming in so much data, we don't know what to do with and the careers out there that deal with data are taking off like you wouldn't believe. What are the tools we need to give students so that they can deal with that?

KK: Yeah.

DK: All right. So all of these things are just updates that we need to do for their math curriculum. And and updating a system that's so complicated, with so many moving parts is difficult. And so that's part of what we work on at the Dana Center. And if anybody's out there who wants to help work on those things and at the same time, make sure all of these systems are equitable, because we know they haven't been in the past, and make sure all students have access to high-quality math instruction independent of your zip code, or who your parents are your economic circumstances. Those are the sorts of things that we work on at the Dana Center.

KK: Yeah, that's really important work. I mean, it's funny to hear you talk about this, because I've been saying the same thing. You know, the math degree that we still give out more or less is what I got 30 years ago, which isn't different than what they were handing out 30 years before that. I mean, things have changed. And we're still sort of stuck that way. It's unfortunate. And they're still doing it on the high schools. You know, my son when he was going through high school, they just marched him in lockstep through algebra, geometry. And he's a musician. I mean, it's great, you know, it's good for him, but I don’t know.

DK: And even some of these topics that are still important, like, I think algebraic thinking is still important. I think a lot of people do algebraic thinking out there all the time. It's just, it's not pencil and paper. It's not symbol manipulation. The most popular place that people use it, their algebraic thinking is when they're working with spreadsheets, right? Writing formulas in spreadsheets is incredibly algebraic. You have these placeholders that are really just variables, things change, some things stay the same. It's highly algebraic. And we could be teaching algebra using that so that everybody, every student coming out of algebra would have a basis for understanding how algebra is going to be used. You know, it's something like 98% of employers want their employees to be able to use spreadsheets. Well, there's a perfect example. We could we could tweak the curriculum to take something that right now is seen as kind of useless and make it very useful and in fact vital. And it's still teaching essentially the same core ideas, but it's really approaching them from a very different perspective.

KK: Yeah, yeah. All right.

DK: And as long as you give me the space Kevin, you know talking about mathematics and music, so I I have to take a little space. I did this set of 12 lectures for the Great Courses on math and music. And it's a sort of a tour through the listening of music. So like when when we record something, and that music eventually gets into your ear, like every step on that process involves mathematics. And over the course of 12 lectures, we talk about rhythm, we talk about harmonics, we talk about tuning, we talk about lots of, we even talk about the digital side of it. So there's a ton of mathematics that goes into CDs, there's a lot of mathematics that goes into compression that we're using now, so that we can, you know, actually hear each other from this far away. And all of that, there's just lots of mathematics embedded in that. Yeah, so it's really fun, it was just a real honor to be able to do that series. So you can find that out there in the Great Courses.

KK: Yeah. Do you talk about why you can't have perfect tuning on a piano or anything like that?

DK: Yeah, in fact, I got to demonstrate tuning. So they brought in a baby grand for me to do this on. And so we tuned an octave and pulled it out of tune a little bit and you could hear the beats. Again, like these things that I saw in high school and in a trig class, but I had no idea they were applicable, right? So I mean, you can actually hear — there are trig identities that tell you, if you play two notes that have really similar frequencies, that's kind of equivalent to something that goes a wah-wah-wah, it has beats in it. And you can actually do that. And so I played that and tuned it, so that was perfect. And then I took a fifth and we did the mathematics to figure out if you want everything to be sort of in tune, how out of tune do your fifths need to be right, which is, you know, fascinating thing. And I know, Evelyn, you've written about this, what was the column called, like, the saddest thing you know about the natural numbers?

EL: Yeah. Something like that.

DK: Like three is not a power of two or something like that.

EL: It is so sad.

DK: It is sad. And, you know, there are mathematical facts at the heart of that, and so much that we hear in music. And I get notes every week from somebody who has had these nagging questions in the back of their head about how music and math are related. And they watch these 12 lectures, and they're just so thrilled to sort of unpack some of that.

EL: Yeah, it's neat. And it's also it's not just, like, the sound waves and the math, which is definitely part of it. But there's also this extra perception in our ears and our brains that is involved and has some math, but that has some, basically, I don't know, wizardry that our brains do to be like, Oh, when I hear this kind of thing, I often hear this kind of thing with it, so it probably came together, I just missed something there.

DK: There’s a there's a whole section on auditory illusions, which are really fascinating, ways in which our brain can trick itself or, or like, because it's been useful in the past, like, our brain does certain things. And so you can intentionally use that to have these auditory illusions, which are really just fascinating. And my favorite fact about this is that is that essentially, your ear is doing a Fourier transform. When you are listening, your ear is doing this Fourier transform. And you know, just sort of physically doing it and breaking down the sound into its constituent frequencies. And that is just a phenomenally cool idea.

KK: That our brains are that sophisticated.

DK: That our brains somehow involved evolved to do something that we didn't describe mathematically until the 19th century, right? Like, oh, now we know what our brains did just sort of by, you know, random chance and little tweaks to random changes in a DNA sequence. And somehow we got to like, oh, oh, yeah. And that's now called Fourier transform.

EL: Yeah. All right. It's a lot of fun. I feel like we could almost do a whole nother podcast episode about one of these facts.

KK: Probably.

DK: That would be so much fun.

KK: Yeah. Okay. Yeah. Well, maybe we’ll have you back for part two sometime.

DK: That'd be great. Yeah.

KK: Well, this has been great fun, Dave. I'm surprised Banach-Tarski made it this long without being somebody's favorite. This is pretty good.

EL: Yeah. Yeah.

KK: Because we've had we've had repeats before, but but this one, I'm surprised. So thanks for joining us. And yeah.

DK: Thank you guys so much. I love the work that you do, and I really appreciate it.

On this episode of My Favorite Theorem, we welcomed Dave Kung from the Dana Center at the University of Texas at Austin to talk about the Banach-Tarski paradox/theorem. Here are some links you might enjoy:
Kung's website and Twitter account
The Dana Center website
Leonard Wapner's book The Pea and the Sun about the Banach-Tarski paradox
A shorter article by Max Levy explaining the theorem
A primer on the axiom of choice from the Stanford Encyclopedia of Philosophy

The Tychonoff product theorem
Kung's course How Music and Mathematics Relate from the Great Courses

Evelyn's article The Saddest Thing I Know about the Integers, mourning the fact that no power of 3 is also a power of 2

Episode 74 - Priyam Patel

11 februari 2022 | 43 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I almost forgot my name there for a second.

EL: It happens.

KK: I realized, like, I was hesitating, and I was like, “Who am I again?” Yeah, you know — so our listeners don't know, but it's 5:30 where I am, which, you know, doesn't sound late. But I've been at work all day, and now I'm tired.

EL: Yeah. Well, you should have made up something. You know, just tried on a different name for fun just to see.

KK: Well, yeah, so even my parents had the deal that if I was a boy, my dad got to name me. So he went with Kevin Patrick. And if I was a girl, my mother was going to get to name me. And should I tell you what I would have been?

EL: Yeah.

KK: Kandi. Kay Knudson.

EL: Yikes!

KK: Now, I'll let you work out why that would have been terrible for lots of reasons. Already, there are multiple axes along which that is terrible.

EL: Great. Yeah, well, my name if I had been a boy ended up with my younger brother. So it was kind of not that interesting. I mean, if you knew my family, you would be like, Okay, well, that's boring. Anyway, yeah. We are very happy today to have Priyam Patel on the show. So yeah, Priyam, could you introduce yourself a little bit?

Priyam Patel: Sure. So my name is Priyam. I am an assistant professor at the University of Utah, and I have been here for three years. Before that I was around everywhere, it feels like, for my postdoc. I was at UCSB for a few years, before that at Purdue for a few years, And I did my PhD at Rutgers, which now feels like ages ago.

EL: Yeah, you’ve been in, like, every region of the country, though, I guess not central timezone, because Indiana is right on the west edge of Eastern.

KK: That’s right.

PP: Yeah. So I was never in the Central time zone. And that's why — in the summer in Indiana, the sun sets at, like, 10:30pm. It's really bizarre.

KK: You could call that Central Daylight if you wanted to, right?

PP: Yeah. Something like that.

EL: Yeah. And as you mentioned, you've been at Utah for about three years. And you you first got here in fall 2019, and I was gone for most of the fall 2019. And then of course, we all know what happened in 2020. So part of the reason I wanted to invite you is because I feel like I should know you better because you've lived here for three years. But, like, with the weirdness of the past three years, I feel like I haven't gotten to talk with you that much. And so of course, obviously the best way to do this is, like, on a podcast that we want to just broadcast to the entire world.

PP: Yeah, perfect. So no private conversation over drinks. Just put me on the podcast.

EL: Yeah. Excellent. So So yes, I'm excited to get to chat with you. And yeah, hopefully we can do this over drinks in a real venue at some point.

KK: Wait a minute, what happened in 2020?

EL: I tried to block it out.

PP: Nothing at all.

EL: For some parts of it, really nothing.

PP: It feels like a whole blur since then. So

KK: I’m not convinced it isn’t still 2020 somehow.

PP: Yeah, yeah.

KK: Alright. Anyway, I'm being weird today, and I apologize. So let’s get to math. So Priyam, you have a favorite theorem. Which is it?

PP: Yeah. So I chose the Brouwer fixed point theorem, which I learned has been done twice already on this podcast.

EL: Yes, I'm very excited to hear more about it because in our emails, you mentioned some aspects of that I wasn't aware of. And so this is very exciting. And this is when people, when we email with people, they’re always like, “well has this been used?” And we're like, “It doesn't matter if it has, you can use it anyway.” We like to talk about theorems because it is interesting, just the different relationships people have with the same math. So for anyone who hasn't been you know, avidly listening and taking notes on every single episode we've done since 2017, can you tell us what the Brouwer fixed point theorem is?

PP: Yeah, so I'm just going to state it for the closed disk because that's the only context that I'm going to talk about it in. But basically, if you take in the plane in our two if you take the closed unit disk, then the theorem says that every continuous map from the disk to itself necessarily has a fixed point. So should I go into detail about what a continuous map? Would that help?

EL: Yeah. Or at least intuitively.

KK: Sure.

PP: So I actually did listen to a lot of the previous podcast episodes while I was preparing. And I like this idea of if you take the unit desk, and you, like, kind of shake it around a little bit, and everything kind of moves in a nice smooth fashion where things don't get sent, like, really far away — so if in a little neighborhood, you’re wiggling, one point is not just going to pop out and end up somewhere else, right? I like that idea of continuity. So if you're wiggling around the disk, the unit disk, and you use any continuous map, somehow one of the points has to stay fixed, so it gets sent to itself. And that's kind of surprising. It feels like if you just move things around enough, something, everything, should get moved off of itself. But in fact, that can't happen. So that's kind of my interpretation of Brouwer’s fixed point theorem.

EL: Yeah. And it's like I guess I always imagine it made of rubber or something. Because you are allowed to, like, stretch and smush a little bit. It doesn’t — because otherwise, you might think, Oh, the only thing you can do is rotate it. So of course, that central point will be fixed. But you could do a lot of other things.

PP: Yeah, absolutely.

EL: And fix some different point.

PP: Yeah, so I think Evelyn has a great point, like, you can spread things out, like you're making it out of like stretchy fabric or material, you can spread things out in one part of the circle, in the unit disk, and then, you know, string things together in another part and that's okay. It's like, you know, just kind of smoothly moving around is the way I think about it.

KK: Yeah, yeah. But something stays put.

PP: Something stays put, which is kind of strange sometimes, actually. And there's like, so many proofs of this theorem, I feel like, and so many different perspectives for proving it. But I do have a favorite proof of that, actually.

KK: Okay, good. Let’s hear it.

PP: So it's unfair, because it uses some algebraic topology. So o be able to get to this point in this in your math life, where you're like, Yeah, this is the proof I like the best, you have to learn some algebraic topology. But essentially, the idea is that when you're in topology, in the field of topology, you're trying to understand when two objects that are made out of bendy, squishable material that you can stretch and shrink, when two of those are really the same. So if you have, let's say, a circle, or a really oblong wiggly circle, those two are the same. It doesn't really matter if one is really beautiful and perfectly symmetric. It's really the same space in topology. So two things that are not the same topologically are the closed unit disk, and just the outer boundary, which is just a circle. Okay, so there's an a thing called an algebraic invariant that you can compute called the fundamental group, that tells you that topologically, formally, these two spaces really aren't the same. And essentially, there's a proof that says, If there wasn't a fixed point, then you could basically take the entire closed unit disc, and shrink every point in the desk to the boundary. This is called a retract. You’re basically saying like, I'm going to retract the entire closed unit disc to just the circle. And retracts are supposed to give you the same fundamental group. And you already know that those two things aren't the same. And so that's my favorite version of this group. And I can slow down on any part of that if you'd like more details.

EL: Yeah, that's really nice. Well, I think maybe a good way to see this is like, you know, that example of turning the circle around, you know, like a record spinning on a record player or something. Like if you took away that central point, everything else can move. And you can also imagine pulling that rubber all the way to the edge, making it into a bike tire or something else like that. (Which is actually topologically different.)

PP: Right, but as soon as you puncture it. So Evelyn's basically saying, let's just take out the center point. But what corresponds to the origin in R2? Well actually, once you do that, there's no contradiction that you derive, right? You can have every point moving. And in fact, that punctured disk and the circle are the same topologically. That retract that you can use to just pull everything to the boundary shows you, actually, that they're the same topologically. So it's just that one— it’s amazing how much like one point can make such a huge difference, right?

EL: Yeah.

PP: Adding in that one point. But yeah, so that's my favorite proof. It's fancy in some ways, but once you know the basic material that leads up to it, it's like a three-line proof, right? Which is kind of incredible.

EL: Yeah, but it's maybe a little bit like, what is the phrase, like using a sledgehammer to kill a mosquito.

PP: Oh yeah.

EL: Once you’ve built all of this fundamental group, then sure, you could just whack that.

PP: Yeah. And it's so funny because in math, typically I am the opposite of a hammer-striker, right? I never use the hammer. I want to understand the nitty-gritty of why you can just explain this using elementary math or something like that, right? But for some reason, when I saw this proof in, like, Hatcher’s algebraic topology book, I was like, Oh my gosh, that just like makes perfect sense to me, like I totally get why now. So it definitely is one of those use a hammer use a sledgehammer to kill a mosquito type of approaches.

KK: I’m actually teaching algebraic topology this year, and so I that is the proof I use for the Bouwer fixed-point theorem. But I use the sledgehammer to prove that every polynomial of odd degree with real coefficients has a root. And I use to use the Lefschetz fixed point theorem to do it. Let’s use the biggest sledgehammer we can find!

PP: It’s awesome. I mean, honestly, if you want for this podcast, if you want to talk about a fun theorem that's called theorem at the end, right? Not just some result, but that's actually named, a lot of the ones you come up with in topology are the fixed-point theorems, right? And I was like, Oh, this one's actually my favorite. And there's a reason for it. So yeah, that makes sense.

EL: So what else do you love about this theorem?

PP: Yeah. So this actually was inspired from talking to a few grad students the other day, but I realized that, you know, I gave them this task, which says, Can you classify all of the isometries of hyperbolic 2-space. Now, that's already a fancy sentence to say. So I can break down all of what that means. But in fact, one of the key ingredients for the approach that is my favorite to solving that problem is to use the Brouwer fixed-point theorem. So I can start off by talking about what hyperbolic space is, and like what metric spaces are. And from there, I can explain what an isometry. It’s kind of similar to a continuous map, but it has a lot more structure and preserves a lot more structure.

So let's start off with just hyperbolic space, shall we? Okay, so the way I think about anything that is a non Euclidean geometry, which hyperbolic geometry is one of those, I have to start thinking about, well, Euclidean geometry first, right? And Euclidean geometry, when I think of that, I think axioms, right? There's Euclid axioms and they're written down. You don't need to know what they are. But the last one is the one that people started saying, let's try to break it and see what happens to these models of geometry that we're sort of studying, right? Like, could we come up with a different interpretation than just Euclidean space. And so if you break the parallel postulate, there's a few different types of geometries you can get that satisfy all the other ones, but they don't satisfy the parallel postulate. And hyperbolic geometry is one of them. So what is a geometry, right? It's a space, like we just talked about, let's say the closed unit disk. And to me geometry, you're studying rigid things like distances, angles. And so you want to have a notion of measuring distance on whatever space you choose. So since we're going to talk about the power fixed point theorem, of course, my space is going to be the closed unit disk. In fact, I'll just start off with the open unit disk for now. So let's just get rid of the boundary. So if I start off with the open unit disk, that is my space. And there is a way of measuring distance on that space. So you can say, oh, put in these two points, I want to know the distance between them. There's a way of measuring distance on it where if you want to go from the center point at the origin, out to one of the boundary points, let's say just (0,1), or (1,0) in the plane, it actually takes you an infinite amount of distance to get there. Okay, so in hyperbolic space, this model of hyperbolic space, which is called the Poincaré disk model, the boundary of the disk is sort of off at infinity. And as you get close to that sort of boundary at infinity, points are getting really, really, really far away. That's what it means to get, you know, closer and closer to infinity, is distances go really big. And so that's the idea of what the Poincaré disk model of hyperbolic to space is. And, of course, if it weren't a podcast, I'd be showing like tons of pictures right now.

EL: Yeah, it is quite attractive. It's just a lovely, appealing model.

PP: Yeah. And like, you can look up all these amazing pictures by Escher. There are these famous paintings where Escher uses the upper half plane model or the disc model, and shows how, like, a bat, or whatever the figure is that he uses to tessellate, a bat of the same area drawn in different parts of the hyperbolic plane can look to our eyes, very different, right? And that's again, coming back to this notion of as you move out towards the boundary of the unit disk, distances are getting really big so the bat would have to look really small to your eye to to have the same area as a bat in the center of the disk. So I highly encourage listeners to Google just hyperbolic space or Escher's hyperbolic paintings, right. And you'll come up with so many things.

EL: Yeah, well, fun fact is that my Twitter profile picture is a tiling of the hyperbolic plane with the Poincaré. with like, a picture of me in it.

PP: Yeah. So I love it. I know Evelyn really loves all of these, like, hyperbolic geometry, topology type thing. So yeah, that's also partly why I chose this topic.

EL: Yeah, you’re definitely speaking my language.

PP: Okay, so that's the idea of what hyperbolic space is. There's so many more things you need to do to sort of gain the intuition of what it feels like to live in hyperbolic space, right? And those are the kinds of things that you build over years of studying it in your life. But the real thing I want to talk about is isometries of the hyperbolic disk model.

So what is an isometry? It's a map of the space to itself. So kind of like that jiggling that we were talking about. But where all of the distances between any two pairs of points remain the same. So a great example that Evelyn already talked about was this rotation around the origin, right? If you rotate around the center point, all of the points, pick any two of them, they actually see the same distance apart. And that's not an easy thing to see, partly because I never told you how to measure distances, right? That completely relies on how we decided to define that, which is the metric on the space, that notion of measuring distances. But if you knew it, and you wrote it down explicitly, you could actually calculate that that rotation is distance-preserving in the Poincaré disk model of hyperbolic space. So that's the idea of what an isometry is.

Okay, so now we want to try to get fancy. And usually, what you do when you have a space is you say, I'm going to try to understand all the isometries of this space. So where does this notion come from? In topology, if we're not talking about geometric structure, and we just kind of care about a space and it's all blobby, and can be stretched and shrunken, we think about all of the symmetries, right? All the topological symmetries of the space. When we're talking about isometries, what we're actually talking about is geometric symmetries of the space, all different ways of moving around this space, where distance hasn't really changed, so you're preserving the geometric structure. It turns out that if you take all of the isometries, you end up with a group. It has a really nice structure, you can compose two of them. But that's not really even important for today, what you really care about always is, can I classify all things of this type? And there are infinitely many of them. It's really hard to classify things when there's a whole infinite set.

EL: Right.

PP: I’m not just putting things in bins, like these are red marbles, and these are blue ones and these are green ones, right? So amazingly, it actually turns out that isometries of H2, the hyperbolic plane, or disk model, only fall into three flavors. They're either elliptic, which are very similar to the rotations that we talked about, or they are the rotations basically. There are parabolics, and there are loxadromics, or sometimes called hyperbolics, which doesn't make sense, because it's confusing. When you're talking about hyperbolic space, calling something a hyperbolic isometry, when you mean a certain type is confusing. So I'll just call it loxadromic, right? And so there's a few things you need to know, like what could the isometry even look like? How could I possibly get equations of these isometries? So you have to work a little bit. But it turns out that you can write down a general formula for a generic isometry of the Poincaré disk model, or the upper half plane model of hyperbolic space. So just as an example, I'm going to switch models. And in fact, when I say I'm going to switch a model of hyperbolic space, what I mean is, I'm just going to go to a different space with another metric on it, but it ends up being the same geometrically. There is a nice map between the two of them where all the geometry is preserved. So I like sometimes the upper half-plane model, because it's really easy to like write down what the isometries are.

EL: Yeah.

PP: So what I'll say is, and I, again, I'd write this down on the board, if I could. But imagine it, close your eyes and imagine it. I'll take four real numbers a, b, c, and d. And all, really orientation-preserving, but let's sweep that under the rug. Well, orientation-preserving isometries of the upper half plane model look like (az+b)/(cz+d), where z is a complex number. Okay, the criteria you need to make sure you satisfy is that ad−bc is 1. Okay, there's another interpretation actually in terms of matrices. Put those four things into a matrix [Editor’s note it’s hard to write a matrix in Word. The top row is the numbers a and b. The bottom row is the numbers c and d.] Well, you're seeing I want ad−bc to be equal to 1, that's SL(2,R). If you multiply the top and the bottom by negative one in the top of the bottom, you're not changing the transformation. So you have to mod out by plus or minus the identity. So we're really looking at PSL(2,R) protect devised special linear space.

EL: Yeah. I actually I do love that because the first time you see these hyperbolic isometries, the az plus b, over cz+d,, and then they're like, oh yeah, ad−bc, you just have this almost spidey sense tingling of like, Okay, that's like a determinant. Why are we doing this? There must be some relationship here with linear algebra.

PP: Yeah. And your spidey sense is totally on point. So I love I love that connection. And I sort of, you know, I always wonder how much detail to go into with these things, since I'm not writing at the board. But I love that, right? Because anybody listening to the podcast should be like, Okay, wait, that's the determinant, just like you did. So, yeah, so there's a little bit of complex analysis that goes into deriving the fact that these are actually isometries, that they map the hyperbolic plane to itself and so on. But once you have a nice general formula, you start to use Brouwer’s fixed-point theorem. So I, this is part of my life, I go back and forth between the models all the time. So I'm sorry if this is getting annoying, but I'm going to go back to the disk model for a second.

Okay. And well, let's think for a second. Right now, what I have is the open disk. And I have a boundary at infinity. And often, when you're working with spaces, you can sort of complete up the space by adding in the boundary. Okay, this is the sort of completion of the Poincaré disk model of hyperbolic space. So now, if I take the hyperbolic disk model with its boundary, I have the closed unit disk, right? And an isometry is way better than a continuous map, but in particular, it is continuous. And so Brouwer’s fixed-point theorem says no matter what map I'm talking about from the disk to itself, the closed disk to itself, I have a fixed point. So if you have a formula, and you know there has to be a fixed point, you should try to solve for those fixed points, right?

KK: Right.

PP: And so I'm going to pop back to the hyperbolic plane model, because that's where we have our nice formula. So I'm just going to try to solve az+b/cz+d=z, right? This is a function, your input is z. Even though it seems weird, because you're like, wait, I'm multiplying by a, then adding b, dividing by… this is really complicated, right? But there are certain points for which you put it in, and depending on a, b, c, and d, you spit out the same number, right? The same complex number. So this is what I tell people to do. This is what my students did the other day. They said, How am I supposed to approach this? And that's the classic proof, you use the Brouwer fixed-point theorem, and you start to solve for az+b/cz+d=z. The key point that you use after you're doing all the algebraic manipulations is that ad−bc is always 1. So a ton of stuff cancels out as you're solving, right? It becomes a very nice equation. But what am I doing, actually? If I cross-multiply, right, multiply by cz+d on both sides, and then move everything over to one side, I'm getting a degree two polynomial in the z coordinate. How do you solve any degree two polynomial? You use the quadratic equation! So the bane of some people's existence when they’re going through high school, they’re like, I'll never use this? Well, first, if you become a mathematician, you're definitely going to use it.

EL: Yes.

PP: But I love when things that you learn when you're so mathematically young still come into play when you're doing really sophisticated math. I think that's really cool. So okay, we write down the formula. And basically, what it comes down to is what's underneath the square root, right? What is the discriminant? Because when you take the square root of a negative number, you get imaginary things, right? Imaginary numbers, complex numbers with non-trivial imaginary part. If the thing under the under the square root is zero, well, you're just getting one root, right? And then if the thing under the square root is not zero, but it's positive, you're just getting two real roots. Okay, so let's think about that. We have three categories: one real root, two complex roots, or two real roots, right?

The thing is that in the upper half plane model, the thing I never talked about, was that you need the imaginary part to be positive. Okay, so the actual categories are one real fixed point, one complex fixed points in the hyperbolic plane, or two real fixed points. Really, when you're going between the models, what ends up happening is in the upper half plane model, the boundary at infinity is the real number line, it's the x-axis in the complex plane. So that ends up being the boundary of the circle, the unit disk. So we're talking about three cases: one fixed point on the boundary, two fixed points on the boundary, or one fixed point on the interior, which is the complex one. And that's it. That's literally the classification. Because if you have anything more than three fixed points, you can show that your transformation was the Do Nothing transformation. It's the identity.

KK: Right.

PP: So it's not just that the Brouwer fixed-point theorem tells you that you can find fixed points, that there is one, you can actually classify all of the isometries based on these three categories, which I think is like, just incredible. And if you want, I can give you a little bit of a geometric interpretation of what the three isometry is the classes of isometry.

KK: Sure!

EL: Yeah, but I would like to pause and say one of the last classes I taught when I was at the University of Utah was an undergraduate, like introduction to topology class. So we touched on some of this a little bit, and it's like, I kind of want to go back and teach it now and use this for that part of it.

PP: Yeah, it's kind of amazing. Even though it has some sophisticated things going on, you can tell some advanced undergrad students about this stuff, and really show them a lot of beautiful pictures. So when I was in Santa Barbara, I did teach a non-Euclidean geometry class. And, you know, of course, I have to do all the other geometries justice as well.

EL: Eh, do you really?

PP: I know, I know. But I mean, when you get to hyperbolic geometry, though, it's kind of like it's limitless, right? The amount of stuff you can kind of tell and teach students. So I do love that aspect of it. But it's not such an advanced sort of theorem that you're just like, What is this using? Where does it come from? It's like, you need to know the Brouwer fixed-point theorem, you need to know the basics of the model of hyperbolic geometry you're thinking about, and that's basically it, right? Okay, so let's talk about the three sort of classic, I guess I would say, the canonical examples that people give for each of the three categories. So for anybody who's listening that's really into math, and that knows a little bit about algebra, every isometry is actually conjugate to one of these. But they're like the model. They're sort of like, the best-behaved one in each category. What do they look like?

Okay, so for the elliptic ones, we're going to start there, because Evelyn's already told us what they look like, right? In the disk model, they’re all just rotations about the origin. Technically, the fixed point could be anywhere, and it's still kind of a rotation around that fixed point. But again, up to this conjugation, you can move that fixed point to the origin. And then so now you're really just asking if I want to just understand the canonical form of this, I just am going to try to understand isometries of the disk that fix the origin. And I'm going to get that these are rotations, right? You can actually go through and derive the formulas, you know, you say I solved for the root, it's complex. Here it is, let me write down what this might mean. And you can really see which Möbius transformations you're talking about. But that's the canonical way that we think about elliptic ones. And I think the elliptic word has to do with that like sort of rotation, right? But don't quote me on that, because I'm not good at words. I'm good at the math and pictures, but not great at words.

Okay, so let's go to something more interesting. So what are the parabolic ones? These are the ones where you have one fixed point on the boundary. And I'm going to go ahead and use the upper half-plane model again, because they think it's a little bit prettier to see the parabolic one there. So I didn't really say what the upper half-plane model was. So let me go ahead and do that. So the upper half-plane model of hyperbolic space, is you take the entire plane, but then you only think about the upper half part, right? So where the y-coordinate is strictly greater than zero. In the disk model, when we approached the boundary, distances got really big, right? What is the boundary for the upper half plane model? It's the real axis in the complex plane, so the x-axis, and the point all the way out at infinity in the plane itself. So in R2, there's actually an infinity, the one point at infinity, that has to get thrown in there. Okay, so what we can do is say, let's have fun and say that the fixed point in the hyperbolic plane model is infinity.

KK: Sure.

PP: So I'm going to take a straight line, it goes from zero, the origin, (0,0) in the complex plane, and it's just going to go straight up to infinity. Okay, if infinity is fixed, and you have to map these sort of straight lines to straight lines, what you can come up with I mean, I'm waving my hands here, you have to actually do some like algebra and manipulate everything and sort of make sure you're reducing this the right way. But what it turns out to say is that these are all translations. So the maps az+b/cz+d, well, c is 0, d is 1. And it's really just z+b, or something like that. Okay, it's like a translation by some by some number. That's approximately what a parabolic transformation looks like. So translation is something we understand from Euclidean geometry, right? It's just that the way that it affects points in — so what do I want to say? Translation in the Euclidean plane, we understand with that metric. We have the upper half-plane with a different metric. It turns out in this case, that translation is still an isometry. But you have to remember, distances look different. So when you're going to see parabolics in the disk model, things get a little bit more complicated. You have to talk about things called horocycles. And that, I would say, it's better to just look up.

EL: Yeah.

PP: This is where a picture would be very, very useful, right? Okay, and now, the queen of them all is the loxodromic isometries. So this is where we start to see a lot of the connections between hyperbolic geometry and dynamics. So when you have a loxodromic isometry, there are two fixed points on the boundary. With a little bit of work, again, what you can see is that if you take those two fixed points on the boundary, there's actually a, sort of like a shortest line segment going from one to the other. Lines look different in hyperbolic space because the metric is different. But essentially, the way that this isometry acts on the disk model is that one of those fixed points acts as a source. The other one acts as a sink, and everything in the disk model is getting taken away from the source and being pulled towards the sink, actually along the axis — that geodesic axis that you have between the two fixed points, it’s acting as translation along that. So this is a phenomenon in dynamics, more generally called north-south dynamics. You have this source and a sink, and things are moving from the source to the sink in a north-south sort of way. And, yeah, that's my favorite one. Of course, it's the most complicated one. It's the one that comes up the most when you're studying surfaces. So yeah, that that sort of is my — I know, that's just my favorite. That's my favorite type of isometry. I think it makes sense when you're working in hyperbolic geometry, because it comes up all the time.

EL: Yeah, well, that's so fun. So another thing we like to do on the podcast is force our guests to pair their theorem with something in the real world.

KK: “Force.”

EL: So what have you chosen as your pairing today? “Invite” our guests to do that.

PP: Yeah, I actually didn't think about this one so hard, because I have a natural pairing in my life, which is climbing. So I love to climb. So I got into rock climbing when I was a postdoc in Santa Barbara.

KK: Good place.

PP: In fact, one of my most favorite, yeah, it is a good place for climbing. And it just so happened that my mathematical grandfather, Mike Freedman actually, is in Santa Barbara as well. He is a very good rock climber. And he took me on my first like, sort of ropes climbing outside adventure. It was a lot more intense than I thought it was going to be. But he's an intense guy. So I kind of knew what I was getting myself into. But it was such a moment of growth for me. And there are a lot of mathematicians that are attracted to climbing. And there's a reason, right? It's problem-solving. But like, with the physical component put in there, right? So when I'm not problem-solving in my office, or at my home office, I'm usually in the gym, problem-solving climbing problems with my friends.

EL: Yeah, well, and that's, I have, I've like, done one little rock climbing thing. I've never done it, but they actually call them problems, right? Like, figuring out a route is called a problem.

PP: Yeah, absolutely. Yeah, it's very heady you know. Of course, if you ask a mathematician to choose a physical sport to get into, they're like, oh climbing, then I can still use my brain all the time.

KK: When I was a postdoc, I did some. There was a climbing gym in Evanston. And yeah, and you're right. It's very good for your brain. And I always thought that too, but man, my fingers just, it hurt so bad. And I was mostly a cyclist at the time, which was also good because it doesn't require you to really use your brain, like riding a bike is so automatic that I could go out for a ride and think about math while I was riding. I mean, I know you want to get away from the math sometimes, but it was actually a good thing to do for me.

PP: Oh, yeah, no, I totally agree. I was really into running for a while. And I loved that sort of time to decompress and add space to your brain, right? So with climbing, I don't have that sort of flow, I don't reach flow quite as quickly as I do in other exercise. But it's very good for getting out the work stress, I’ve got to say. You just, like, work really, really, really hard physically. And it is very rewarding. I think it's the type of thing where, just like math, in my opinion, you know, “natural ability” is not actually the thing I think that determines how well somebody does in math over time. I think it has a lot to do with how hard you work, right? And if you're in training, taking care of your body, learning things, watching climbers, being very observant, you tend to pick it up pretty quickly. And there's a lot of big, burly guys in the gym who struggle, and then they're sort of surprised when, like, petite women get up and sort of just crush the problem, which I mean, I don't mind that. You know, I think it's a good lesson for everybody. So yeah, I also think, you know, for me, climbing and math are both very dominated by certain genders and races, it's a very white dominated sport, it's a very white male dominated sport, often like math is. Math spaces are very dominated in those same ways. And a lot of my work in math has been to promote diversity and equity and justice, really, in my math communities wherever I exist. And it turns out that that extends to my climbing communities as well, because I co-founded a group called Color the Wasatch, which is an affinity group for people of color climbers in the Wasatch Valley. So it's been really great, and I've learned a lot from that. And it's very similar. When you have people around you that have similar experiences to you, it can be so enriching, and it can be such a relief to just sort of feel yourself relax, and feel just comfortable in your own skin wherever you might exist. And that's a big one that I sort of learned very recently. I’ve sort of always been into sort of activism in the math community. But this was a real first big thing I did in my community outside of my work as a university professor, and it's incredibly rewarding. And it's kind of taken off. It's been great.

KK: Cool.

EL: And we also like to give our guests a chance to talk about anything, you know, you'd like to plug. And actually, I think I just saw on Twitter, some people tweeting about the Roots of Unity conference that is happening. Is that — I don't know if conference is the right word for it — happening this summer. And I think this episode will probably be published in time for people who learned about it to apply,

KK: I think next week, this will be out.

PP: Yeah, that’s great. Because the application deadline for the Roots of Unity workshop is actually February 15, which is great. As you said, it's going to be out in time, the podcast will be out in time. So the Roots of Unity workshop, I am actually co-organizing with phenomenal women in math who I really looked up to, actually. And we designed this workshop to sort of support people who might not see people that look like them at their home institutions early on in their graduate career. So there's amazing programs out there, like EDGE, the EDGE network for people going into grad school, there are amazing research-focused conferences, like the Women in Numbers group, we actually have a Women in Geometry, Groups and Dynamics group now. And we felt that there was a sort of gap in between. And it's very hard, speaking from personal experiences, to be the only woman of color in your department or in your graduate cohort. And so we're really aiming to support anybody who would benefit from this kind of kind of program, but especially gearing it towards people who don't get that same training or preparation or encouragement at their home institutions, especially women of color. So it's going to be a professional development and research development workshop. And one of the things that we're doing is we're sort of helping grad students learn how to read papers, because like, gosh, we just are given papers and said, Go read this, right? And there is a skill to reading a paper well, I think. And so that's one of the big things that we're focusing on, is that transition between early coursework that feels very much like an extension of the undergraduate curriculum, and then into this whole new world that really requires a big pivot mentally, of reading papers and coming up with research problems and having a good network of support and collaborators as you do that. So that's occurring in June of this year at the IMA, so the program is you know in the process, the schedule is in the process of being set. But yeah, applications are open, and we would love to see lots and lots of applications.

KK: Excellent. All right. Where can we find you elsewhere online?

PP: So I am on Twitter. What is my handle? Evelyn, do you know what my handle is?

EL: Um…

PP: I think it might be priyam886. [Editor’s note: It is!]

EL: Maybe?

PP: That might be it. But you know, y'all can post it if you want on the website, eventually. My website actually has a link to the Roots of unity workshop. So that's patelp.com. And I also have a little page actually about Color the Wasatch there as well. So if people are interested in the climbing aspect of things also, that's that's all on my website.

KK: Oh, cool. All right. Well, this has been a lot of fun. I always like learning new things about the Brouwer fixed-point theorem. You know, I've taught complex analysis, and so I've thought about — I think I've shown my students sort of how linear fractional transformations, which are these isometries, are acting on the upper half plane, but I never thought about it in terms of isometries. So this is good for the next time I do it.

PP: Yeah, absolutely. So those linear fractional transformations are like the basic ingredient you need, right? Once you know those, you say which ones are actually going to map the upper half plane to itself, and then which ones are going to be distance-preserving, and everything falls out from there, so it's really nice.

EL: Yeah, this is a lot of fun.

KK: Yeah. Thanks.

[outro]

On this episode of My Favorite Theorem, we're revisiting the popular Brouwer fixed-point theorem with Priyam Patel of the University of Utah. Below are some links you might enjoy after you listen.
Patel's website and Twitter profile
Our previous episodes about the Brouwer fixed point theorem with Francis Su and Holly Krieger
A pdf of Allen Hatcher's algebraic topology book (available, legally, for free!)
The Lefschetz fixed-point theorem
Douglas Dunham's page about Escher and hyperbolic geometry
A blog post Evelyn wrote about putting pictures into the hyperbolic plane
Information about the Roots of Unity workshop (application deadline: February 15, 2022; if you're listening to this in later years, poke around and see if it's happening again!)

Episode 73 - Courtney Gibbons

13 januari 2022 | 42 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined today by my fabulous co-host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City where we are preparing for another snowstorm this week after we had one last week, which is great because we are so low on water right now and we need every bit of precipitation. So even though I'm from Texas, and I don't naturally love shoveling snow or being below 50 degrees, I am thrilled that we're supposed to get snow tonight.

KK: So when I lived in Michigan — you know, I grew up in North Carolina, so snow was a thing, but we didn't shovel it. We just sort of lived with it — and I had a neighbor across the street who was in his 70s. And he had a snowblower and he let me use it and I thought this is amazing. So if you and John haven't invested in a snowblower yet, you know, maybe it's time.

EL: But we'll see. Climate change means that we might have to do less and less snow shoveling.

KK: Well, it's true. Actually, I remember growing up, you know skiing was a thing in North Carolina and I think you might still be able to but like the natural snow ski resorts, kind of they have to manufacture all their snow now. It's, it's things have changed even in my lifetime, but it's not real. As we're told, it's not real. Sorry to editorialize. Anyway, let's talk math. Today, we are pleased to welcome Courtney Gibbons. Why don't you introduce yourself?

Courtney Gibbons: Hi, I'm Courtney Gibbons. I will see far more snow up here in Clinton, New York, then either of you, I think.

EL: Definitely.

CG: I just sent in my plowing contract for the year. So that's awesome. Make sure that I don't have to shovel or snowblow my own driveway, which is not long. I'm a professor. I'm an associate professor of mathematics at Hamilton College up here in beautiful Clinton, New York. There is a Hamilton, New York, but that's where Colgate is. So don't get them confused.

KK: Oh, right.

CG: Yeah. It’s weird. It's a strange thing.

EL: Yeah. At least it’s not the whole Indiana University of Pennsylvania thing because that is not okay.

CG: Yeah, no, no. I think there was an incident on one of our campuses where, like, Albany sent some kind of emergency response squad to the wrong Hamilton. But they're only 20 minutes away, so it was a quick thing. Yeah.

KK: All right. Well, welcome. So, I mean, well, maybe we just get into it.

EL: Well, I will say that we have talked to Courtney before on the podcast, although extremely briefly, when we, I guess this must have been the joint meetings that 2019 a decade ago. (Hahaha.)

KK: Yeah.

EL: We had people give us, you know, little, like, minute or two sound bites of their favorite theorems. And she did talk about a theorem, although I understand it's not the theorem that she's going to talk about today, which, I mean, you don't have one and only theorem in your life? Come on, Courtney.

CG: I am a lover of many theorems. I think back then I mentioned Hilbert’s Nullstellensatz, which is the beautiful zero point theorem that links roots of polynomials to factors of polynomials over algebraically closed fields. It's beautiful. It's a really nice theorem. I initially thought I was going to talk about Hilbert’s syzygy theorem today, which by the way, syzygy, excellent hangman word.

EL: Yeah.

CG: Unless you're playing with people. You've used that word on before, in which case, their first guess will be Y. But I decided today I wanted to talk about Emmy Noether’s isomorphism theorems, in part because they're usually just called the isomorphism theorems. And Emmy Noether’s attribution gets lost somehow. So I wanted to talk about those today because I'm a huge Noether fan. I mean, I'm also a huge Hilbert fan. You kind of have to be a big fan of both. But these theorems are super cool. They're theorems you could see in your first course in abstract algebra, and that's actually where I first saw them. I'm a commutative algebraist and I do a lot of homological algebra. So I love arrows. I love kernels. I love cokernels. I love images. I love anything you can set up in an exact sequence, and I think this was my first exposure to a theorem that was best explained with a diagram. And I remember at that moment being like, “This is what I want to do! I want to draw these arrows.” And I'm lucky because I got to grow up to do what I want to do.

EL: That’s kind of funny because I loved abstract algebra when I took it in undergrad, and I think as it got more to, like, you know, having all these kernels and cokernels and arrows, that was when I was like, “I just can't do this,” and ended up more in geometry and topology. So, you know, different, different things for different people. That's fine. So yeah, let's get into it. So what are these theorems?

CG: Excellent, well, they are often numbered. I grabbed a couple books off my shelf, and it wasn't consistent, but Rotman and Dummit and Foote, kind of numbered them the same way. So the first one, which is usually the first one that you see, it's true for rings and groups and modules. I most often use it for modules, but I'll state it for groups. And it says that if you've got a homomorphism F from a group G to a group H, then the kernel of that homomorphism is a normal subgroup of your group G. Or if you're working with rings, it's an ideal of your ring, or you know, a submodule of your module. And you can mod out by it. So you take G mod the kernel, and it's going to be isomorphic to the image of your homomorphism. And so if you've got a surjection from G to H, and you are like, “I kind of want to build something isomorphic to H, but built out of the parts of G,” you're like, “Cool, I can just take the kernel and mod out by that and look at the group of cosets of of that normal subgroup.” And you've got this — you're done! You've built this cool isomorphism. And I advertise it to my students as, like, a work-saving thing. Because usually to build an isomorphism, you've got to show one-to-one/injective and onto/surjective. And this is like, well, take the thing you want to be isomorphic to, try to imagine it as a better group, a nicer group, mod the kernel of something, and build that homomorphism. Make it surjective and then you get one-to-one for free from this theorem. So I love this theorem. It's a really, it's a nice theorem. I actually use it. I don't reference it, but I think when you do the first step of finding a free resolution, which is which is what I do, it’s like my bread and butter, I love doing this. If you're calculating it by hand, you take a module, and you surject onto it with a free module. You look at the kernel of that thing, and then you build a map whose image is that kernel. And the big deal here is that your module M is isomorphic to the cokernel of that image map, which is the same thing as what you get from the isomorphism theorem. It's that first free module mod the kernel. So this gives you a nice presentation for a module. You can do this in certain nice cases. And I always sort of give a little thanks to Emmy when I start building free resolutions. I'm like, “I know that this is your theorem in disguise.”

EL: Yeah.

KK: I don’t think I actually knew that attribution, that Emmy Noether was the first person to explicitly notice these. But you know, she's the one who figured out that, you know, homology is a group, right?

CG: Yeah, exactly. So she was thinking about rings and groups. And, you know, a lot of the terminology is thanks to her and Hilbert. When you think about integral domains, I think that was what Hilbert initially called rings. I dug this up at one point for my students, they're like, where don't come from?

EL: That makes sense, right? Because I think of a ring as something that's like the integers.

CG: Yeah, it is really. Yeah. And like when people started generalizing to super bonkers weird examples, like, you know, the ring of quaternions and stuff, you're like, okay, so everything isn't quite like the integers. But we've got these integral domains, integral being the “like the integers” adjective and domain, I think of like, where stuff lives. The stuff that's like the integers lives here.

EL: Yes.

CG: Which is nice. But yeah, these these are, these are attributed to me. And it really bugs me to see them without her name attached since we have Hilbert’s syzygy theorem and Hilbert’s Nullstellensatz and it's like, but what about Emmy?

KK: Yeah, she she, she doesn't get the recognition she deserves. I mean, everybody knows like, she's like a mathematics mathematician. But yeah, she she definitely doesn't get credit a lot of the times.

EL: Yeah, I will say I am so, so tired of the headline, “The most important mathematician you've never heard of,” and I read this, and I know that I know more mathematicians than your average person. But like, if it's Emmy Noether, I’m just like, come on. You can't say you’ve never heard of her.

KK: Even the physicists.

EL: A lot of people haven’t heard of her. But yeah.

CG: But it's also interesting the way she's talked about. Because at the time, of course, it was difficult to be a woman in math even though she somehow was able to be a professor, although unpaid. But you know, you look at the way people described her and there's that one guy, I forget, who was like, “I can testify that she was a mathematician, but I can't testify to the fact that she was a woman.” You know. And it's like she was accepted because she was so unfeminine. And it was not threatening to the status quo of men doing math to have this, in their words, coarse, rough, simple soul wearing men’s shoes, blah, blah, blah, great heft, among them, because it's like, well, she's basically a man. So that always bothered me, too. I'm not a particularly feminine person, but occasionally, I do feel like I get a little bit of a brush off because they're like, “Oh, you're so cute.” I'm not cute! I'm a big strong mathematician!

EL: Yeah.

CG: That’s okay. I'm aging and rapidly getting less cute.

EL: You’ll become invisible soon.

CG: Yeah, I know. I'm really I'm excited for that. Yeah, I think I'll enjoy that better than the like, “Oh, aren't you a cute little mathematician?”

EL: Yeah. All right. So…

KK: So there's a first isomorphism theorem. So there must be a second.

CG: There’s a second, there's a third, there's a fourth, there's probably a fifth, then at some point, you just start calling them nth. And I mean, I think I've only seen four. I'm only going to talk about the first three, because I actually don't really know what the fourth one says. There's no quiz at the end. But there's homework. The fourth, look up the fourth. Okay, so the second isomorphism theorem is kind of a special case of the first one. You can prove it using the first one. It says that if you've got a group G, and you've got a normal subgroup, and you've got — we’ll call it N for normal — and you have another subgroup doesn't have to be normal, so we'll just call it S for subgroup, or Su or whatever. This says a couple of things. It says that the product of S and N is a group (and you form the product the way you would expect; you just multiply pairwise elements together, things from S with things from N). The intersection S intersect N is a normal subgroup of S, which is a really nice exercise. And you've got an isomorphism between the product SN mod N, and the quotient group S mod S intersect N. So this is a way of sort of saying like you've got this one group, maybe you don't understand SN really well, and you're working with SN mod N. Instead, replace it with a group you might understand better, which was your original group S mod S intersect N, which is a nice, normal subgroup. So it gives you some opportunity to work with a nicer group to understand the one you don't know. And the proof of this is you build a surjection from S to SN mod N: you’re going to just sort of send S to its coset representative over there and argue that it's onto. And then you calculate that the kernel is S intersect N and then you're like, boom, first isomorphism theorem, take it away. Which is, which is really cool.

KK: It’s kind of a corollary.

CG: Yeah, it's kind of a corollary, but the diagram is super cool. It looks like a diamond because you've got S and S intersect N and then SN and and then you've got all these things all over, the arrows. I love the arrows! And you can put the group G at the top and you put the identity at the bottom, and then you've got like a nice lattice. So I think it even is sometimes maybe called the lattice theorem. But that may. Yeah, I think so. But I, I wouldn't bet your final exam on that if you need to say something.

EL: Yeah. Well, and lattices are a different thing also. So I would be confused if I called it the lattice theorem. I’d think it was about like lattices in the plane or something.

CG: Right. Yeah. I think the idea is it looks like like a lattice. And most algebraists are like, “Oh, cool. We can call that a lattice.”

KK: It is a lattice though, right? The set of subgroups is a lattice. I mean, it’s a partially ordered set with meets and joins and blah, blah, blah.

CG: It is. Yeah, and what's funny is that the person I learned abstract algebra from in undergrad studied lattice-ordered stuff. I forget exactly. It was algebraic stuff. Now I'm of course blanking on it. That was Marlow Anderson, and I am the retirement replacement for Bob Redfield at Hamilton, who studied lattice-ordered subgroups. So I inhabit this space where lattices surround me. It’s kind of exciting.

KK: Yeah. But you're right. This one's a little more obscure. I can't remember ever using this in life except for, like doing homework?

CG: Yeah, I don't recognize a place in my life where I've used this one. If I have I was ignorant of the fact that that was what was making things work, which is not unusual. This is why I need to have collaborators. So they can be like, “Ah, Gibbons.” Hope only gets you so far.

KK: Right? All right. All right, number three.

CG: So number three, I think this one is low-key my favorite even though again, this is not one — actually no, I did use this. We used this in my modern algebra class this semester. This says let's say you've got a group G, and you've got a normal subgroup K in G. And you've got a normal subgroup N in K, so you're normal all the way up. This says that if you take G mod N mod K mod N, which kind of looks like — if you think of it as fractions, like if you're taking 1/2 then dividing by 3/2, this basically says you can “cancel” the denominators, and you're going to get G mod K. Which is really nice, because if you think about what G mod N mod K mod N is, G mod N is a group of cosets and K mod N is a subgroup of cosets. And now you're making cosets out of cosets. And that's a level of abstraction that you don't really want to work with directly, and so having this isomorphism that's like, that’s the same thing as G mod K, just work over there, that’s it's a really powerful thing. And in my class, we're doing a bunch of Galois stuff. Like, our first course in algebra follows that the story of why the quintic isn't solvable by radicals, basically, why there's no nice quintic formula that looks like the quadratic formula. And when we get into the solvable groups thing, and you're building these chains of subgroups, and you want to do stuff with them, we got on this sort of sidetrack of like, well, what if you wanted to mod out by one of the modded out things? Yeah, there’s an isomorphism theorem. This is so cool.

EL: So yeah, I remember seeing this and possibly having it like a homework question or something in abstract algebra, probably in grad school. And I remember feeling like I was getting away with something. When you could just like, like, cancel the denominators, which is not what this actually is.

CG: It’s one of those things where like, I'd be like, sure, that should work. And that's the hope I'm talking about, like, you’ve got to check the details. But again, this one is one you prove with the first isomorphism theorem. You set up your surjection from G mod N to G mod K, and you show all the stuff that's going to go to zero there was the stuff in K mod N. So this is another nice one, but it just looks so cool. You're like, “I can cancel the denominators. Yes, life is good.” You know, some people worry that mathematics is fundamentally flawed, and we don't know where the error is. And I'm like, it can't be. Look, this worked out. We can cancel denominators. That’s got to be on solid ground! It worked out exactly how we think it should.

EL: I love that. That's a great attitude to go through life with.

CG: W could worry about the fundamental flaws, or we could enjoy the wins where denominators cancel. I'd say we've done pretty well for ourselves. We've built this in a sensible way.

KK: I sort of feel like even if we do find the place where it's bad, it doesn't mean that bridges are going to fall down. Right? I mean, most of what we’ve built it seems okay. I mean, yeah, yeah.

EL: It will probably only break something like what Kameryn talked about with us last month.

KK: Some weird set theory.

EL: Yeah, who cares about that?

CG: Let it burn.

EL: Just kidding, Kameryn.

CG: No, I mean, isn’t there, I don't follow this very much. But I thought there was a modern push to sort of replace the foundations of math, the set theory stuff with category theory instead.

EL: Homotopy?

CG: Yeah, homotopy type theory, which in my mind is just like, you know, an evil cousin of category theory. But I, I don't know. I like homotopy. But I don't know what the homotopy type theory is.

EL: I know. I’m afraid to say it.

CG: It is a little scary to say.

EL: Yeah, I have not been able to figure out the correspondence between what I think of as a homotopy, which is like, “Look, you could drag the little loop around on the doughnut,” and homotopy type theory. Don't tell anyone.

CG: Well, and as someone who studies homology, I'm like, why would we do homotopies? They're kind of yucky. Like homology makes it nicer, right? Homology is the abelian version.

KK: It’s abelian, right.

CG: I mean, I guess, you know, if you actually want to describe stuff in the real world, he can't just live in commutative land, but who wants to live in the real world? That’s what the physicists do.

EL: Yeah, that's why we do math, to not have to do that.

CG: Exactly. Exactly. The real world is full of snow. And climate change deniers.

KK: Sure. Yeah. So it seems like you've had a long love for these theorems, right? This goes way back, right?

CG: This goes way back. For me, I was not actually intending to be a math major. I failed seventh grade math. My elementary school report cards were like, “She's very creative. Not so great at math.” “Asks too many questions,” I think was one of them. My first existential crisis was like, what happens if we keep adding forever, and we run out of names for the numbers, and I was like, I can't go to school. They asked me to add as high as I can, and I got to 100. And I was like, I'm going to stop here because at some point, I know what to do, but I don't know what to say. And that was pretty stressful. For me, I hadn't worked out like the time calculation, like how long it would take me to get to the point where I didn't know the names of the numbers anymore. But that was at a time where like, most of my classmates thought 31 was the biggest number because we only ever counted on the calendar.

So I was really surprised in college. I mean, I was also a college dropout who went back. I was surprised in college to get caught up in math and actually find that I could ask these questions. And my professors weren't like, “Oh, my god, shut up.” They were like, “What a cool question!” And they would actually talk to me, and I think it was the relationships more than the math that drew me in at first. By the time I transferred I was starting in multivariable calculus, having finished my first year elsewhere. And I was like, it's cool. We're gonna parameterize the way a leaf falls from a tree, mathematical poetry, all that, but like, it didn't set my soul on fire. And then I take differential equations, which really didn't set my soul on fire, although it was fun. I had fun time in it. But when I got to algebra, I felt like the language I was trying to speak my whole life was finally available to me. It was all about relationships and these like, very picky, not like picky in the sense of real analysis, but you could talk about really subtle differences in things. Like the difference between an ideal and a subring to me, was this really compelling thing, because they're so similar, but they're just a little bit different. And that little difference makes such a huge difference when you start talking about quotient objects and cosets and things like that. I really loved it. And then when I saw the arrows come out, I was like, I'm hooked. I want to build the isomorphism theorem of friendship. This is the way I want to describe everything in my world. And I just got worse from there. I had a brief flirtation with topology in grad school, and then I realized that you could be a commutative homological algebra person and steal all the cool diagrams, like the Meyer-Vietoris sequence, and all of that, and you could do with polynomials! And, and I was like, all right, that's, that's for me. That's what I'm gonna do.

KK: Very cool. I mean, I'm a pretty algebraic topologist myself. I mean, not not so much anymore. But earlier in my career, I was doing topology, but I was doing, like, homology of groups. So there's some geometry there, but not really, right. You’re really just thinking about algebra all day long.

CG: Yeah, I love that. I love that the tools of homology. I was tempted a little bit by geometric group theory, because it's so pretty and so fun. And I think some of the best, most fun conversations I had in grad school before starting my research work were with my friends who were taking that class too. We could just draw something on the board and argue about it for hours. And that’s one of the things I love most. I think I'm a mathematician — some people are mathematicians because they want to uncover these deep, beautiful, abstract truths. And I'm a mathematician because I like talking to people about math. So all my research work, my job is really like, if I didn't do research, I wouldn't have math to talk to people about, so I've got to keep doing the research and ideally doing with my friends, so we can talk about math. I love talking to my students about math. I love talking to you guys about math. I love talking my family about math, and they're like, “Oh my god. Please don't explain again what an algebraic geophysicist does.” Which is at one point what my mom decided. I said, “I do commutative algebra, which is close to algebraic geometry.” And in her mind, it became algebraic geophysicist.

KK: That sounds cool though.

CG: Which, that sounds like a cool gig. Yeah, like I would do it if I knew what it was. Yeah, like homology of rocks. Yeah, yeah.

EL: Figure out when the next earthquake is gonna be using commutative diagrams.

CG: Yeah. Maybe that's the the new topological data analysis stuff where you're doing persistent homology.

KK: That’s right. Well, you can use that to try to understand the structure of various things like you're looking for cavities inside of these. Yeah. So I mean, there are people who've done these analyses. It's real.

CG: All right. Maybe that's my mid-career pivot. Yeah. Algebraic geophysicist. Okay, joint appointment in the geology department. I get to go on their cool trips where they go hiking. I’m in.

KK: Right. Okay, so part two of this podcast. So well, we don't have a part three. Maybe there is a part three? Anyway, we’ve got three things here. We ask our guests to pair their theorem with something. So what have you chosen to pair the isomorphism theorems with?

CG: Well, I decided the best way to do this would be a clickbait-y Buzzfeed listicle kind of thing. So I'm going to start with theorem number three. Theorem number three was the theorem where you are cancelling denominators. I think it's a really fun and satisfying theorem, and so what I recommend you pair this with is a small batch craft beverage, alcoholic or non, of your choosing because when you use this, you should indulge all your senses and just be overwhelmed with joy that this works out and you could just cross out the denominators. For me, that would be a nice local, unfiltered wheat beer, just sit back, cross those little denominators out and just be like, “Yes, life is good. We have built a good thing.”

EL: That sounds great.

KK: Actually, one of the things I might be looking forward to the most about the joint meetings being in Seattle. [Transcriber’s note: Whoops! The 2021 Joint Mathematics Meetings were canceled/postponed after all.] And Evelyn knows this. My favorite distillery happens to be in Bainbridge Island. And so I think I'm going to make a little side trip to procure their excellent whiskey that I cannot get here. I can only get it there. So this is good. This is a good pairing.

CG: Oh, excellent. Are you taking orders?

KK: Well, I don't know. For you? I guess I could ship you something.

CG: Yeah. I’m not going to make it in person this year. [Narrator: She wasn’t the only one.] But I do love a good whiskey.

KK: Their whiskey is spectacular.

CG: We’re going to have to have an offline conversation about this and compare tasting notes. All right. Theorem two. It's about multiplication and intersections. And what goes better with multiplication than bunnies?

EL: Okay.

CG: And did you know there's a huge intersection of math people with bunnies, especially on Twitter?

KK: Yes.

CG: I think you've got to pair theorem number two with bunnies, you've got to it. It's not a theorem you use super often, but if you do use it, you have to immediately go out and acquire a pet bunny and join the math bunny Twitter crowd. It's just, I don't think there's any way around that. I think that's one of the laws of the universe.

KK: Okay.

CG: I should apologize to the math bunny Twitter folks, because they can infer that this is not the best thing, right? Number one is always the best thing you can you can have. But I hope they'll forgive me when they hear that my pairing for theorem number one, the OG, the original isomorphism theorem, is friends. When you use this theorem, you should think about your friends, you should write a note to your friends, you should talk to your friends. I've got friends on the brain this week because I was just thinking about a couple of friends of mine who passed away pretty young one was in his late 30s and one was in his mid-40s. And you know, I thought about Emmy's life and how she had a pretty tumultuous life, right, trying to make it onto the faculty at a university in the first place and then being shipped out of the faculty to Bryn Mawr’s gain. She ended up at Bryn Mawr.

EL: Basically dodging Nazis.

CG: Yeah, dodging Nazis. We talked about “math is apolitical,” but is it? Bryn Mawr lucked out, I guess? Because they got — I mean, America in general lucked out because we got all these amazing mathematicians who were fleeing Nazi Germany. But that sort of seems to me, and I'm not a math historian, but it seems to me that that must sort of mark the point where mathematics in Germany really took a backseat and why English has become the predominant language of mathematics.

EL: Yeah, I mean, the American mathematicians, the great, the big names from the early 20th century studied in Germany. And now it does very much go the other way. There's even a quote about, like, the center of gravity of research crossed the Atlantic because of the Nazis.

CG: Yeah, so I mean, that must have been such an awful thing to experience, being expelled and the uprooting of all of that. And then she died fairly young. She died at 53. I think it was complications from surgery, a cancer surgery. And, you know, it just it makes me wonder what did she have planned? What were her mathematical plans? How was she about to revolutionize physics again? Her most famous theorem probably is the theorem that talks about conservation of stuff under different types of symmetries. Like when you do something, something is conserved, which is, I guess, super important to physicists. I've never used that one in my own work. Right. But it seems like 53, she must have had so much more stuff planned for herself mathematically, physics-y stuff, personal stuff.

EL: Well, and how would Bryn Mawr have been different? And how would American — you know, would she have stayed at Bryn Mawr? Would she have ended up at a different university? Because I think she was only in the US for a couple of years before that happened. Yeah, she barely even got to teach anyone. I mean, I guess it should really be reversed. People in the US barely got to learn from her. And yeah, it's really tragic when you read about it. I think it was maybe an ovarian cancer surgery or something.

CG: I think that's right.

EL: I'm trying to remember if she's one of the people who seemed to be recovering fine, and then just kind of dropped dead a couple days later, or something like that. I know that I’ve read about it. I don't know for sure.

CG: I read up this morning, actually, to make sure that I had her age right. And it seemed it seems like the internet consensus at least is it was a viral infection. So the surgery went fine. But she had incurred some infection after the surgery and just dropped dead. You think of Maryam Mirzakhani. Also someone who was taken really young with so much math on the horizon. But it's sort of hopeful, right? Like, you know, Emmy really didn't get the recognition she deserved. But Maryam Mirzakhani, of course, got at least some of the recognition she deserved, which is, I guess we're making some progress.

This weekend, my friend and collaborator Nick Baeth passed away from a really rapid pancreatic cancer. I think he got the diagnosis maybe six weeks before he died.

EL: Gosh.

KK: That’s awful.

CG: And I wish I had finished writing the note to him that I was saying to him that it was just such a — he and I are math siblings. He was my math “older brother,” although we were in grad school at different times. But we met through our advisor, Roger Wiegand, who — his students become basically family. I know all my “siblings,” pretty much now and we hang out and they're great. But that's how I started collaborating with Nick. And we started working on some semigroup stuff, which is a little outside of my wheelhouse, but related to some questions that I want to answer in my research life. And it was just so much fun to work with him. We'd started up a new collaboration, and we paused it, obviously, six weeks ago. But I thought I had enough time to tell him just what a wonderful surprise it was to find such a like-minded person to collaborate with, somebody who was super generous with ideas, super generous with coauthorship. It really felt like if you'd had a good conversation with Nick about something, he'd be like, “Do you want to be a coauthor? Can you prove this thing? Let’s actually keep doing it.” And I know that's not always the norm in math, and people are very protective of their ideas, not without reason. But it was just such a joy to work with somebody who actually felt like it was more fun to do math with people, and that was why you did math, than you have to go chasing down these big, outstanding results. Not that what he worked on wasn't important. I mean, he had a Fulbright in Austria. It was pretty cool. And I've just been thinking about that. And you know, this other, my graduate office-mate who died, like, a year and a half ago, who was also really young, and just thinking about for someone like me, who does math, to talk math with friends, I need the reminder that I should tell my friends that I enjoy talking math with them, and I enjoy being friends more often. And so for me every time I sit down and calculate a free resolution, and I do my little “Thanks, Emmy,” I'm going to also jot a note to somebody, just a short note like, “Hey, I was using Emmy Noether’s first isomorphism theorem today, and I thought I'd say hi.”

EL: That’s great. I love that.

KK: So you know, maybe there was a third part of this podcast. We always like to give our guests a chance to plug anything, if they like. Where can we find you on the internets? Or do you have something you really want to promote?

CG: Oh boy. You can find me on Twitter, I am addicted to Twitter. I am @virtualcourtney. And that is my online presence as opposed to my corporeal presence, which is at corporeal Courtney, although I don't know if that's on Twitter? Anyway, I tweet a lot there. My pinned tweet, there is a 10-minute pep talk that I recorded for my students during pandemic times. This is a pep talk I give pretty frequently to students who are maybe in their first math course where it's not just calculus 1-20. We're starting to develop proof techniques, or we're dealing with things that are more abstract than things they've thought about before. Or we're encountering the isomorphism theorems and they're like, “Oh my God, what do these arrows mean?” And they're having this crisis of confidence, like “am I cut out to be a mathematician?” And my little 10-minute pep talk goes into my own bumpy route to becoming a mathematician. I'm plugging it a little bit self-consciously. I put it as an email attachment to my students. And I had put it on YouTube and meant to make it unpublished but clicked the wrong button. And then suddenly it had been shared, and I was getting emails about it and thousands of views. And I was like, I guess I'll leave it up. So I'll plug that for anybody who's feeling a little bit, I don't know, shaky in their mathematician identity, or, you know, not sure that they're allowed to have a mathematician identity. I certainly feel like the last person who was “supposed” to become a mathematician. My initial plan was to be a French major. And I did take my graduate language exams in French. So it's good that I studied enough French to know the subjunctive. But yeah, so I'll plug that and then I'll say one more thing. I am working on writing an open-source, remixable, free WebWork-enabled algebra textbook that takes my predecessor Bob's idea of using Galois theory for the first course in modern algebra, and makes it into an active pedagogy kind of thing, sort of in the model of Active Calculus.

KK: Oh, cool.

EL: Oh, cool.

CG: So I am on sabbatical next academic year, and that's my main project. And if anybody wants to, test WebWork problems, or help out when I'm like, “Okay, why did I think I could write a textbook?” I'm happy to talk math with old friends, new friends, math book friends, WebWork friends, anybody who wants to help make that thing actually happen.

EL: Yeah, and selfishly keep me in the loop about it, I'd love to know about it when it comes out here. You're actually convincing me, and several of our guests are always like, working on me. Like, I should really learn algebra a little better and feel a little more comfortable with some of these ideas. CG: Sure. I’ll keep you in mind as a beta tester. Over the years, I've developed all these active learning worksheets that are the basis for the book. They look fun. My students are — well, they used to smile. Now they’re masked, so I just assumed they're smiling all the time. But they would smile and argue when they were working on them, which is what I had hoped would happen.

EL: Oh, that’s great. Yeah.

CG: You should rope some friends in to learn algebra with.

EL: Yeah!

CG: Because I think that's the most important thing for sticking with it is having pals.

KK: Well, good luck. You know, as somebody who, I’ve written a couple of books, you always reach this point where it's like, “Why am I doing this? What did I get myself into?” But once you push through that, it's fine.

CG: I’m in the planning stage where I have a big wiki essentially, like a big table with, like, here’s chapter one’s outline, here’s all the resources for everything in here and now, it's a matter of actually connecting the things. I mean, PreTeXt is its own special thing, learning learning PreTeXt. But yeah, I think it will be a good experience. I feel a little bad because you know, both my algebra Professor Marlow Anderson and my predecessor Bob Redfield have these great books. And I'm like, “No, I'm not going to use those. I’m going to write my own. Stick it to you guys.” But I think what I haven't seen is a mixture of the active pedagogy with the Galois theory as the thing that links rings, groups, and fields. So that's what I want to create and put out there.

KK: Cool. All right. Well, this has been a lot of fun, Courtney. Thanks for joining us today.

CG: Yeah, absolutely. I have sort of been like, “When are they going to ask me?” for a little while, not thinking like, maybe I could just reach out and be like, “Hey, ask me.” So I was really excited when you did. And I've loved listening to the podcast. And I appreciate you both for for making it happen. It’s awesome.

EL: Well thank you.

KK: Yeah. All right. Well, take care.

CG: Bye.

[outro] >

On this episode of My Favorite Theorem, we were delighted to talk with Courtney Gibbons, a mathematician at Hamilton College, about Emmy Noether's isomorphism theorems. Below are some related links you might find useful.
Courtney Gibbons's website and Twitter account
The Wikipedia article on Noether's isomorphism theorems, which includes a helpful chart describing differences in labeling the theorems
An article about Emmy Noether by astrophysicist Katie Mack and her biography on the MacTutor History of Mathematics Archive
Evelyn's 2017 article in Undark about the effect of Nazism on German mathematics in the 1930s
Our episode with Kameryn Williams
Active Calculus, a free, open-source resource for teaching calculus

Episode 72 - Kameryn Williams

10 december 2021 | 23 min

On this episode of My Favorite Theorem, we had the pleasure of talking with Kameryn Williams from Sam Houston State University about Gödel's condensation lemma. Here are some links you might find interesting after you listen to the episode.
Their website and Twitter account
The Skolem paradox in the Stanford Encyclopedia of Philosophy
Gödel's incompleteness theorems in the Stanford Encyclopedia of Philosophy
Akihiro Kanamori's paper about Gödel and set theory
A shorter and more accessible paper by Kanamori and Juliet Floyd on the same topic
Before the Law by Franz Kafka

Episode 71 - Emily Howard

11 november 2021 | 40 min

Evelyn Lamb: Hello and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer based in Salt Lake City but currently podcasting from my parents’ house in Dallas, which is actually not any warmer than Salt Lake City right now, unfortunately. This is your other host.
Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I'm in my faculty office. I'm usually — I'm the chair of the department. But I'm hiding out in the faculty office today. Actually, I was looking for better wireless. And it seems to be working a little better in here. But it's so weird because I have nothing in this office, like nothing. It's very strange. So anyway, how are things going for you?
EL: Not too bad. Yeah, seeing my family, which is nice, and very excited about today's episode. So let's get right to it! We're happy today — both of us are music lovers, and we're very happy to introduce our guest, Emily Howard, a composer. Emily, do you want to tell us a little bit about yourself?
Emily Howard: Yeah. So I'm based in the UK, in Manchester. I'm originally from Liverpool. And I'm a composer. I love writing for large ensembles, large acoustic ensembles, such as the orchestra. I also write vocal music, choral music, and also chamber music. So a lots of different areas. And I suppose probably the reason that you've got me on here is that I've got a real interest in mathematics. And actually, I have a degree in mathematics and computer science, my undergraduate is in mathematics and computer science. And I suppose that, you know, definitely it's one of the key influences on my work.
EL: Yeah, I was listening to — it might have been the BBC Proms a few years ago — I was listening and saw this piece that I think was called Torus. And I thought, “You don't accidentally name something Torus.” So I decided to try to find out more about this person who had named her composition Torus. And so yeah, I found out that you had a math background, and thought it would be just really fun to talk to you on the podcast. So yeah, can you talk a little bit about the — I know, you've done some collaborations with mathematicians, you know, written pieces, like kind of in conversation with mathematicians in the composition process, and I would love to hear about that.
EH: Yeah, so I mean, I suppose actually, in 2015, I had I think it was a Leverhulme fellowship at the University of Liverpool, working within the mathematics department. I had been invited by Lasse Rempe-Gillen. He's a professor in dynamical systems. And I think he'd been in touch because he had himself played the violin in an amateur orchestra in Liverpool, and they had performed a piece of mine called Mesmerism. Actually, it was after Ada Lovelace. Ada Lovelace used to dabble in all types of things, including mesmerism, sort of a form of early hypnosis. And the piece, I mean, that piece was a piece for solo piano and orchestra, and he was playing in it. And I think he thought it will be great to invite me as someone with a mathematics background back — after 15 years away in the music world— actually back into the maths department. And, I'm so glad that this happened, because going back in and speaking to lots of different mathematicians in a different way, rather than, I don't know, having to take exams and study, I was more being an observer, having amazing conversations about people's research. Regularly I'd speak to someone in one area of mathematics and another, and I suppose I felt — what I realized was that mathematicians often don't understand what each other are speaking about. So I didn't feel so bad about it, that I could sort of dip in and I suppose it helped me to take a more global approach, or to take in more general ideas, because I think that's one thing, perhaps, you lose if you don't practice mathematics regularly, you know, you lose this very detailed approach, and that's kind of annoying. It’s really annoying in many ways, but I also find it annoying when you can’t completely understand something because you'd have to spend quite a few years really thinking about it in depth. But then something is also gained from sitting back and taking a look at everything and absorbing it in a very different way.
So anyway, we worked together on a set of chamber pieces. You can find them on my website. One’s called Leviathan, another is called, well, these are all pieces, exploring ideas from lattice research in dynamical systems. And I suppose they're all based on thinking about perturbations, and thinking about — I’ve got some etudes as well, they’re called Etudes in Dynamical Systems — just trying out very particular things with datasets, but also conceptual ideas about reaching stasis and loops from something. I also spoke a lot with colleagues in mathematical biology and also in topology. And I remember, I think this is where, kind of, talking about Torus, it was kind of born, the idea, because this period of time gave me more confidence to actually do something with mathematics. Just to go back back a bit, after my undergraduate in mathematics, I'd spent a couple of years doing a Masters at the Royal Northern College of Music in composition, and then more years at Manchester University doing a PhD in composition as well. And I suppose, when I was doing this, I really wanted to use these ideas or concepts from mathematics, but I didn't really have a musical technique to do that. I suppose that grew. And at this point, it sort of all merged, and I was able to perhaps achieve something that I wanted to achieve with these mathematical ideas through music. And so yeah, just to go on, you mentioned Torus. And I'm immensely proud of Torus. It's a big orchestral work, it's like 23 minutes long, based on the mathematical shape, a torus, a doughnut.
And this is when I had actually just met a mathematician, Marcus du Sautoy, who I’ve worked with quite quite a lot. He's based at Oxford, and we met because of this thing. And when we spoke first, actually, he was writing about, no, he’d written a play about a torus. And I was sort of saying, “Well, actually my piece is about a torus,” so we really bonded on this. And one of the things that interested me most was, you know, the, the BBC Proms are held at the Royal Albert Hall in London. And the first thing he said was, “Well, that is a torus.” And I just thought, How funny. It really is, because you can run round and round it, you know, it's just just really nice to, you know, hear this.
KK: So I'm especially fascinated here, because my son is a composer. Well, he just finished his undergraduate degree, and now he's in graduate school for composition. And so I'm always — first, I'm encouraged that composers can actually make a living, right, because whenever I would tell a mathematician, one of my friends, “Oh, my son's studying composition,” they say, “Well, how’s he going to live?” And I say, “Well, I don't know,” but clearly it’s possible to figure it out. So I'm super encouraged. Yeah, so Evelyn, you had a real question, though.
EL: Oh, well, I was just saying this is an existence proof then for your son.
KK: Yeah, that's right. That's right.
EL: Yeah, well, I was — you know, music is a very non representational art form, unlike a canvas where you could — I mean, it's still hard to represent mathematical ideas on a painting. But can you say anything about how you do use the form of music to represent mathematical ideas?
EH: I suppose — it's really interesting that you say that, because I completely agree with you actually, that it is very difficult to do that. And I wonder if, even if I, especially in abstract music, if we take aside — you know, if it has text in it, that's a very different thing — but in an abstract form, actually, I wonder if I ever tried to absolutely represent something from mathematics within music, whether in fact, anyone would know. And I suspect, actually, there are cases when I've worked very closely with people and they really know what I'm doing. I think they can certainly tell, but actually, I don't think that that's necessarily happens. And also, it's definitely not what I'm trying to do. Absolutely not. So I suppose my aim is not to represent mathematics. My aim is to — I mean, I love hearing about mathematics, and I'm completely inspired by processes and systems and patterns. And I suppose what I'm doing is taking them, and that’s a catalyst for my creative process. And so I do think something comes through, but I think it's more that I couldn't make what I'm making without doing this. And it's more that something new is occurring that came from thinking about these things. But it's definitely not a representational thing. I mean, that's definitely the case for maybe the more large-scale pieces we could talk about, like Torus.
But I think also there have been a couple of pieces when I have tried my very best to represent ideas, and one of them would be this set of dynamical system etudes. That was one of them. And then another one would be the music of Proof, which is, I wrote a string quartet because Marcus and I were discussing proof and different ways you can prove things: proof by contradiction, by induction. He wanted to, to put those to me, and then I would create responses to different kinds of proofs. And so I suppose that’s as representational as I've been, and actually, it was really useful, because in doing that, then I gained a whole set of like, I thought about — I've never tried to write music before as though I'm solving a mathematical proof. But actually, in doing that, that led me to new places. So that's kind of what I'm trying to do, find new ways to do things and make new sounds.
EL: Yeah, maybe respond more than represent you?
EH: Definitely.
KK: I mean, certainly, some music is sort of, you can tell, it's not deliberately mathematical, I don't think. I think of like, like Steve Reich's minimalism percussion pieces, right?
EH: Clapping Music.
KK: Yeah, they’re so cyclical. And you know, you can kind of, if you think about it as a mathematician, you can kind of imagine, well, if I saw this on the page, it would almost look like — our listeners can't see my hands doing this — but you can imagine sort of, you know, intersecting sine waves and things like that. So I can see how you could do that.
EH: And to take that further, I wonder if all music in some way can be — well, I say reduced, and I don't like to reduce — but you could certainly represent ideas in very complex music, I think mathematically probably. Not that anyone necessarily has. But perhaps the secret to lots of things is in really complex music represented. I mean, I don't know.
KK: Well, there’s a whole journal of mathematics and music. I mean, you could you could certainly, I sat on a PhD committee for composer on campus here, who really was trying to do these very strange time signatures that were sort of approximations of pi and things like that.
EH: Wow.
KK: And I put the question to him. I said, “Okay, I mean, you can make a machine do these. But, you know, can a human do this?” And he said, “Well, no, not really.” But it was interesting.
EL: Can humans do math anyway?
KK: This is not a philosophy podcast. I don't know.
EL: Yeah. Thankfully,.
EH: Just to say on that on that subject. I mean, I've got certain things — there are certain things you do put in notation that are absolutely beautiful mathematically, you know, but in fact, yes, they're not really possible. But they they do make a performer think in a certain way, and it will give you certain results. So I mean, I think there can be, I think they're beautiful, and they can be there. But you just can't expect perfection, perhaps.
EL: Yeah. All right. So we always like to ask our guests, what is your favorite theorem?
EH: I mean, we've kind of been speaking about it. That's, that's the main problem with this. I mean, actually, I would say, rather than a theorem, it's definitely these shapes. So I mean, let's take the torus, but I've got this fascination, I'd say, with thinking about mathematical shapes and thinking about, you know, them being far away, and also thinking about being on them. And I suppose, I mean, so you've got a torus. I mean, so, if you think about the difference between, say, flat geometry and a sphere with the spherical geometry, and then, I mean, there's a pseudosphere, and then I would call, like a negatively curved geometry, I've got a piece actually called Antisphere because I've worked out that an antisphere, there's a word antisphere, which is the same as the pseudosphere, and the word antisphere is a lovely word, and I don't like the word pseudosphere. So I called the piece Antisphere. And I suppose I've just got this — because, you know, my music is usually notated. So I find it very interesting as a process to be in my mind wandering around these shapes. And I suppose, I also think it’s very useful for music because they don't necessarily need to be 3d, I mean music, potentially, you could perhaps hear really high-dimensional mathematics. Because, you know, you could be traveling around and you're not necessarily visualizing it.
So my answer to your question is that I like these mathematical shapes. And I've been thinking about them and really studying them and using them to influence these large-scale orchestral work. So that's Torus, Sphere, and Antisphere. And going forward, I'm now looking at, well, I'm trying to look at Thurston’s eight geometries and pushing myself in this direction with the aid of a number of very kind mathematical friends, because they're very difficult. And yeah, so that's kind of the direction where I’m going.
And I find, you know, I suppose that's what we were saying earlier about the representation thing. So there are a number of stages. So I'm thinking about this tours, I'm thinking about traveling around it. Shall I give an example of the compositional process for writing Torus?
KK: Sure, please.
EL: Yeah.
EH: Okay so Torus is a piece, yes, it was for the BBC Proms. It was performed first in 2016, by the Royal Liverpool Philharmonic Orchestra with Vasily Petrenko. Now I’d say, I wrote it over at least a year or so. And I'd also been thinking a lot about this just previously, thinking about traveling on this torus. And what I was thinking about when I wrote it was the idea that you're traveling round and round one direction of the torus, you know, and I took these very consonant chords, see, I've got major sixths. If you hear it, they’re a very constant chord. And I start with this major sixth, and I travel up one and down one and up another — they're going up in semitones, and down, and they reach a point and they come back again. So that the harmony is very much also shaped like a torus all the way through. And it's skewed, when you hear it, it's more complex than just listening to it timed because I've got a real combined fascination with exponential functions and the idea of really big things becoming absolutely minuscule, but also kind of being related. So, you know, this kind of journey around the torus, the first loop in this orchestral work is about five minutes long. And if you do listen, you'll hear it in the strings. And to my mind, I've got this sort of ever-expanding enclosing torus idea in the strings, but, but maybe on the surface of it, as you go round, you hear different things each time you go round. And that's perhaps in the wind and in the percussion, and in the brass. So it’s almost like, you're on a landscape and it's changing.
Now, the piece works by this — suppose the radius of the doughnut-shaped Torus, sort of shrinking. So you go around it, and it gets quicker and quicker. And you'll hear — I think it's around about maybe 17 minutes in, there’s a viola solo. Because it's really big, and it sort of shrinks down, I think there are seven, sort of around this this way of the torus, and you get this viola solo that encapsulates these major sixth idea because you can just play all of that on one solo viola. And then there's this sort of huge shift in thinking in the piece. And, again, that almost came from thinking about dynamical systems and just completely changing the goalposts, and then you're traveling very fast. And in my mind, we've flipped to that we’re now thinking about going on the other direction of a torus forever and ever. Do you know what I mean?
EL: Okay.
EH: You’ll hear that. It’s a huge moment. And that music seems to completely change. And I don't think anyone listening to it will hear a torus shape, because you can’t, because you're hearing my ideas about journeying around it. Do you see what I mean?
EL: Right.
EH: But actually, it really helped me to think about this thing and you know, it is absolutely about this sort of journeying around on it.
EL: Yeah, I don't know, I'm almost imagining like you could, maybe — I wish I had like a, some kind of inner tube or something. But like, maybe the first loop around is on the top where it's further apart, and then you're kind of almost falling into the middle where the You know, the two sides of the torus, or the hole of the tours is kind of small. And then you you kind of flip the other way and start. The next time I listen to it, I'll have to imagine that kind of journey.
EH: The thing is, I’m sure now I've told you that, you probably will hear it, you know? And I think as well, I mean, it's not, it’s never so obvious, because I'm also doing a couple of other things as well. So each journey around this torus — so I said the first one's about five minutes — you’ll also hear these major sixths alternate in the string, so it’s almost like there's one side and there's another side. So you hear this sort of, they appear to travel like this, and then on the other way, they're sort of all, everything is together, so it's rhythmic unison. And it and there's a sort of written-out rall, so it gets slower and slower. And then you'll hear, it comes around again. So I mean, there’s that and there, and then sometimes as well, one side of the torus. That is going on in the background in my mind, and that's definitely happening. But in fact, there's something else on the surface, like there are these very loud things going on sometimes. And they obliterate this thing, but it's always there. And a bit like, say, a painter might have a layer of something to start the piece, this is my layer, this sort of journey. And then it builds up.
KK: And you totally, you had Evelyn at viola solo.
EL: Yeah, yeah, I do play viola.
EH: Great. You’ll hear that.
EL: Yeah, you're pushing all our buttons. Kevin’s son’s composer thing, my viola thing. You’re just excellent. Yeah. What I was wondering is, do you know, why were you drawn to the torus specifically? Do you know?
EH: It’s a really good question. No! I’m not sure I do.
EL: Well, you sort of mentioned the Sphere and Antisphere and Torus are kind of this set. And those can be the three different two-dimensional geometries. But I'm wondering whether you kind of already liked the torus and just were excited that you could use it here, or if you kind of sought it out because it is this model of flat geometry.
EH: That's very interesting, because I think I was trying — when I was writing Torus, I was already thinking about writing Sphere as well. And, you know, in many ways, there's something too perfect about a sphere.
EL: Yeah. It constrains you. It’s very limiting.
EH: It's a problem. And the other thing is, I was having to think of a title. And actually, the title Torus is just the most beautiful word. And there, you know, I think it's the fact that the sphere was too perfect. And there was a way to sort of have two very different things on this torus. But I think it's also true to say that I hadn't thought until later on to do the set. It emerged, you know. So I was sort of playing around and suddenly, I thought, when I'd written Torus, I thought, “Well, actually, that's quite a really wonderful way to think about things.” And then then came Sphere. And that's a shorter piece. It’s a five-minute piece for chamber orchestra. I suppose, I was thinking more about spherical geometry. As I said, I found that one more difficult because it had to come straight after I was writing Torus. However, a couple of years later, that's when I decided to write this Antisphere. And that's for the London Symphony Orchestra with Simon Rattle. And it was just performed a few months before lockdown. So I was very lucky to have that happen. I'm really grateful. And I'd say that, I mean, it's interesting, because somehow, I feel that my link to the maths in it is stronger with Antisphere. And I think that's just because I've gained experience and maybe confidence. I suppose with Antisphere, I was thinking about harmony. I like using quarter tones. They’re present in a lot of my music over the last, say, 10 years. We’re used to maybe the 12 semitones of the scale, and the quarter tones are just in the middle. And I suppose I was thinking about, you know, the classic angles in a triangle, which add up to 180. And then on the sphere it’s more, and then on the negative curves, it’s less. And actually in Antisphere, I use that so that you get these chords that we might all recognize, like a major chord, but actually it's been shrunken in some way.
EL: Oh!
EH: And so it's weird. And it sounds weird. And also, there's a section in there, it’s a very fast section of a kind of circle of fifths idea. But actually, I think it's a circle of 4 3/4. And they're great, because they last longer. Obviously, you know, these players are amazing, because they can do this in an orchestra ensemble.
EL: I was going to say how did you get — I think I would struggle to play quarter tones on purpose. I’m sure I’ve played some by accident.
EH: I mean, I was really blessed with this string section, but it is an amazing experience. And I think you do feel because you do feel this — I mean, the association of a major chord, I mean, I don’t think I’m using major chords, but whatever I’m using — you can feel it, but then you can certainly feel that something's happened to it. And I really liked that. And another thing I did was the rhythmic thing. We've mentioned Steve Reich, and I like the thought that you've got a regular pulse in our world. But if you're somewhere else, or looking in on a different kind of — well, the thought is that you're the pulse might, if it was like 1-1-1. But in fact, it could go 1-2-4-8. So actually, therefore our perception of this piece of music is not one of a regular pulse. In fact, I have created it as though it is, but it isn't. So we hear a short section and a longer one. And then and by the time you get to here, it's so long, you perceive it as something completely different. And I love that as a compositional process. So I did that. In Antisphere, I've actually taken a chord sequence that I used in Mesmerism, this this early Piano Concerto I wrote in 2011. And that’s, quite regularly, you know, there are a few chords, at the opening, you hear solo piano playing these chords, and they're quite regularly spaced. But in Antisphere, what I've done is I've taken them as the basis, these piano chords, as the basis of these sort of chords that start off like a bit and actually end up as three minute sections. And loads of weird things going on in between you like this with the percussion, there's loads of weird metals and sort of resonant sounds. And so, yeah, as I said, it gives a completely different sort of perception of what's going on.
EL: Yeah, I don't think I have listened to Antisphere yet. But I am now going to seek it out because I really want to hear this — especially if I've got in my mind this idea of, like, hyperbolic triangles with their, you know, curved in — or they look to me like they're curved in. If I actually believed enough, they would look like straight lines to me, because I would really embody the hyperbolic metric.
EH: But and this so this is happening in pitch, but it's also happening in sort of rhythmically and timbre in lots of ways as well. I did write an article, Orchestral Geometries, which I will post, you know, we could put there. And actually, I've been lucky enough to have wonderful recordings of the three pieces, and also the scores as well alongside so yeah.
EL: Oh, wonderful. So yeah, another thing we do in this podcast, which we kind of already done, is we ask our guests to pair their theorem, or mathematical object, with something. And so yes, I kind of assume that you would be pairing it with your compositions based on these, on the torus and these other shapes. But yeah, do you want to add anything else about that, or any other pairings? You know, if it's just a nice cup of tea or something?
EH: Or a doughnut? I was going to pair them with that. And actually, I was going to pair them perhaps with, you know, we could we could play a little clip from one of them. So this clip is a little bit from the orchestral work Torus, and I think, it’s sort of about two thirds through where everything changes. So before then you've been rotating round, the kind of doughnut shape of the sphere, no sorry, the doughnut shape of this torus. And then you go into this viola solo and then you hear the huge perturbation, and that's when you change as though you're rotating. At least when I was writing, I was thinking about rotating around the other part of the torus.
[music clip]
KK: So I really, now I'm sort of curious to know how you’re — I mean, I know you haven't thought about it yet. But pursuing the sort of 3D geometries, that's going to be weird. Have you seen these, people have tried to visualize these things using VR. Have you seen any of these explorers?
EH: Yes, yeah, I have. And actually, if you have any more, I would love to have them because it's very helpful to see them. We've been looking online, and there are quite a few. You know, there are some amazing websites and people, actually programs for doing it. And we were kind of exploring it. But I am interested. And it's a new pursuit, you know, so I'm really interested to, well, think about it more. I think it'll take some time as well.
I'm currently writing a piece. And I was thinking that it will be very nice, now having written these orchestral geometries, to embody sort of a process of moving between the different geometries a bit. So that you could, I mean, no one need necessarily know this, but musically, there might be a difference between this spherical and the — but they could, so I'm really interested in that. So I'll do that. But I also I'm interested in this, is it H3 [hyperbolic 3-space]?
EL: Yeah.
EH: So yeah, I'm interested in that. And it's kind of frustrating, because I'd really like to understand why. And it's quite difficult, I think. I'm not sure. I think it's probably well beyond me. But, you know, the thought of this dodecahedral space and moving around and the twist, I'm really interested in the fact that it would you know, that the spherical that these twists, change what dimension, it’s really interesting.
EL: Yeah, yeah. And I'm wondering how you can use like, the specific — what am I trying to say, like, the opportunities that an orchestra gives you to sort of, you know, can you, like, pass off ideas from one section to another, can that give you a twist? Or, you know, something like that, how you could, I guess, use the tools that you have at hand to kind of explore it in a different way than you might if you were trying to draw it on a piece of paper?
EH: Yeah, I think, I mean, it’s probably better doing that than drawing on a piece of paper actually. Like, I feel there must be something about the way I think that I like — these thoughts, and these kinds of systems and shapes immediately present musical ideas to me. So, absolutely, you know, what I would be interested in is a process that you could audibly hear becomes something different when it's spherically curved, and becomes something different when it's, you know, hyperbolic, and making that more and more extreme. So the math underlies that, and perhaps, maybe it's a process of me imagining going through this, these these dodecahedron and things. But in fact, that will be a layer, and then after that, perhaps then, I do something more musical, as in this has given me this data, but I accentuate it in ways, and perhaps also go against it and things. Because I do think as well with art, usually, you're asking a question rather than answering anything, which is a real difference, I think. Because mathematicians, you're so interested in truth, or mathematical truth, and you’re really bound by it; it’s really important. But I always feel that I'm more interested in what your process is in these wonders, and then I'm just using that to sort of leap somewhere unknown.
KK: Well, so are we. We're leaping into the unknown all the time, we just, then want we want the answer once we get there, right?
EL: I do think there's a similarity in that, you know, the way mathematicians approach things a lot is, you kind of set up these axioms, and this is the rule system I'm going to be working within. And I think that forms of art do that as well, you know, say, this is the aesthetic system that I'm working in, or this is, and maybe they're not as rule-bound as mathematics is, because when you say like, these are the axioms, or these are, you know, whatever, I'm doing, then you're just really stuck with them. And with art, a lot of the times it's about breaking the rules. But I do think, you know, you kind of set up these, sometimes composers can set up compositional rules where, like, Okay, well, I'm writing a fugue, which means that I have, you know, I have this kind of structure. And the allowed things to do are like transposing it or flipping it backwards, or things like this, and say, like, well, I'm working within this form, in this way. So, I mean, I did a lot of music in college, and I was kind of torn between going into math and music. And I think the way that I thought about them kind of tickled the same part of my brain, that’s why I was interested in both things, and ended up, you know, in math instead of music professionally, but I do think we've made these aesthetic or form kind of rules in music or art. And now we're going to work within them, just the way mathematicians do with axioms.
EH: Yeah, I mean, I completely agree with you on that. And I also would say as well that I agree, and actually working closely with professional mathematicians has really kind of opened my eyes to how much they are going, because I don't think you learn that, you know, these are the answers, actually it is a lot of guesswork. And it's really, yeah, leaping, as you say.
EL: Yeah, well, this, this has been a lot of fun to talk with you about this. I really hope our listeners will go find your pieces. And we'll definitely link to your website and the article you talked about and everything. Is there anything else that you want to share, you know, things you want to suggest that they look into or read or any concerts coming up that people could actually attend or livestream?
EH: Yes. So actually, next week, the BBC Philharmonic is playing sphere in Manchester. And that should be on Radio 3, as well. I’m based at the Royal Northern College of Music in Manchester. I’m a professor of composition, and I run, I direct, the PRiSM lab. PRiSM is a research center for Practice and Research in Science and Music. And I'm lucky enough to work there also with colleagues. Marcus du Sautoy joins us and David De Roure, as well. And I suppose we're interested in mathematics meets music meets science meets AI, and there are lots of different types of composers. We have a blog, and that's where this orchestral geometries blog will be. And lots of very exciting, very different things going on there. So I just wanted to mention that and maybe I could also give you a link for that as well.
KK: Sure.
EL: Yeah, that would be great.
KK: We will include it. All right. This has been fantastic. Thanks for joining us, Emily. It's really been great.
EH: Thank you so much for having me. It's been a pleasure.
[outro]

On this episode of the podcast, we were delighted to talk to composer Emily Howard, who uses her mathematics background in her compositions, about her favorite mathematical object, the torus, and the orchestral work it inspired. Below are some links you may enjoy after you listen to (or read) the episode.
Emily Howard's website
Her page about the composition Torus, including a recording by the BBC Radio Orchestra
Her article Orchestra Geometries
The November 11, 2021 BBC Radio concert featuring Howard's composition Sphere
PRiSM, the Royal Northern College of Music Centre for Practice and Research in Science and Musicthat Howard directs
A website visualizing the eight Thurston geometries for 3-dimensional space
An article by Evelyn about the pseudosphere (or antisphere)
Our episode with Emily Riehl, who is relevant to this episode because she is both an Emily and a violist

Episode 70 - Joel David Hamkins

22 september 2021 | 40 min

Evelyn Lamb: Hello, and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, and this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. I like your background, Evelyn. Our listeners can't can't see it. But it looks like you had a nice camping trip somewhere in Utah.

EL: Yes, that's actually a couple years ago — we’re going to describe this in great detail for all the listeners who can't see it, no — this was a camping trip in Dinosaur National Monument on the Utah-Colorado border, which is a really cool place to visit.

KK: Excellent. It’s beautiful over there. Yeah. So how's things?

EL: Oh, not too bad. A bit smoky here, we're getting a lot of wildfire smoke from the west coast. And it, you know, makes some of those outdoor activities that are so fun, a little less fun. So I hope it clears out.

KK: Right. Yeah, well, my big adventure lately was getting my son settled into his new apartment in Vancouver. And it was my first time on an airplane in a year and a half. That was weird. And then of course, Vancouver, such a lovely city, though. We had a good time. So he's all set up and starting grad school and his nice new adventure.

EL: Yeah. And he's kind of, almost as far as he can be in a populated place in North America from you.

KK: That’s correct. That's, like, almost 3000 miles. That's far.

EL: Yeah.

KK: It’s great. Anyway, enough about that.

EL: Yeah. Well, today, we're very happy to have Joel David Hamkins join us. Joel, would you like to introduce yourself and say a little bit about yourself?

Joel David Hamkins: Yeah, sure. I am a mathematician and philosopher at the University of Oxford. Actually it’s something of an identity crisis that I have, whether I'm a mathematician or a philosopher, because my original training was in mathematics. My PhD was in mathematics. And for many, many years, I counted myself as a mathematician. But somehow, over the years, my work became increasingly concerned with philosophical issues, and I managed somehow to turn myself into a philosopher. And so here in Oxford, my main appointment is in the philosophy faculty, although I'm also affiliated with the mathematics department. And so I don't really know what I am, whether a mathematician or a philosopher. I do work in mathematical logic and philosophical logic, really all parts of logic, and especially connected with the mathematics and philosophy of the infinite.

KK: Well, math is just kind of, it's just applied philosophy anyway, right?

EL: This might be a dangerous question. But what are — like, do you feel a big cultural difference between being in a math department and a philosophy department?

JDH: Oh, there's huge cultural differences between math and philosophy, I mean, on many different issues, and they come up again and again, when I'm teaching especially, or interacting with colleagues and so on. I mean, for example, there's a completely different attitude about reading original works. In philosophy, this is very important to read the original authors, but in mathematics, we tend to read the newer accounts, even of old theorems, and maybe for good reason, because oftentimes, those newer accounts, I think, become improved with greater understanding or more connections with other work and so on. Okay, but one can certainly understand the value of reading the original authors. And there's many other issues like that, cultural differences between math and philosophy.

EL: Yeah, well, I, when I was at the University of Utah, I taught math history a couple times. And I was trying to use some original sources there. And it is very difficult to read original sources in math. I don't know if part of that is just because we aren't used to it. But part of it, I do feel like the language, and the way we talk about things changes a lot really quickly. It makes it so that even reading papers from the 1920s or something, you sometimes feel like, “What are they talking about?” And then you find out Oh, they're just talking about degree four polynomials, but they're using terms that you just don't use anymore?

JDH: Yeah, I think that's absolutely right. And oftentimes, though, it's really interesting when there's an old work that uses what we consider to be modern notation. Like if you look at Cantor's original writings on the ordinals, say, it's completely contemporary. His notation, he writes things that contemporary set theorists would be able to understand easily, and even even to the point of using the same Greek letters, alpha and beta to represent ordinals, and so on, which is what we still do today. And so it's quite remarkable when the original authors’ notation survives. That's amazing, I think.

EL: Yeah. So we invited you here to share your favorite theorem. What have you decided to share with us today?

JDH: Well, I found it really difficult to decide what my favorite theorem is. But I want to tell you about one of my most favorite theorems, which is the fundamental theorem of finite games. So this is the theorem, it was proved by Zermelo in 1913. And it's the theorem that asserts that in any two player finite game of perfect information, one of the players has a winning strategy, or else, both players have drawing strategies if it's a game that allows for draws. So that's the theorem.

EL: So what is the definition of a game? This is, like, the most mathematical thing to set, you know, like, yeah, okay, games, we've all played them starting from when we were, you know, two years old or something. But now we need to sit down and define it.

JDH: Yeah. Well, that's one of the things that really excites me about this theorem, because it forces you to grapple with exactly that question. What is a game? What is a finite game? What does it mean? And it's not an easy question to answer. I mean, the theorem itself, I think, is something that you might think is obvious. For example, if you think about the game of chess, well, maybe you would think it's obvious that, look, either one of the players has a winning strategy, or they both have drawing strategies. But when it comes to actually proving that fact, then maybe it's not so obvious, even for such a game as chess. And then you're forced to grapple with exactly the question that Evelyn just asked, you know, what is a finite game? What is a strategy? What is a winning strategy? What does it mean to have a winning strategy, and so on?

So there's a wonderful paradox that surrounds the issue of finite games called the hypergame paradox. So if you have a naive account of what a finite game is, maybe you think a finite game is a game so that all plays of the game finish in finitely many moves or something. Okay, so that seems like a kind of reasonable definition of a finite game. And then there's this game called hypergame. And the way that you play hypergame is, say, if you and I are going to play hyper game, then the first player, maybe you go first, you choose a finite game. And then we play that game. And that's how you play hypergame. So the first player gets to pick which finite game you're going to play. And then you play that game. And if we said every finite game is a game, so that all plays end in finitely many moves, then it seems like this hypergame would be a finite game, because you picked a finite game, and then we play it, and then the game would have to end in finitely many moves. So it seems like hyper game itself is a finite game. But then, the paradox is that if hypergame is a finite game, then you could pick hypergame as your first move.

KK: Right.

JDH: Okay, but then we play hypergame. But we just said when playing hypergame, it's allowed to play hypergame as the first move. So then I would pick hypergame as my first move in that game. And then the next move would be to start playing hypergame. And then you could say, hypergame again. And then I could say hypergame, and so on, and we could all just say hypergame, all day long, forever. But that would be an infinite play. And so what's going on? Because it seems contradictory. We proved first, that every play of hyper game ends in finitely many moves, but then we exhibited a play that didn't end. So that's a kind of paradox that results from being naive about the answer to Evelyn's question, what is a finite game? If you're not clear on what a finite game is, then you're going to end up in this kind of Russell paradox situation.

KK: Exactly. Yeah. That's what I was thinking of, the sets that don't contain themselves. Right.

JDH: Right, exactly. It's actually a bit closer to the what's called the Burali-Forti paradox, which is the paradox of the class of all ordinals is well ordered, but it's bigger than any given ordinal. And so it's something like that. So if you think a lot about what a finite game is, then you're going to be led to the concept of a game tree. And a game tree is the sort of tree of all possible positions that you might get to. So there's the initial position at the start of the game, and the first player has some options that they can move to, and those are the sort of the child nodes of that root node. And then those nodes lead to further nodes for the choices of the second player, and so on. And so you get this game tree. And the thing about a finite game is that — well, one reasonable definition of finite game is that the whole game tree should be finite. So that in finite infinitely many moves, you're going to end up at a leaf node of the tree, and every leaf node should be labeled as a win for one of the players or the other, or as a draw, or whatever the outcomes are. And so if you have the finite tree conception of what it means to be a finite game, then hypergame is not a finite game, because at the first move, there are infinitely many different games you could choose, and so the game tree of hyper game won't be a finite tree, it will be an infinite tree. It will be infinitely branching at its first node.

And so if you have this game tree conception, then a finite game can mean a game with a finite game tree. And then we can understand the fundamental theorem of finite games. So my favorite theorem is the assertion that in any finite game like that, with a finite game tree, then one of the players has a winning strategy, or both players have drawing strategies. And what does it mean to be a strategy? I mean, what is a strategy? If you think about chess strategies or strategies as they're talked about conventionally, then oftentimes, people just mean I kind of heuristic, you know, the strategy of control the center or something, but in mathematics, we want a more precise notion of strategy. So controlling the center isn't really a strategy; that’s just a heuristic. It doesn't tell you actually what to do. So a strategy is a function on the game tree that tells you exactly which moves to make whenever it's your turn. And then a play of the game is basically a branch through the game tree. And it conforms with this strategy if whenever it was your turn at a node in the game tree, then it did, what the strategy was telling you to do at that node. So a strategy is winning if all the plays that conform with that strategy end up in a win for that player.

KK: Okay, so is this the point of view Zermelo took when he proved this?

JDH: So yes, Zermelo didn't have, he didn't quite have it all together. And this is maybe related to the fact that we don't actually read Zermelo’s original paper now when we want to prove the fundamental theorem of finite games because we have a much richer understanding, I think, of this theorem now. I mean, for example, in my in my proof-writing book, I gave three different proofs of this theorem. And Zermelo didn't have the concept of a game tree, or even a finite game in terms of game trees, like I just described. Rather, he was focused specifically on the game of chess, and he was thinking about positions in chess as pictures of the board with the pieces and where they are. But nowadays, we don't really think of positions like that, and there's a kind of problem with thinking about positions like that. Because if you think about a position in chess, like a photograph of the board, then you don't even know whose turn it is, really, because the same position can arise, and it could be different players’ turns, and you don't know, for example, whether the king has moved yet or not, but that's a very important thing, because if the king has moved already, then castling is no longer an option for that player.

KK: Right.

JDH: Or, or you need to know also what the previous move was, in order to apply the en passant rule correctly, and so on. So if you just have the board, you don't know whether en passant is allowed or not. I mean, en passant is one of these finicky rules with the pawn captures where you can take it if the previous player had moved two steps, and your pawn is in a certain situation, then you can capture anyway. So I mean, it's a technical thing, it doesn't matter too much. But the point is that just knowing the photograph of the board doesn't tell you doesn't tell you whose turn it is, and it doesn't tell you all the information that you need to know in order to know what the valid moves are. So we think of now a position in a game is a node in the game tree, and that has all the all of the information that you need.

So Zermelo’s proof was concerned with games that had the property that there were only finitely many possible configurations, like chess. There are only finitely many situations to be in, in chess. And he argued on the basis of that, but really, it amounts to arguing with the finiteness of the game tree in the end.

KK: Yeah. So is this at all related to Nash's equilibrium theorem? So I was scrolling Twitter the other day, as one does when one has nothing else to do, and I saw a tweet that had, it was a screenshot. And it was the entire paper that Nash published in the Proceedings of the National Academy proving his equilibrium theorem. I mean, it's a remarkable thing that it's just a few paragraphs. Is there any connection here?

JDH: Right, so actually, this kind of nomenclature, there are three different subjects connected with game theory. There’s game theory, which, Nash equilibrium and so on is usually considered part of game theory. And then there's another subject standing next to it, which is often called combinatorial game theory, or it's sometimes also called the the theory of games. And this is the the study of actual games like Nim, or chess, or Go, and so on, or the the Conway game values are part of this subject. And then there's a third subject, which I call the the logic of games, which is a study of things like the fundamental theorem of finite games, and the sort of the logical properties. So to my way of thinking, the Nash equilibrium is not directly connected with the theory of games, but rather is a core concept of game theory, which is studying things like the stability of probabilistic strategies, and so on, whereas combinatorial game theory isn't usually about those combinatorial strategies, but about sort of logically perfect strategies, and optimal play, and so on. And that isn't so much connected to my way of thinking with the Nash equilibrium.

KK: So fine, I hand you a finite game. It has a winning strategy. Can you ever hope to find it?

JDH: Oh, I see. Is it computable? Well, yeah, this is the big issue. Even in chess, for example, I mean, chess has a finite game tree. And so in principle, we can, in the sense of computability theory, there is a computable strategy you can prove, because it's a finite game. So the strategy is a finite function, and every finite function like that is definitely computable. And we can even say more about how to find the strategy. I mean, one of the proofs is this backpropagation proof, you look at the game tree, and you propagate. You know that the leaf nodes, the terminal nodes are labeled as a win, and you can propagate that information up the tree. But it turns out that the game tree of chess is so enormous…

KK: Right.

JDH: That it wouldn't fit in the universe, even if you used every single atom to represent a node of the tree. And so in that sense, you can never write a computer program that would compute the perfect chess play. It's just too big, the strategy. If you're really talking about the strategy on the whole game tree, then the game tree is just too enormous to fit in the universe. And so it's not a practical matter. But theoretically, like in terms of Turing computability or something, then of course it's computable. There are some other issues. For example, I've done a lot of work with infinitary games. I mean, this connects my interest with infinity. And it turns out that there are some, some positions, say, in infinite chess, which I've studied, we identified computable positions in infinite chess, for which White has a winning strategy, but but there's no computable winning strategy. So if the players play computably, in other words, according to a computable procedure, then it will be a draw. So it's very interesting, this interaction between optimal play, and computable optimal plays not always, they don't always align.

EL: So I've got to ask you to back up a little bit. What is infinite chess?

KK: Yeah.

JDH: Oh, I see infinite chess. So imagine a chessboard without any border. It just goes forever in all four directions.

EL: So you start with the same, you know, don't start with infinite number — or, yeah, I’ll let you keep going.

JDH: So of course, infinite chess. I mean, we can imagine playing infinite chess, you know, at a café or something, but it's not a game that that you sit down in a café to play. It's a game that mathematicians think, “What would it be like to play if this if the board looked like this, or if it looked like that?” And so there's no standard starting position. You present as a starting position, and you say, “Well, in this position, it has a very interesting property.” So Richard Stanley asked the question, for example, on mathoverflow. And that sparked my interest in this question. Well, one of the things that we proved was that you can have positions in infinite chess that White has a winning strategy, so it's going to win in finitely many moves, it’s going to make checkmate in finitely many moves. But it's not made in n for any n, for any finite n. In other words, Black can make it take as long as he wants, but it's hopeless. In other words, Black can say, “Oh, this time, I know you're going to win this time, but it's going to take you 100 moves.” Or he could say, “This time, it's going to take you a million moves.” For any number, he can delay it that long, but still White is going to win infinitely many moves following the winning strategy. So these are called games with game value omega, and, and then we produce positions with higher game, higher ordinal game values, omega squared, omega cube, and so on, the current record is omega to the fourth. So these transfinite game values come in. And so it's really quite fascinating how that happens.

KK: Well, knowing that you like logic and philosophy so much, now I see why you like these games, right? This is kind of a logic puzzle, in some sense. Yeah. Yeah.

EL: Well, and what made you choose this theorem as your favorite theorem?

JDH: Well, I mean, there's a lot of things about it to like. First of all, what I mentioned already, it forces you to get clear on the definitions, which I find to be interesting. And also it has many different proofs. I mean, as I mentioned, there are already three different proofs, three different elementary proofs. But some of those proofs lead immediately to to stronger theorems. For example, there's a slightly more relaxed notion of finite game, where you have a game tree, so that all plays are finite, but the tree itself doesn't have to be finite. So this would be what's called a well-founded tree. And then these would be called the clopen games, because in the in the product topology, the winning condition, amounts to a clopen set in that case. And then the Gale-Stewart theorem proved in the ‘50s, is that infinitely long games whose winning condition is an open set, those are also determined: one of the players has a winning strategy, open determinacy. And then Tony Martin generalized that to Borel determinacy. So Borel games also have this property. So infinite games whose winning condition is a Borel set in the product space are determined, one of the players has a winning strategy. And then if you ask, Well, maybe all games are determined, in the sense that one of the players has a winning strategy, you whether or not the winning condition is Borel or not. And this is called the axiom of determinacy. And it's refutable from the axiom of choice for games on omega, say, you can refute it. But it turns out that if you drop the axiom of choice, then the consistency strength of that axiom has enormous strength. It has large cardinal strength. In large cardinal set theory, the strength of the axiom of determinacy is infinitely many wooden cardinals, if you've ever heard of these large cardinals. And for example, under AD, the axiom of determinacy, it follows that every set of reals is Lebesgue measurable, and every set of reals has the property of Baire. And so there's all these amazing regularity set theoretic consequences from that axiom, which is just about playing these games.

And so what I view as the whole topic, the fundamental theorem of finite games just leads you on this walkway to these extremely deep ideas that come along much later. And I just find that whole thing so fascinating. That's why I like it so much.

EL: So I guess I want to pull you out of these extremely deep things into something much more shallow, which is, are there games like, games that kids play, or that like, people play, that are infinite games by their nature?

JDH: Oh, I see. I mean, I've studied quite a number of different infinite games. For example, well, I have a master's student now he just wrote his dissertation, his master's dissertation on infinite checkers, but we're in the UK, so we call it infinite draughts. But I guess ordinary checkers is usually just on an eight by eight board and so that doesn't count. So, I've done some work on Infinite Connect Four, I mean, infinitary versions of Connect Four and infinite Sudoku and infinite Go. And there are infinite analogs of many of these games that are quite interesting.

EL: I guess what I'm wondering is, so, in my mind, maybe there's a game where you can, like, go back and forth forever. And there's no rule in the game that says you can't just like walk towards the opponent and then walk backwards. And then the game tree might not be finite, or maybe I don't quite understand how the game tree would work.

JDH: No, no, you're absolutely right about that. Yeah, in that situation, the game tree would be infinite. I mean, it's related — for example, in chess, there's this threefold repetition rule. If you repeat the situation three times, then it's a draw. But the actual rule isn't that it's automatically a draw, but that either player is allowed to call it a draw. Right? And that's a difference. Because if you don't insist, if both players choose not to call the draw, then actually the game tree of chess would be infinite then because you could just keep moving back and fourth forever. And there's also another rule, which is not so well known, unless you're playing a lot of chess tournament play, which is the 50 move rule. And this is the rule in chess tournaments that they use, where if there's 50 moves without a pawn movement or a capture, then it's a draw. And the reason for that is — I mean, of course, I view both of those rules as kind of practical rules just to have an end of the game so that it doesn't just go on forever in the way that you described. But when we were deciding on the rules for infinite chess, we just got rid of those rules, and we thought, look, if you if you want to play forever, that's a draw. So any infinite play is a draw is the the real rule, to my way of thinking. And and the reason why we have the threefold repetition rule, and the 50 move rule is just those are proxies for the real rule, which is that infinite play is a draw. But it doesn't quite answer your question. I'm sorry, I don't know any children's games that are just naturally infinite already. Name the biggest number.

EL: Yeah, well I’ve also been sitting here wondering, like, can you make a game — like other children's games that aren't maybe on a board? Tic tac toe, of course, is the first thing I think, but then what about Duck Duck Goose [for the Minnesotans: Duck Duck Grey Duck] or Simon Says, or these other children's games that aren't really board games? You know, can you even work those into the framework of these finite games, or infinite games, or not? I don’t know.

JDH: There is this game of Nim where you play with stacks of coins and you remove coins from one stack or another. It's a beautiful game with a really nice resolution to it in terms of the strategy, but that game has infinitary versions where the stacks are allowed to have infinite ordinal heights. And basically, the classic proof of the Nim strategy works just as well for ordinals. And so if you if you think about the sort of the balancing strategy, where you look at the things base two, and so on, well, ordinals have a base two representation in ordinal arithmetic also, and you can still carry out the balancing strategy even when the stacks have infinite height.

EL: It never occurred to me to think about infinite ordinal height Nim.

KK: Well, you never have that many toothpicks, right?

EL: Yeah. Yeah. limited by my environment, I guess. I must say that I've spent a lot of my mathematical career sort of avoiding things with with too much infinity in them, because they're very intimidating to me. So maybe we have kind of different mathematical outlooks.

KK: Right. Yeah. So the other thing we like to do on this podcast is invite our guests to pair their theorem with something. So what pairs well, with the fundamental theorem of finite games?

JDH: Well, there's only one possible answer to this, and I worry that maybe I'm cheating by saying, of course, I have to pair it with the game of chess.

KK: Sure.

JDH: Because Zermelo’s theorem was really focused on chess, and he proved that, look, in chess, either White or Black has a winning strategy, or else both of them have drawn strategies. And I never played a lot of chess when I was a child, but when I had kids, they got involved in the scholastic chess scene in New York, which is quite hyperactive, and fascinating. And so my kids were playing a lot of chess, and I went to hundreds of chess tournaments and so on, and so I started playing chess and I learned a huge amount of chess from, they had such great coaches at their schools and so on. But actually, I'm a pretty mediocre chess player even after having played now for so many years. And one of my coauthors on the infinite chess papers that I wrote, is quite talented chess player. He's a national master, Cory Evans. He was a philosophy graduate student when I met him at the City University of New York, which is where my appointment was at the time. And so I got to meet a lot of a lot of really talented chess players, and it was really great working with him on that infinite chess stuff, because I realized that that actual chess knowledge is really focused on the 8 by 8 board, and that once you go to these much bigger boards, the the chess grandmasters even become a little bit at sea. And so I would know what I'm trying to do mathematically to create these positions with high game values, and I would show them this crazy position with, you know, 20 bishops and hundreds of rooks and so on. And I would talk a little about, and he would say, hang on, this pawn is hanging here, it's totally unprotected. And it would completely ruin my position. So their chess ability, their chess reading ability was such that they could look at these crazy infinite positions and point out flaws with the position. And that was really something that was important for our collaboration. These chess positions are so finicky, these huge, infinite ones. And so many details are running on whether the things are protected properly, and whether — because oftentimes, you have to argue that the play has to proceed according to this main line. And if you want to prove the theorem, you have to really prove that. And if there's some little upset that means that the flow of play isn't exactly like what you thought, then the whole argument is basically falling apart. And so it really was depending on on all of that. So I really had a great time interacting with a lot of these talented chess players. It was really fantastic.

KK: I’m a lousy chess player.

EL: A lot of interest among chess players in the mathematical study of the game of chess? Even leaving aside the infinite versions, but, you know, the finite version. I assume there are theorems being proved about regular chess. Do players care about them much?

JDH: Well some of them definitely do. And actually, there's a huge overlap, of course, between chess players and mathematician

EL: Oh, yeah, that's true.

JDH: And so maybe maybe a lot of the interest is coming from that overlap. But, for example, there was a problem that I had asked, I think I asked it on Mathoverflow. Take chess pieces. On an empty board, take a full set of chess pieces and just throw them at the board. So you you get some position. What's the chance that it's a legal position? So in other words, a random assignment of pieces. And you can you can make some calculations and prove some interesting things about the likelihood that it's a legal position. In other words, a legal position, meaning one that could in principle arise in a game, in a legal game, right. And

EL: Do you happen to recall any ballpark, you know, is this, like a 1% chance?

JDH: It’s way less than 1%. It’s exceedingly unlikely.

EL: Okay.

JDH: If you allow, if you insist on all the pieces, because then there haven't been any captures. So the pawns have to be sort of perfect. There has to be one column and opposing. And already just because of that, that already makes it extremely unlikely to happen if you have all the pieces. And then some other people answered on Mathoverflow, I think giving better bounds when you don't have all the pieces and so on. But it wasn't quite open. But I think the general conclusion was that it's extremely unlikely that you get a legal position.

KK: Well, that makes sense. Given the complexity of the moves, it would be pretty remarkable if a random placement would would actually work.

JDH: There are some amazing — there’s a book by Raymond Smullyan, about the “chess detective,” and he has these many instances. It’s sort of like he gives you a chess position, and and you have to deduce, what was the previous move? Because these positions are often extremely strange. Like you think, “How could that possibly arise?” So there's sort of logic. I mean, he's a logician. And so there are sort of chess logic puzzles to figure out what the previous move was. And there's often a story associated with the game that, you know, so it must have been Black, who was the murderer because… It’s really some fascinating work that way. I really like that.

KK: Well, this has been informative. I've certainly learned a lot today.

EL: Yeah.

KK: So we like to give our guests a chance to advertise. Where can we find you online? And if there's anything you want to promote, we're happy to let you do it.

JDH: Oh, I see. Well, you can find me online, I have a blog, jdh.hamkins.org. And also, I'm on Twitter, and also on Mathoverflow. And I just published a number of books. So one of them I mentioned already, it's called Proof and the Art of Mathematics. And this is a book for aspiring mathematicians to learn how to write proofs, Proof and the Art of Mathematics with MIT Press. And I have another book, a philosophy book called Lectures on the Philosophy of Mathematics, also with MIT Press. And that is a book that I use for my lectures on the philosophy of mathematics here in Oxford. And it's, I would say, a kind of, grounded in mathematics perspective on issues in the philosophy of mathematics.

EL: Yeah, and if I can praise you a little bit, I will say something that I have enjoyed ever since I've been following you is that you — some of some of the things you write are about, like very technical, you know, deep mathematical things. But you've also had some really neat, like, puzzles that you've shared with children and stuff like that. I remember I was working on a Math Circle project one time about paper folding and cutting and you had a fun, I think it was like you show someone a configuration of holes in a piece of paper and say, can you fold the paper so that you just have to punch one hole in this folded paper to get the holes looking like this? Or something like that. And so it kind of spans a big range of mathematical sophistication, and what level you want to jump into something. So I think that's something fun and other people who might be looking for activities like that might enjoy it.

JDH: Thank you so much. I'm so glad to hear you mention that project. Those projects are all available on my blog if you click on the math for kids link, which is one of the buttons on my blog. And they all arose because I was going into my daughter's school every year, or a couple times a year, with these different projects, including that one and a number of other ones. So have about a dozen or more math for kids projects on my blog.

KK: Very cool.

EL: Well, thanks for joining us. I enjoyed talking about chess, a game that I have probably played, you know, 10 times in my life.

JDH: Well, it's a pleasure to be here. Thank you so much for having me.

KK: Yeah. Thanks.

[outro]

In this episode of the podcast, we were happy to talk with Joel David Hamkins, a mathematician and philosopher (or is that philosopher and mathematician?) at the University of Oxford, about the fundamental theorem of finite games. Here are some links you might enjoy perusing after you listen to the episode.

His website, Twitter, and Mathoverflow pages
On his website, check out Math for Kids for some fun activities for all ages
His books Proof and the Art of Mathematics and Lectures on the Philosophy of Mathematics
The Wikipedia page about the fundamental theorem of finite games
The PBS Infinite Series episode on infinite chess
The Mathoverflow question and answers about legal chess board positions

Episode 69 - Ranthony Edmonds

14 augusti 2021 | 36 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where there is a family of quail that live outside my window. And they don't know that I'm here so I can watch them scurrying around in the bushes. There are at least five young ones right now.

KK: Cool.

EL: They are so cute. Oh, it's just like, sometimes — they're not here right now, which is good. Because otherwise, I would just be like, staring out my window. Looking at these cute little quail.

KK: Oh, see, so in Florida, we're in actually the boring birds season because you know, it's just, this is the locals. So I see my cardinals and the titmice and all of that, but it's still fun. I still feed them. I'm out there every day. They’re eating me out of house at home. It's true. It's a good thing. All right, well, so today we are very pleased to welcome Ranthony Edmonds. Why don't you introduce yourself, please?

Ranthony Edmonds: Hi. Yes. So I'm Ranthony Edmonds. I'm a postdoctoral researcher at The Ohio State University.

KK: The Ohio State.

RE: The “the” is very important in Columbus. We take this very seriously. And I've actually become one of those people who corrects and makes sure that they add the “the” in conferences and notes and things like this. It’s very obnoxious. Yeah. So I'm a postdoctoral researcher at The Ohio State University. I study commutative ring theory, classically, specifically factorization theory. And I'm in the midst of this sort of interesting transition, where I am looking into applications of algebraic topology. So I spent the last year learning a bit about topological data analysis, and I’m specifically interested in applying that to redistricting. So I, you know, I my interests are kind of broad, but specifically, to kind of give you some keywords, commutative ring theory, topological data analysis, redistricting. And, of course, you know, my general mission is to increase access to mathematics for Black Americans and members of other traditionally underrepresented groups in the mathematical sciences. And I'm trying to do that through a combination of inclusive pedagogy. academic research, and community engaged scholarship.

EL: Nice, and I was perusing your website before this to familiarize myself a little bit, and I saw that you're working on a kind of a history-related project about Black mathematicians at The Ohio State University historically?

RE: Yes. I think a lot of people had a lot of different reactions to what happened last summer with with George Floyd and the protests that swept over the country. And one thing that I sort of questioned is this idea of, well, if we're going to try to improve access to mathematics for Black Americans, for other traditionally underrepresented groups, how do we really begin to do impactful work if we're not really aware of what's happened historically? And I've always been interested in history. I think a lot of this comes from a previous project I'm still doing with the Hidden Figures story and sort of using that to center discussions about diversity and equity in the discipline. But yeah, I just love math history. I have a team that's very interdisciplinary, and we're looking at the history of the math department, specifically at Ohio State this summer, and then on into the fall. So there are really two things that we want to do. One is, you know, a lot of the narratives of these the pioneers who graduate from the department with PhDs, with master's degrees, they're kind of just hidden. There’s not a lot of recognition about the work that they've done. But we have discovered that there are seven Black PhDs who have graduated with a doctorate degree in math from Ohio State. And we’ve got two former university presidents among that midst. We've got lawyers, authors, you know, program officers in the NSF, just people who have gone on to do really prolific things, and yet are still somewhat kind of unacknowledged by the university themselves, and then just in the wider math community, and I think that there are a lot of hidden stories out there. And I think when I reflect on the Hidden Figures story, this is what made that so impactful is because people didn't know. So I think that there's a lot of work out there that's being done by wonderful people that people just don't know about. And so what we're trying to do is to highlight and amplify those stories, one. And then two, examine and contextualize their experiences at the university. So what was happening when they were students here? What influenced their trajectories after graduation, where they went to work, if they went to industry or academia? I think if we think about trying to get more people in graduate school or get more people at the Faculty level, well, we should start by thinking about how we're serving our undergrads who are in that actual population and how we've done that historically.

So we're doing a lot of things. I'm working with some people in strategic communication, some people in our Office of Diversity and Inclusion here at Ohio State. Also we have a connection with the National math Alliance. So they are an organization that I'm very intimately familiar with from my time in graduate school at the University of Iowa, but they are really focused on trying to increase the number of minorities entering PhD programs in the mathematical sciences. And this is pretty broad, right? It's not just math. It’s statistics, it’s economics, it’s something that requires quantitative training as an undergraduate. So we're working with them. And we're also working with a local museum, the Ohio History Connection, and there is a specific branch which is, they call themselves Afro-Am, but the official title is the National Afro American Museum and Cultural Center. And they're located in Wilberforce, Ohio. We're working with them with some of our archival research, and also our community programming. So we've learned a lot of really interesting things. We’ve sort of broken up the history of the department starting at 1963, when the first Black male PhD, his name was William McWorter, graduated from Ohio State up to the present and just identifying individuals who earned degrees during that time period and interviewing them, as well as sort of contextualizing what are the big things? How did selective admissions affect Black student enrollment in general and specifically in the math department? How did you know the protest of the ‘60s and ‘70s impact the campus environment? We have students who are wonderful who are helping us look at these different questions. And then there's actually like a lot of cool math. For instance, the the first Black male PhD from Ohio State, his name was William McWorter. And he was part of this camp along with like Axler and others that was like “death to determinants.” We don't need them, why are we teaching them to students? He felt like it was a very crippling tool pedagogically, in that students just used them for computations and had no idea what they were. And which makes sense, because I think I've experienced that on the student end of things.

EL: I’d say guilty as charged there.

RE: So he wrote a couple of papers for that were published in Math Magazine about determinant-free methods in linear algebra. And specifically he competed came up with an algorithm for computing the characteristic polynomial of a matrix and computing eigenvalues and eigenvectors of a matrix without using the determinant. And I say, “a matrix,” there are obviously conditions imposed upon it, but it was really cool. So I'm working with a student this summer, and we're reading through this paper, and then we want to create a lesson plan related to that algorithm. Because it mainly focuses on dependency relationships, like do you understand the difference between, like, given a list of vectors, can you determine if they're linearly independent or dependent? And then it requires doing that via elementary row operations. So it's just sort of hitting some of the high points from introductory linear algebra, without getting into the weeds of what the determinant really is. So we're working to create a lesson plan from that, and then hopefully, that'll be incorporated back into the honors track here. It was when he taught here. And disseminating that. Our main goal is learn it and then disseminate it, you know, so other people can can learn from what we're figuring out. So there's this history component, and then there's a lot of math that we're uncovering from the history that's just really interesting in its own right, that we hope to, over time turn into lesson plans that other people can use for their classrooms.

EL: That’s really cool. Like, so I've done a little bit of dabbling in math history and stuff. And it's always really interesting to me how much the language has changed. And you'll see an abstract for a paper written it decades ago and realize, like, we just talk about things differently now, and it's kind of hard to dig down and figure out, Okay, how would I think about what they're doing here? You know, they have these names for different curves that aren't names I use anymore, and like, how do I translate it? It's like almost a translation project. Even going back just to the ‘60s, maybe.

RE: Yeah.

EL: So that must be really interesting. And I think that's a great project for students to do. So yeah, that sounds so cool.

KK: It is. You're very busy. And you know, your list of mathematical interests is super interesting to me too. I mean, I'm not a ring theorist, but the whole TDA and redistricting.

RE: Yeah. We’ll have to talk a bit after the podcast. But yeah, it is really interesting. And I think, you know, it's part of this whole approach of just trying to humanize mathematics. We're studying it, and we're getting into the nitty details, but we're also thinking about how people came to be mathematicians, and how this has actually been affected historically, especially for Black Americans, by policies. You know, a lot of the PhDs that we're studying about were supported by NSF fellowships, and this is a direct response to the space race that was happening. And they saw the influx of federal funding. And so it's all really interesting. I feel like I'm learning a lot about — even though it's focused on Ohio State — I feel like I'm learning a lot about the math community. And one thing that is really cool about us is sort of how we do our lineage. Right? And so the math genealogy websites are really cool, because you can sort of track back, very easily, Oh, this person who would who would they have been working with? You know, I think in another discipline, if you are trying to figure some information out about the person that you want to know, who their academic “siblings” were, that might be actually difficult to discern, but we have the Math Genealogy site where we can get that information easily.

KK: Yeah. All right. So this podcast, though, is called My Favorite Theorem, so we asked you on here for a reason. So, Ranthony, what is your favorite theorem?

RE: Yeah, so I am taking it back, back, back, back. So I actually would say that my favorite theorem is the fundamental theorem of arithmetic.

KK: Okay.

RE: It’s very classic. And the reason that I like it is because it's sort of the first introduction to really meeting math in disguise, because I think a lot of people are at least aware of the concept of it in grade school, even if maybe we don't get into the implications. And so, you know, the fundamental theorem of arithmetic, it states that, given an integer, so positive whole numbers greater than one. So yeah, greater than one, excluding zero, you know, it can be written uniquely as the product of prime numbers. And that this decomposition into primes is unique except for the order. So in practice, it means give me a number, like any number that's an integer, and I can factor it uniquely into small pieces called primes. And that’s it. That's the only way I can factor this number. And it gives it a unique signature. And it's telling us that in the same way that atoms are the building blocks of ordinary matter, these prime numbers build up the integers. And I love it, because there are a lot of implications in the work that I do in factorization theory that can all sort of be traced down to this fundamental idea. And I also love it because when I talk to younger students about ring theory or things like this, I always start with the fundamental theorem of arithmetic. And I tell them when they're drawing factor trees — at least that's how I learned, I'm not sure how you guys — is that what you remember? You had the number and then you do the branches?

KK: Yeah.

EL: I loved doing that when I was a kid. I don't know if you two were also like that. That it was kind of a soothing little exercise. Like, write down a big number — not too big; I wasn't super ambitious — but like, write write down a number and just do a little tree figuring out, you know, yeah, that kind of thing. I don't know. I thought it was fun.

RE: Yeah, I usually start off with when I’m talking with — I don't want to say little kids, right — with general audiences. I'll start off by asking people to pick their favorite three-digit number. So I guess maybe, do you guys have a phone or calculator handy?

KK: Sure.

RE: So this may not pack the same sort of punch, but I ask people to pick their favorite three-digit number. And then I ask them to create a six-digit number by taking that three-digit number and repeating it twice. So I usually use 314 because it's an approximation for pi. I was also married on Pi Day. And so 314 is is my number, and then I create a six-digit number, so that's going to be 314,314. Yeah. Okay.

KK: I chose 312.

RE: Okay, all right. And so I want you to take your six-digit number and divide it by 11.

KK: Okay.

RE: Okay. I have 28,574 right now. And then I want you to take that number and divide it by 13.

KK: It’s amazing that you're getting integers here.

RE: Yeah.

EL: Or is it?

RE: So now I have 2198. Okay, and so now I want you to take this number and divide it by your original three-digit number.

KK: Yep.

RE: And did everyone get seven?

KK: Yes.

EL: Yay!

RE: So yeah, so it's like this really cool thing where if you take a number and you multiply it by 1001, it has the effect of creating a new six-digit number. That's your original original number repeated twice. And so essentially, because we know that 1001 factors uniquely into primes, which is guaranteed to us by the fundamental theorem of arithmetic, you know, 1001 is 7×11×13. And so if you divide away 11, then divide away 13, if you divide away that original number, no matter what it was, you're going to be left with 7. And so it's really exploiting this property of the integers. This is really cool. And so I don't know, I just really love the theorem.

So why, I guess maybe, do I care about it? In terms of the mathematical sense, besides the fact that it's cool? It’s because there's a lot of deeper underlying mathematics here. So it's like, we have this statement that tells us, given any integer, we can decompose it uniquely into the product of prime numbers. And so like I mentioned before, these prime numbers are acting kind of like the atoms of the integers. And so in factorization theory, this is sort of the name of the game. We're really interested in, how do we decompose a mathematical object into its smallest pieces? And this is our very first introduction to this idea. It's in elementary when we're breaking numbers into primes. And then typically, when we kind of level up, the next thing we try to break down are polynomials, right? And so in algebra, whatever level in which you had it, you have a polynomial and you want to break it down too. And so it's like, okay, we want to factor it. And the question is, how do you know when you're done factoring? So you know, with a prime number, you circle it, and it's like, we have our, you know, but with polynomials, it’s a little bit more hazy. There's not a list of, well, there are, but a list of just all the irreducible polynomials that ever are. And so the question is, is there some sort of fundamental theorem that exists for the set of polynomials over the reals? So if we had something like x4−1, I remember in algebra that this was a difference of squares, and so there was a pattern. So I could break this into x2+1 and x2−1. And then this was always a tricky one, because it was like, aha, another another square, x2−1. So you can keep going. But then the question is, you know, do you circle x2+1 or not? Is it irreducible? And the question depends on the setting. Like, it depends on if we're working over the reals, or if we're allowing complex numbers, because if we allow complex numbers, then we can suddenly say that x2+1 is (x+i)(x−i). But if not, then, you know, maybe we're done.

So feasibly, it's like, well, we don't want to have to come up with a fundamental theorem for every single set of polynomials that exist. That's not very efficient. So we kind of generalize this idea of the integers into something called a commutative ring. And we generalize this idea of primes into irreducible elements. AndI think that living that abstraction is what I've spent most of my mathematical career looking at, like how things decompose, but I think tracing it back down to, you know, what we're really trying to do here is to come up with really nice notions that generalize the fundamental theorem of arithmetic. So this is probably why it's my favorite theorem, because I feel like if you keep going down to just the bare bones of what it is we're trying to do, the best example I think is there in that theorem, and also the best things are the things you can talk about.

KK: Yeah, and it's kind of like the first real theorem you learn.

RE: Yeah.

KK: Because, you know, I mean, you start learning mathematics in elementary school, and you learn how to add and subtract, but there aren’t really — well, there are theorems there, or definitions, maybe, but this one, you learn how to do it somewhere like fifth grade, maybe?

RE: Yeah, you’re really young, but I don't know that it was given the name.

EL: Yeah, I didn’t know the name of it, I think until I was probably in grad school, maybe college?

RE: Yeah.

EL: Still, but you learn it. Maybe you don't learn it as a theorem.

KK: Yeah. You learn an algorithm, right? Essentially.

EL: Yeah.

KK: Yeah. How do you do it? I mean, what do they teach you to do? Like, start dividing by primes, maybe?

RE: I guess I felt like at that point, well, this was when I was still just using a lot of memorized facts. And that was math to me. And so I guess I had my list of things that I thought were prime. And then maybe if they threw in a big number, I'd have to think about it. Like if they threw in like a 37. It's like, Oh, wait, what's happening? But 2, 11, 13? You know, we were pretty good to know. But yeah, I think I had no idea that it was a theorem. I do remember learning it, though. And so my favorite things are when I'm learning something, especially in a more advanced mathematical setting, and it takes me back to a very young me who just didn't know that there was a lot more to this when I was first exposed to it.

KK: Mm hmm. Yep. And hopefully didn't fall to the Grothendieck trap of thinking that 57 was prime, right?

RE: No. So basically, I've done a lot of work looking at unique factorization. And so because I work with zero divisors, which I don't know that I need to get into the nuts and bolts, but I thought a lot about what makes a unique factorization domain tick. Because I think a lot about settings where we don't have those nice properties. And so a unique factorization domain is the is the exact generalization of the fundamental theorem of arithmetic. So the fundamental theorem of arithmetic, you know, we've got a setting, the integers, where everything factors uniquely into primes, and in a unique factorization domain, it’s commutative ring, which is a generalization of the integers and the nice properties that they have. And it's a commutative ring, where everything factors uniquely into atoms, so we're generalizing primes now into atoms. And so there are some really nice results related to polynomial rings, where if you have a ring that has unique factorization, then the polynomial ring extension also has that same property, and vice versa, too. And so in the world that I live in, there are a lot of times where these factorization properties don't extend. And so I spend time thinking about what can we do to try to make them extend. So yeah, I think a lot about factorization theory and commutative ring theory. And so a lot of this is sort of based on this very gold-star standard of a factorization setting, which is a unique factorization domain. It's the nicest place that you can live, where factorization is just really well-behaved. You don't have to distinguish between primes and irreducibles, it's just a beautiful place to be. And so I call this a utopia and it's really mimicking or generalizing the fundamental theorem of arithmetic and the results there.

KK: So another thing we like to do on our podcast is ask our guests to pair their theorem with something. So we hear you might have multiple pairings. You’re only obligated for one.

RE: Okay, so originally, my first thought with a pairing was was alcohol, and I don't really drink that much, but I do love mead. So there is a meadery here. Okay, so mead is like it's like when they fermented grapes to make the wine, they ferment honey. So it's sweeter. And so there's a meadery here in Columbus called Brothers Drake. And I believe that they have a cousin, or a brother? You know, another brother meadery that's in California. But that's really broad. I'm not sure which part in California, but they have an apple pie mead. And it's my happy place when I do you know, partake in a little bit of something. So, I think the apple pie mead, and just any mead in general, if you would like to try it, especially when we get into history and stuff. I feel like this is like a very historical drink. Yeah.

EL: Yeah. Like, I don't know, I think of, like, dank castles and that kind of thing. Probably a lot of like, Disney fantasy, you know, people coming from battles and drinking their mead or something.

KK: Right.

RE: Yeah. I think a lot about Thor because I just am a big Marvel Universe person. And so I feel like Thor and Loki would just be having some mead, you know, catching up. But okay, so I was trying to think of what else would go with my theorem that wasn't alcohol. And so because I feel like the fundamental theorem of arithmetic is a very classic thing. So I would just like to pair my theorem with two things. One, sleep. So this is like a shameless plug for everyone to attempt to get eight hours of sleep. This is something that I tried really hard to do last year. And it was really crazy how much I fought against. Like, “I don't have time for this because XYZ,” but it took me maybe a whole semester, and I finally am now sleeping eight hours a night no matter what. And so this is a nice pairing with math, is sleep because I think that it's really good to do math when you're rested and your head is clear. So that's one. And then the second would just be like nice long walks. I love nature. I love cycling and strength training, but you just can't beat a good walk. And so for those who are able and mobile, I just think taking the time to go on quick walks during the day, even if it's just, like, 10 minutes, in between a meeting or something it’s a really great thing to do. So those are, I guess, like, my classic pairings. So what I say is apple pie mead, eight hours of sleep, and a walk.

EL: A nice long walk. This sounds like a great day.

KK: This is amazing. And you’ve got your bike there in the background.

RE: Yeah. Oh, my gosh, we're getting to know each other.

KK: Yeah. Well, when I was a postdoc, I was a very serious cyclist. I mean, I spent a lot of time, it was good therapy for me to get on the bike. Like if I was stuck on my math, I went for a ride.

RE: Yeah.

KK: But I lived in Chicago at the time, so you can't really ride in the winter.

RE: Yeah. That's an interesting city. I guess what like, did you go maybe to like suburbs and ride out there?

KK: Yeah, I was in Evanston. So I was at Northwestern. So that was good, because you could just head north, and then you're out in the country pretty quick. But you know, I had a group I rode with and all that, but just very good therapy all the way around. And then I had a kid and moved to Detroit. And those two things will just kill your cycling.

RE: What about what about you, Evelyn? Do you have — because you were talking about birds in the beginning, and I see that your background is very scenic.

EL: Yes, this is a cold day at Bryce Canyon National Park down in southern Utah. It's extremely hot in all of Utah right now. So this was kind of nostalgic, like, bringing some cold weather into it. But yeah, I love biking and taking walks and stuff. I'm really lucky in the neighborhood I live in. Basically, if you go north from my house, which is also uphill, you end up in less than a mile going into this extensive trail network that can get you all over the place if you're willing to go for a long walk. And it's like, I live less than two miles away from the state capitol building in downtown Salt Lake, but the fact that you can get up into nature so quickly is amazing.

KK: Well, it's right up against it.

EL: We’re built into the foothills here. And it's great. So yeah, I love taking walks in nature and I I've never tried that kind of mead, but there's actually a local like fruit wine place here that has a whole mead series in addition to fruit wines. And it's really cool because they have some that are sweeter and then some that are less, where they fermented, like, all of the sugar and it's amazing some of these, they almost taste like a Chardonnay or something. Yeah. Because you think of mead, honey wine, it's going to be super goopy and sweet, and depending on how much you ferment it and stuff, it actually has all sorts of different flavors so yeah, it's a cool place. I think they've got some like apple and honey cider mix things so I should check those out. Yeah,

RE: You definitely should, but I do agree that there are so many different like flavor profiles. The meadery here, you can go and do samples and they have like, music nights, pre-COVID, I think they're starting to resume this, and, like, empanada nights, which was a very personal weakness of mine. But yeah, I love — like, some of it just is too strong for me, right? Because I was I was leaning towards mead because I was like, okay, I don't know if I'm a hard alcohol drinker. But it's not all just sweet. It's not all just like juice with alcohol. I really like it. And so the last time that we tried was called Purple Rain. And I believe that the guy said that he, and it depends also on the barrel on which you're aging the the the mead, but he did something, like it was like using some sort of like blackberry something, and they accidentally like made too much and it was like overflowing the barrel when they came to check in on it. And so he called it Purple Rain. So I thought that was was pretty cool. Back to the cycling comment, my bike was actually stolen out of my garage at the beginning of COVID, and I was so upset. So at the beginning of COVID and work from home, when I realized that we'd actually be here for a while, I started nesting. I did a lot of things to my office space. So you see this black peel-and-stick wallpaper that I put up and actually turned out really nice.

EL: And a beautiful — I was hoping to see the rest of that picture that you’re tilting up now because yeah, I was thinking that looks really cool.

RE: Yeah, I got art. So this is um, I Gosh, I want to say it's just an Israeli painter named Itay Magen, and I just really love. It does a lot of really vital prints, colorful art. And so this came as a canvas. And then I realized when you buy canvas prints, you actually have to go get them mounted, which can be a little pricey. And then I ordered this Blackboard that you see, but the point that relates to the bike is that I also put together some shells, there's a landing gear, so you're kind of blinded, but I was painting and staining the shelves in my garage, and I left it open for a little bit, just to let the air sort of, you know, vent because of the paint fumes, and my bike was stolen. I got a little too trusting living downtown, you know, moving here from Iowa, you know, I kind of learned my lesson in the big city, I guess. So I got a new one. And I'm still getting to know this one a little bit better. But I took it out for the first time last week, and I have new clipped-in pedals. I got a different pedal than I had last time. I've been practicing just getting clicked in and out just at home because it's a little bit more challenging. So yeah, but I love doing outdoor things. And I think it's really nice to get fresh air, just for balance. And then also it does help, I think, with math. There's a tendency, I think, to try to double down, like, “no, I got to get this result. And then like sleep can happen or life can happen.” But I found that, you know, actually taking the breaks is really helpful.

EL: Yeah, the number of times where, you know, you're stuck on something, and then you actually let yourself sleep and wake up and realize, Oh, I can approach this in some different way — I wish I learned better from that rather than continuing to torture myself sometimes.

KK: Yeah, yeah. All right. Well, this has been great fun. So where can our listeners find you online?

RE: Yeah, so you can find me online on Twitter. My handle is @RanthonyEdmonds, let's see, with regards to the OSU Black math history project I mentioned, we will have a website, probably by September. But in the meantime, you can contact us at [email protected] if you're interested in telling a story related to your time, you know, at Ohio State or affiliated, or just you just want to tell a math story. You know, that's the place to go. And then I think lastly, I'll start posting a lot soon on Twitter about another project that I'm working on just by the end of the summer related to redistricting and communities of interest, and sort of synthesizing community input so that when the redistricting process happens at the end of this year, we are taking into account communities of interest, which is this sort of traditional redistricting principle that says that communities with shared interests should be kept together in the mapping process. But what are those communities look like? Where are their boundaries? What are their key characteristics? We're working with the MGGG Redistricting Lab along with Ohio Organizing Collaborative and their independent citizens redistricting commission to really collect a lot of public input related to communities of interest. And so I'm focused on what's happening here in Ohio. But this is an effort happening over 10 states this summer, as we prepare for redistricting in the fall and all that's going to happen with the release of the census data. So I guess just stay tuned. Some good places that aren't my Twitter profile will be Common Cause, Ballotpedia, and of course, here in Ohio, the let's see, don't let me lie, ohredistrict.org. And so this is where you can find the Ohio citizens redistricting commission information. And so this is an independent commission that is sort of focusing on modeling good redistricting practices. And we're working closely with them this summer. But like I said, this is definitely happening in over 10 states. And so I'll start posting about this soon. But in terms of not me, specifically, just, you know, look some things up about what's happening in redistricting. Try to get involved and make your voice heard, because it affects all of us and it's really important, but I don't want to go on a separate tangent. This is supposed to be like a closing plug. So follow me on Twitter @RanthonyEdmonds. Email me if you're interested in telling your story related to Black math history at [email protected]. And then just you know, ohredistrict.org and Common Cause are really great resources for learning more about redistricting that's happening this year.

EL: Excellent. That is a fantastic set of resources. Thanks so much for joining us. This was a lot of fun.

KK: It’s really great.

RE: Thanks for having me.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking to Ranthony Edmonds from The Ohio State University about the fundamental theorem of arithmetic. Here are some links you might enjoy after you listen to the episode:
Edmonds' website and Twitter account
An interview with NPR about her Hidden Figures-based course about mathematics and society
Math Alliance, a program that supports mentorship for early-career mathematicians from underrepresented groups
Ohio History Connection and the National Afro-American Museum and Cultural Center
An article by Evelyn about why 1 isn't a prime number, which mentions the distinction between prime and irreducible
The Metric Geometry and Gerrymandering Group (MGGG)
Ohio Organizing Collaborative
Ohio Citizens Redistricting Commission
Common Cause

Ballotpedia

Episode 68 - Rekha Thomas

8 juli 2021 | 27 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, and this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn?

EL: I’m doing okay. I was trying to think of something interesting to talk to you about at the beginning of this and life isn't very interesting, but in a good way. So that's good.

KK: Yeah. Yeah. You know, monotony is underrated, isn't it?

EL: Yeah.

KK: We just had our 29th wedding anniversary on Sunday.

EL: Congratulations.

KK: Thank you. That's our big news. We went out and sat at a restaurant for the first time in nearly a year. So we were very excited

EL: Excellent. I hope that, you know, this podcast is going to go on for decades and decades, and someone is going to be catching up on old episodes, you know, in 15 years and say, “Why do they keep talking about all these very boring things that they're doing for the first time in a year?” So that'll be great. You know, last time it was haircuts, this time it’s restaurants.

KK: That’s right. That's right. Yeah.

EL: But whether they are saying that or not, they will be very excited that we are talking today to Rekha Thomas, and Rehka, would you like to introduce yourself?

Rekha Thomas: Hi, thanks for having me on the show. So I am a professor of math at the University of Washington in Seattle. And I come originally from optimization. My PhD is actually in operations research. But I have worked in a math department ever since I graduated. And my work lies somewhere at the intersection of optimization, applied algebraic geometry and combinatorics. And I very much like problems that have an applied background, or a motivation. Not necessarily because I work in applied things, but because I very much like the problem to be motivated by something real. And in the last 10 years or so I've been working quite a bit in the mathematics behind questions that come from computer vision. And this has been especially fun, but optimization in general does have that applied side to it. So I like problems like that.

KK: Right. You know, I talked to my colleagues over an industrial and systems engineering, where they do a lot of that work all the time. And I think, “You guys are basically mathematicians. Why don't you come over here?” Heavy overlap there. Yeah.

EL: Well, and I just want to say, how we came to invite Rehka onto the podcast.

KK: Yes.

EL: I think it's pretty cool. So one of her students sent in an email to our little submission form on Kevin's website, saying that he thought that his linear algebra teacher was great and would be a great guest for My Favorite Theorem. And this is especially remarkable because this is in the semester, we're all teaching has been online and stuff. And I just think to be connecting with students enough when you're doing Zoom teaching, that they actually reach out to a podcast to get you as a guest on a podcast, just you must have really made an impression. So I think that's very cool.

RT: Thank you. I was very honored by that piece of news. I thought, “Okay. We finally made contact.” That was great. But especially for an undergraduate to take that initiative to write to you was very touching.

EL: Yeah. So you know, not to build things up too much, but I'm sure that this is going to be a great episode.

KK: Yeah.

EL: So with that introduction done, what is your favorite theorem?

RT: Yeah, so I announced my favorite theorem in that class and told them about your podcast. So when you invited me, I thought, “Oops, is that really my favorite theorem? Is that what I really want to speak about?” And I think yes, so I'm going to stick with that theorem. And this is a theorem from linear algebra. It was a linear algebra class. And I think it's one that's not very well known to pure mathematicians. So it goes by the name of Eckart-Young theorem or the Eckart-Young-Mirsky theorem, but apparently the history is more complicated. I've been reading about this so I can tell you about it if we get to that. So the theorem is basically says the following. So if you have a matrix, say a real matrix of size N by N, and we know its singular value decomposition [SVD] — which is a very special decomposition of the matrix as a sum of rank-one matrices — then the closest rank-K matrix to the given matrix is just made up of summing the first K rank-ones in that decomposition. So if I wanted the closest rank-one matrix, I just take the first rank-one in the singular value decomposition of A. If I wanted the closest rank-two matrix, I take the first plus the second rank-one matrices in the singular value decomposition, and so on. So there is this neatly arranged set of rank-one matrices that add up to give you A, and if you truncate that sum anywhere at the K-th spot, you get precisely the closest rank-K matrix to the given matrix. This is the theorem.

KK: Closest in what sense? What’s the metric?

RT: Yeah, so that's a very good question. So closest in either Frobenius norm or spectral norm. So these are standard norms on matrices. Frobenius norm is just, you think of your matrix as a long vector, where you just take every row, let's say, and concatenate it to make make a long vector, and then take the usual Euclidean norm. Okay. So the sum of Ai,j squared, square root. [Maybe a little easier to read in math notation: (Σ Ai,j2 )1/2.] Spectral norm is the largest singular value, so it is the biggest stretch that the matrix can make on a unit vector.

KK: Okay.

RT: So in either norm this works.

KK: That’s pretty remarkable.

EL: Yeah. So just doing a little bit of mathematical free association, is this, can this in any way be thought of as like a Taylor's Theorem, like, you're kind of saying like, Okay, if you want your approximation to be this good, you go out this far, if you want it to be this good, you go out this far, maybe that’s — I don't I don't know if that's a good analogy.

RT: No, I think it is. So I think there are many constructs in mathematics like this, right? Like Fourier series is another one, where you have a break down into pieces, and then you take whichever part you want. Taylor series is similar. The SVD is special in the sense that the breakdown, this breakdown into rank-one matrices is actually tailored to the matrix itself. Like, for example, as opposed to, say, in Fourier series, where the the basic functions that we are trying to write the function as a combination of, they are fixed, right? It is always cos(θ) + i sin(θ), or cos(Nθ) + i sin(Nθ). So it's not particularly tailored to the function. It's just a fixed set of bases, and you're trying to write any function as a combination of those basis functions. But in the SVD, the basis that you construct, the factorization that you get, is tailored to the actual data that's sitting inside the matrix. So it's very, very nice and is incredibly powerful. So it's similar, and yet, I think, slightly different.

KK: Right. In other words, yeah, that's a good explanation. Because, as you say, with Taylor series, you're choosing a basis for a subspace of the space of all smooth functions or whatever. Whereas here, you're taking a particular matrix. Does it matter what — so if you change the basis of your vector space, do you get a different SVD?

RT: No. So what the SVD is — there is a very nice geometric way to think of the SVD, which may answer that question better. So the SVD is sort of a reflection of how the matrix works as an operator. So what it's telling you is if you take the unit sphere in the domain, so let's say we have an N by M matrix, so the domain is Rn, take the unit sphere, under the map A, the sphere goes to an ellipsoid.

KK: Right.

RT: In general, a hyperellipsoid, right? And this ellipsoid has semi-axes. The singular values are the lengths of those semi-axes. Oh, so it tells you, yeah, the length of the semi axes, and the unit vectors in the directions of those semi-axes are one set of basis vectors. So that's one set of singular vectors, and the pre-images in the domain are the other set of singular vectors. So if you're willing to change basis to the special bases, one in the domain, one in the codomain, then your matrix essentially behaves just as diagonal matrix. It just scales coordinates by these singular values. So it's a generalization of diagonalization in some ways, but one that works for any matrix. You don't need any special condition. So they're very canonical bases that are tailored to the action of the matrix.

KK: I’m learning a lot today. So I have to say, so before we started recording, I was mentioning that I think students need to take more linear algebra. But the truth is, I need to take more linear algebra.

RT: I think we all do.

EL: Yeah, I mean, I am really realizing how long it's been since I thought seriously about linear algebra. So this is fun. And it goes really well with our episode from a few months ago. I believe that was Tai-Danae Bradley who chose a singular value decomposition as her favorite theorem. So we've got, you know, a little chaser that you can have after this episode, if you want to catch up on that. So Rekha, you said that there was a bit of a complicated history with this theorem. So do you want to talk a little bit about that?

RT: Yeah, sure, I'd be happy to. So I always knew this theorem as the Eckart-Young theorem. I only recently learned that it had Mirsky attached to it. But then in trying to prepare for this podcast, I started looking at some of the history of the singular value decomposition. And there's a very nice SIAM review article by Stewart, written in 1993, about the history of the singular value decomposition. And according to him, singular value decomposition should be attributed, or the first version of it, is due to Beltrami and Jordan from 1873 — so the Eckart Yang theorem is from 1936, so almost 60 years before — and they were looking at more special cases, they were looking at square real matrices that are non singular, perhaps. So they, you know, people were interested in that special case. Then there were several other people, like Sylvester walked on it. Schmidt from Gram-Schmidt, he worked on it, Hermann Weyl worked on it. So from 1873 to 1912, this went on. And this article says that this approximation theorem that I mentioned, the Eckart-Young theorem, is really due to Schmidt from Gram-Schmidt fame. And he was interested not so much in matrices, but he was studying integral equations, where you have both symmetric and asymmetric kernels, non-symmetric kernels. And he wrote down this approximation theorem. So really, Stuart claims that that this theorem should actually be attributed to Schmidt. And then in 1936, Eckart and Young actually wrote down the SVD for general rectangular matrices. So that is in that paper, for sure. And they seem to have rediscovered this approximation theorem. So, that is my understanding of the history of how it is, but I did not know this till two days ago. And I'm not really a bonafide historian in any way. So, but this is what I've understood. It's an interesting story.

EL: Yeah, that sounds like an interesting article. I mean, I guess in the history of math, there are an uncountable number of places where unravelling back to where the idea first appeared is more complicated than you think.

RT: Right.

KK: And also, this sort of gets at your interests more generally, how you like things to sort of come from an actual application. Well, if this really came from integral equations, right, that's really an application.

RT: Absolutely.

KK: So it's working on lots of levels for you.

RT: That’s right. So he, of course, did this in infinite-dimensional vector spaces. And approximation will allow you to approximate an operator as opposed to a matrix, right? And apparently, that really elevated this whole theory from just a theoretical tool to something that's actually widely used. It became much more of a practical tool. And I guess, in modern day, the SVD, and versions of the SVD in exists in all kinds of mathematical sciences. So in signal processing, and fluid dynamics, all sorts of places. So it's, in some sense, one of our biggest exports from the math world, and yet we don't quite teach it normally, to math people. So yeah.

KK: Right. So was this like a love at first sight thereom? Was this the sort of thing that came up a lot in your work, and that's why you're now so enamored?

RT: So I did learn of at first in the process of writing a paper. I did not know about this theorem before, maybe about 10 years ago. But I think this theorem sort of perfectly fits me, which is why I love this theorem. So for different reasons, right? So first of all, it's an optimization problem, it's about minimizing distance from a matrix to the set of rank-whatever matrices. So it's an optimization problem. The space that you're trying to minimize to, which is the space of rank at most k matrices, that is an algebraic variety. So it can be written as the set of solutions to polynomial equations. So there's the applied algebraic geometry side, or at least the algebraic geometry side. And it's not a very simple variety. It's actually a complicated variety. So it's an interesting one. And lastly, this problem is sort of a prototypical problem in many, many applications. So a lot of statistical estimation problems are of this flavor. You have a model, which, let's say is our rank-K variety, so the rank being some measure of complexity. And then you have an observation that you have in the field with instruments, and it tends to be noisy, so it's not on the model. So that's your observed matrix. And now you're trying to find the maximum likelihood estimate, or the closest true object, that fits the observation. So this is a very standard problem that comes up in many, many applications. So in some sense, I feel it really lives at this intersection of optimization, algebraic geometry and applications, which is sort of what I do.

KK: That’s you.

RT: Yeah. So that's one reason that I think this theorem is so cool. And the other thing is, I think it's a very rare instance of an optimization problem where the object, the observed matrix, knows the answer in its DNA. It doesn't need to know the the landscape that it's trying to get to, which is your space of matrices of rank at most K. it doesn't need to know anything about that variety. Just inside its own DNA, it has this SVD. And from the SVD, it knows exactly where to go. So this is completely unusual in an optimization problem, right? Like even if you're minimizing, say, a univariate function over the interval [0,1], I really need to know the interval [0,1], to figure out what the minimum value is. The function doesn't know it. But this is, I think, kind of a gem in that sense. You don't need to know the constraint set. And then lastly, it appears all over the place in applications. So in things like image compression, you know, the Netflix problem is sort of a version of this, distance realization, you know, things that would come up in areas like molecular modeling, or protein folding, and so on. It's all — many of these problems can be thought of as low rank approximation to a given matrix.

KK: Very cool. Yeah, I'm now I'm thinking about that variety. It's not a Grassmannian. But it's sort of like, is it stratified by Grassmannians? Let's not go down this path.

RT: It’s just — yeah, it's a set of solutions to the equations that you get by setting all the rank, whatever minus to zero.

KK: Right.

RT: So the matrix cannot be rank more than K. So you set all the K+1 by K+1-minus to zero.

KK: That’s complicated.

RT: Yeah, yeah. It's a complicated variety with singularities and so on. Right.

KK: So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with this theorem?

RT: So I thought about this. This, to me was a very interesting challenge that you posed. So one thing that I always think about when I teach this theorem or teach the SVD in general, is there’s sort of a layer cake analogy. Okay, so I have always drawn this layer cake picture in my class. But when I started thinking about what I should tell you on the podcast, I thought, “Okay, it's not quite a layer cake, like you would buy in a store.” But it's sort of like a layer cake. So there is a layer cake analogy going on here. And that is simply we can think of a matrix. So if you have, say, an M by N matrix, let's start by just thinking of it as a rectangular chessboard lying on the floor. And then every entry of the matrix is creating, let's say, a building on each square. So you have, you know, the buildings have different heights, depending on the entry there. And then in that sense, what we're doing is, if you think of what a rank-one matrix is in picture that I'm trying to draw, then what it is, would be this sort of city block with skyscrapers that we've constructed, where when you look from East Avenue or West Avenue, you see one skyline. And then you see, like, various up and down versions of the same skyline as you look across. And then similarly, if you stand on North Avenue or South Avenue, you see one skyline. And then you see these up and down versions of that skyline as you go in the other direction. So that's a rank-one matrix. And then the whole matrix is built of these puzzle pieces, if you like. They’re all rank-one matrices. And what we're doing is sort of, take different puzzle pieces, you know, the first puzzle piece captures the most amount of energy in the matrix, then the first and second, the next amount, and so on. So that's sort of one geometric thing. And so thinking of a pairing, that’s just a geometric thing, not not exactly a pairing. But in my mind, another way to think about the whole thing is you could think of your matrices as, say, living in a universe. We have this M by N, universe. And then each of these landscapes, these, you know, matrices of rank, at most K, they form landscapes inside this universe. They're nested one inside the other. And you could almost think of your matrix as sort of a flying object. And if it needs to make an emergency landing on one of these landscapes, it knows exactly where to land. It doesn't need any, you know, radio control from the ground, right? There's no air traffic control on the ground who needs to tell this matrix where to land. So that's sort of my geometric vision of what is happening. I love geometry. So I always try to make pictures like this. But the closest physical phenomenon that I was thinking that maybe we could match with this is with the way migratory birds work, right? Like these migratory birds, they have sort of an inherent genetic compass in their head that tells you where they should land. So Florida being one of the biggest places.

KK: Yes.

RT: And that's exactly before they fly over the Gulf, right, where there's a long stretch of water so they know exactly where the end of the land is, or where the beginning of the land is when they fly in the other direction. So I think that's some amount of this sort of DNA information that's in their head. So there’s either genetic information — of course, they also use celestial signals like the sun and the stars and so on. But yeah, so that that, that to me was the best pairing I could come up with, just thinking of matrices as having this inbuilt computer inside them.

EL: I love that!

KK: I do too. As an avid birdwatcher, I'm really into this pairing a lot. This is really nice. Yeah, and luckily for the matrices, it doesn't get messed up. You know, I mean, I get various — I subscribe to Audubon and things like that. And I just read an article recently about how light pollution is really a problem for migratory birds. Especially, you know, they fly over New York City. You think they're not there, but they are, and you can catch it on radar data and all of that. And it's really become a problem for them. And especially with climate change, they're getting messed up on all these things — not to get off into birdwatching, but this is a really excellent pairing. I love it.

EL: Yeah, well, I know Kevin is quite the birdwatcher. I have only recently been getting a little more into it. And so I will think about flying matrices the next time I go and look at some birds. I recently discovered a new birdwatching place not too far away from where we live. And there have been some great blue herons nesting there. They're probably leaving soon, but they were there for the spring. And so that was cool as a very beginning birdwatcher to suddenly have your first serious like, “I'm going to watch birds at this place” have these nesting great blue herons at them. It really raises the bar for subsequent birding outings.

KK: They’re impressive birds, too. I mean, I've seen them. We have lots of them here, of course, I mean, they're walking around campus sometimes. But yeah, they'll catch up like a big fish swallow it whole. And it's really pretty remarkable. So yeah. All right. Well, Rekha, so we always like to give our guests a chance to plug anything they're working on. Where can we find you online?

RT: Oh, so I have a basically just by webpage. I'm not a social media person at all.

KK: Good for you.

RT: That’s basically where you can find me.

KK: You're part of a wonderful department. I've visited there several times.

RT: Okay.

KK: Great department, great city. And we'll be sure to like the your homepage. Anyway, thanks for joining us. I learned a lot today. This has been great.

EL: Yeah. And can I just, I don't do a little bit of tooting our own horn was saying that I love this podcast because we get to do things like now every time I decompose these matrices I’m going to think about migratory birds. And, you know, it's just, like, building all these little connections. I love it. Thanks for joining us.

RT: Thank you.

[outro]

On this episode of the podcast, we were excited to talk to Rekha Thomas, a mathematician at the University of Washington, about the Eckart-Young-Mirsky theorem from linear algebra. Here are some links you might find interesting after you listen to the show:
Thomas's website
Our episode with Tai-Danae Bradley, whose favorite theorem is related to Thomas's
Stewart's article about the history of singular value decomposition

Episode 67 - Liz Munch

10 juni 2021 | 33 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knutson, professor of mathematics at the University of Florida. I am joined today by your fabulous and glorious other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And I assume the “fabulous and glorious” means my new haircut, which I actually got professionally cut for the first time in quite a while recently.

KK: Good for you.

EL: I had been playing around with, you know, scissors — not official hair cutting scissors, just you know, the scissors that are lying in the kitchen drawer — and, like, my clippers and stuff for a while, but I went in and I look very sleek and chic right now.

KK: You do. You look very sleek. I’m getting a haircut tomorrow, I look less sleek.

EL: Not too bad, though.

KK: You know, I still have the plague beard though. I don't know, I can't decide what to do about that. And my hair has gotten longer, but you know, thinning, and I don't know what to do anymore.

EL: My husband actually decided to try for the ponytail. So he had been very excited about, like, two weeks after the second shot going and like getting a real haircut for the first time in a while. And then by that time, it was long enough that it's almost ponytailable. So he's like, “I've never had a ponytail. I think I want to try this.” So so now we're like at opposite, you know, going in opposite directions.

KK: I have a ponytail and my wedding pictures and then that’s the last time.

EL: Okay.

KK: Anyway, today we are pleased to welcome our good friend Liz Munch. Liz, you want introduce yourself?

Liz Munch: Hi. Yep, so I'm Liz Munch. So I am an assistant professor at Michigan State University. I'm in the departments of Computational Mathematics, Science, and Engineering, which is a mouthful, but we just say CMSE. And I'm also in the Department of Mathematics.

EL: All right, and what is your recent hair story?

LM: It’s the kind of, like, what can I do rolling out of the shower to like, make it do anything? And it's kind of like, wherever it lands. I did get a haircut, though, recently. It was very exciting. I had — it was getting so long that it was going out of control. And usually what happens is I I go get a haircut and I do like extreme versions. I'm like, “I hate my hair!” And then I chop it off to my ears. And then I grow it out again. And then “I hate my hair!” and I chop it off to my ears. This time, she convinced me not to chop it off all the way, but I'm due for one. So we're gonna work on getting that one soon.

KK: Looks good. Looks good. Yeah. So CMSE, that's interesting. You have this joint appointment. How's that? Yeah.

LM: I like it. It's different. So at least for me. So I do very interdisciplinary math, very applied math, but not your usual applied math. And so for me, it's a it's a really nice setting to be in. So CMSE is — so the faculty in CMSE are all sorts of different backgrounds. So there are mathematicians, but there are also statisticians and biostatisticians and biologists and plant biologists and geologists and physicists and engineers, and I'm sure I'm forgetting people. But it's basically rigged to be interdisciplinary. And so it's made a lot of fun, because you can find these interesting projects that are sort of in the intersection of complicated math and interesting stuff to do there with applied projects I would never have thought of before. So at least for me, that's been a really good fit for trying to do interesting applied math research explicitly outside of the usual academic silos.

KK: Yeah. Well, that's nice, because I think a lot of us in math departments often want to engage in these activities, but you know, the physics department’s in another building, and it's more effort, right? You know, it's true, there’s this idea of these collision spaces. So you know, if you have colleagues in all these different disciplines just right down the hall, it might lead to more interesting stuff. So that's a nice model. I wish we could do more of that. Anyway. So you have a favorite theorem, I hear.

LM: I do have a favorite theorem.

KK: You going to tell us what it is?

LM: I was going to talk about max flow min cut.

KK: Okay.

LM: Which is sort of my, I don't know — so the reason I like max flow min cut, is partially because, again, I do interdisciplinary applied mathematics, and I tend to fall into the theoretical computer science land a lot, just based on a lot of things we want to do. Because I work in TDA, topological data analysis research, and so a lot of things come down to here's a complicated thing I want to compute, and I’ve got to go figure out how to do it in a computer in a reasonable amount of time. So this is an example of that.

So max flow min cut. So here's the game. So you get yourself a directed graph. And this graph has a source and a sink. So you've got your S vertex source and your T vertex sink. And you essentially have capacities on each of the edges. So my directed edges, I now have an amount of stuff I can push across the edge. So I like to think about this with, like, water tube flows kind of thing, right? So I've got how much capacity each of the edges can take. And so the game is, try to see how much stuff you can push along from the source to the sink, basically playing nice with those capacities, so flowing from one side to the other. So if you're at any vertex, all of your inward flow amounts should be equal to all of your outward flow amounts, because you can't have anything hanging out at the vertex. And so your other restriction is your flow values — this is essentially like a second choice of weighting on all your edges — your flow values have to be less than your capacity values, right? So you can't flow more across an edge than the capacity allows.

KK: Sure.

LM: And so the trick is, what's the most amount of flow you can get across? I hand you one of these graph setup problems, what's the most flow you can get across these things? So the max flow min cut, as the title would suggest, is that you also need to know something about the minimum cut. So what's what's a minimum cut? So same starting input information. What you can do is you can try to divide your collection of vertices into two piles. And the rule is that you have to have your source in your sink in different piles. And the cost of whatever choice of piles has to do with, essentially the cost of the capacities that go from one pile to the other, right? So you sort of add up those values, there's some negatives if things are backwards, etc. But that's the cost of a cut.

KK: Okay.

LM: And so the game with that part of the problem is you want to make that as small as possible, right? Can I reorganize these vertices in such a way to make that cut cost low? And so the max flow min cut theorem says the capacity of the highest possible flow value is equal to the cut, the minimum possible cut value you could get. And so this is cool, because this means that you could answer your problem either by going and hunting for a maximum flow value or by hunting for a minimum cut value. And so it gives you two very different ways of looking at this problem to try to solve something. And these show up in all sorts of different application settings, right? So you can imagine, like, railway networks, where you're trying to move stuff around, you could try to do things with electric grids, and you can do things with water flow. There are lots of places where this thing would show up. And so having access to two different ways of looking at the problem is super useful.

EL: Yeah. So I have never thought about this before. So I'll just kind of — I’m not even sure if I can formulate the right question. But I'm trying to think, my mind immediately goes to extremes. So like, if you put everything except your source in one pile, or everything except your sink in one pile, is it obvious that usually that's not going to be good?

LM: I guess it kind of depends on the sort of setup you have, right? So I guess the simplest version would be like, okay, let's start with a graph that's just a path, right? And so in that case, I can stick a bunch of random numbers as capacities on my path, and so the maximum flow I can get across this path is going to be something like the minimum weight capacity that I've got, right?

KK: Yes.

LM: That minimum weight capacity is also going to be if I chop my path graph in half at that particular edge, that's as low as my cost for my path can go.

EL: Sorry, just to slow down a little bit for me. So that capacity is like, this one edge can only hold, you know, whatever amount of whatever stuff?

LM: Yeah.

EL: Okay. Great.

LM: This is kind of like the bottleneck in the system, right? Like I could have pushed, like, 30 gallons back here. But if this particular tube only allows for five gallons, it doesn't matter how much I can push in the backend, right? It's got to be, it's limited by this one, right?

EL: Yeah.

LM: And so then you basically take that, that path problem and scale it up. So then you've got all these other possible paths where you could push stuff through. You could imagine a vertex where you have, like, a capacity of 10 coming in, and maybe five edges coming out. And so your flow could be 10 in and then it got split up and there's all sorts of things. And so the way you prove this, so this is the Ford–Fulkerson algorithm from the 1950s, I think? It basically comes from looking for paths like that. So the whole problem comes down to basically that path example I was giving you. So you essentially rig up this modified version of your graph, and then look for paths from your start to your sink, where you try to see places where I could push some amount of flow across that line, across that path. And what you do is you essentially update some sort of flow that you're trying to build as big as possible. So if I've got a path where the bottleneck value was five, I now push value of five across each of the edges in that path. And then I update this other graph I can use to keep track of things. This is called a residual graph. Basically, then I update this flow, and then I go look for a new path. And then I update this flow, and then I look for a new path. And it all comes down to just finding these paths that have some sort of bottleneck in them and updating accordingly.

KK: That seems slow.

LM: Oh, yeah. Oh, yeah. This is not nice.

KK: So there must be better algorithms for doing this. But that’s the easiest to explain algorithm?

LM: Yeah, right, easiest to explain, and the one you get the proof off of. So that's the the Ford–Fulkerson one. And of course, I did not check this beforehand, but I believe the running time has something to do with the value of your maximum flow. And there are some caveats in here. So if you have integer-valued flows, so I'm only allowed to put integers on each of my edges, that process will terminate, which is a good sign, right? It will terminate at the best possible flow, which is a good thing, right? And worst-case scenario, you basically updated by one every time, and so the the order of the number of times you have to do that update procedure has to do with essentially the maximum flow value.

KK: Okay.

LM: So yeah, it's not pretty, but it works. This is another one of these examples where in practice, it tends to not be that bad. But of course, you can construct the nasty examples.

KK: Sure. Right. So yeah, the theory is bad, but the practice is good. So that's fine.

LM: Yeah.

KK: So do you use this much? Does it come up for you?

LM: I haven’t in a long time. The reason I was thinking of this was a while ago, I was teaching this summer program for bright high school kids, where we were doing a topology class for high school kids that basically have seen calculus. And so one of the reasons I like this was I was remembering I had rigged up this thing where I created a graph on the ground. And I put capacities on it involved in the — sort of like, I don't know, I think it was squares of paper or something like that — and messing around with getting them to actually manually push stuff around on this graph and start thinking about flows and how you can update information. Yeah, it hasn't come up in my research recently. But I think in general, just the messing around with problems where you can have multiple ways to find a solution, that's been really interesting lately, because trying to rig, trying to look at a problem from a different angle where you might have an easier ability to go solve something than something else, right? So like, if you told me go find the minimum cut in this particular graph, the knee jerk reaction is like, “Okay, let's test all possible ways of splitting the two piles!” which you really don't want to do, right? Bad idea. Never do that, right? And I don't have any good intuition for how — if you had to be that problem, I would start going at any sort of algorithm that would get me anywhere. But because we've got this duality idea, right, I can go back and do this pushing flows through a network thing that gives you access to a way to solve a problem that wasn't there before.

EL: Nice. And so do you remember, like really loving this theorem when you first saw it? Or is it something where your appreciation has grown over time?

LM: Ah, there's probably — now I;ve got to remember where I saw it. I'm going to guess I saw it in an algorithms class in my PhD. Yeah, I don't know. I don't know that I appreciated it at the time. I feel like a lot of the beginning of my math career was like, “Okay, this theorem exists, but I'm not sure why I care.” Right? It's taken me much longer to start getting to the like, “Oh, the point is the duality. The point is accessing a problem from different sorts of viewpoints.” And having the ability to sort of question the vantage that you're looking at something from because there might be a better way of doing something. And “better” here is now a subjective term, right? Like, mathematically, the theorem says, I don't care. But practically, right, there is a better and a worse way of doing something. And so you're going to end up having choices, as you go about doing math research, that are going to be better or worse in a practical sense,

KK: So it's clear listening to you that you work in an interdisciplinary environment, right? Because, you know, the mathematician loves the duality, but you know, practically, the engineer wants to find the most efficient way to do it. Yeah, very cool.

EL: So I wanted to come back to something you said a little while ago, and I think it just percolated through my brain. So you said that, like, if your your capacities on these are all integers, then the process terminates. And the fact that you needed that “if” actually just made me really nervous. So is it the case that if you don't have integers, and possibly even rational numbers or something like this? So like, maybe if you had like a bunch of irrational or transcendental values on these, that it might not terminate? And then you’d be sad.

LM: Yep, yep, yep. Yeah, then you'd be sad.

EL: Or maybe you would be sad. I don't know.

LM: Well, yeah. Okay. So I actually, I'm not sure of the answer if it's rational. I know that if you allow irrational values on the edges, you can definitely have examples that don't terminate. And basically what ends up happening — okay, so the way it is pushing the flow through your residual graph, there's a setup that allows you to essentially undo pushing through edges. So these residual graphs get this sort of backwards edge that's added to them. There’s the flow that you've already pushed through that you could undo and the flow that's available, which is the capacity minus the flow that you've done to this point. And so finding a path in this graph could push new stuff through, or essentially undo pushing things through and move it to some other path in your graph. So what ends up happening if you — again, if you create incredibly contrived examples — is that finding these paths in this graph can keep updating and then undoing the update. And you end up with this vicious cycle where you kind of like spin stuff around in circles, and you just never get there. And it's got all sorts of weird behavior, like not only it, like, doesn't decrease, it doesn't convert, like, it's, it's all kinds of nasty.

KK: Right? I would imagine with rational, though, you can do the standard trick of just scaling everything to be an integer. Like, that's probably okay.

LM: Oh, yeah. Because especially if you've got a finite starting graph, yeah. Yeah. Right.

EL: Oh, “if you've got a finite…” you're putting all these “ifs” that I wouldn’t even think you’d need!

LM: Let's have everybody be finite before I say something false.

EL: That’s fine with me.

LM: I am going to bet somebody thought about that.

EL: Oh, I'm sure. But, you know, since you started saying this with water pipes, I just immediately thought of, like, okay, a city water system. And I just know that that is finite. So yeah, in my mind, this is always going to be finite.

KK Right. And you'd sort of create one source by having it all come in your case, Evelyn from the Wasatch, right?

EL: Nice job.

KK: And then it all goes to the water treatment plant. That's it. That's your whole, you’ve got this crazy network.

EL: Or that eventually to the the Great — the Lessening — Salt Lake as it dries out.

KK: Yeah. All right. So also on this podcast is, we ask our guests to pair their theorem with something. And you mentioned what it was going to be ahead of time and I'm really intrigued. So let's look. What pairs well with max flow, min cut?

LM: Okay, so we're gonna pair this with the cross-strung harp, which I promise is going to make sense. So give me a moment. So just for some backstory, so my undergraduate, I was actually a harp performance major. That was my previous life. And I sort of backdoored my way into the math department.

EL: So I have to interrupt say that my sister is a harpist, was a harp performance major, is now working on a PhD in music theory, but teaches harp and has a harp studio and yeah, that’s such a cool coincidence.

LM: You probably might already know some of the stuff I'm about to say. So that's super exciting. Because, yeah, most people you talk to have no idea what I'm talking about. This is great. Yeah. So I spent the first part of my professional life, I was going to be a professional harpist. I got about halfway through and was like, “No, that's not for me.” I got into, started taking math classes. Ended up in a math PhD program, unbeknownst to me. And here we all are, right? Yeah, so the thing with harps is if you look at a regular harp, what you're actually looking at are, like, the white keys on the piano. So no chromatics, no sharps and flats. It's basically in one key, right? So there was an issue in, sometime around the 1800s, where they were trying to figure out how to make a chromatic scale possible on a harp. So prior to that, there was sort of a trick where you could put things like levers on each of the strings and they would shorten each string by a little bit. And so you could basically get, if you tuned your string in natural, you could get the sharp. If you tuned your string flat, you could get the natural, but that was it. So you could do the key of B — the key of F and the key of D, and then things got hard. And so in the 1800s, they were trying to figure out how to deal with this problem, and so there were two solutions that showed up at about the same time. So there was the cross-strung harp, which is not what you've seen before, and the double-strung harp which is what you've probably seen before. So if you go to an orchestra concert, somebody most likely has the cross-strong harp, or sorry, a double action harp, which is, you have two sets of essentially these twisty peg things that go on top of the string. And so now you engage a pedal, and the pedal, if you push it down once it engages one of these twisty bits, and if you engage it a second time, you get two of them. And so now you have three notes on every string. And so you tune it in such a way that now you've got flat, natural and sharp on every string.

EL: And there was much rejoicing.

LM: And there was much rejoicing, right. So this was invented by a watchmaker who was trying to solve this problem, because the mechanics of this are absolutely nuts. Like, I don't even know how the inside of my harp works. At the same time, so there was another option that was starting to show up, which was the the cross-strung harp, which was literally — you're going to have to Google this later — but it looks like two harps with two sets of strings that meet at an X in the middle sort of interlaced.

EL: Whoa!

LM: Yeah. I've seen these things. My brain goes fuzzy. Like, you can't focus on any of the strings, right? But the idea is that now you can move your fingers up or down to get sharps and flats. And so now you have access to all the strings. And it involves different techniques and things like that. And so there were these two companies. So it was Erard at the Erard company. So Erard was the watchmaker that invented the double action harp. And the Pleyel company had invented the cross-strung harp, and they were fighting about who was basically going to win the business.

KK: Harp domination.

LM: Yeah. There's some there's some catty stuff in the harp world, let me tell you. And so each of the companies, so there are actually two very famous pieces for harp that were commissioned by each company to prove that their harp was better. And so the Pleyel company, commissioned the Debussy dances, and the Erard company commissioned the Ravel Introduction and Allegro. And these two pieces are basically made to be — I'm not going to say easy, but like, manageable on your harp and not good on the other one, right? Right, so fast-forward 100 years, and the double-action clearly won out, right? So that's what everybody has now. But we still play both pieces. So if you go to college for harp performance, you're going to play both. And so the Ravel isn't, you know, again, it's not easy, but it is manageable, because you are playing it on the instrument it was written for. And then you get to the dances. And it is a combinatorial nightmare, because basically, what you're trying to do is figure out these fingerings, and how to play these different chromatic notes at the same time, when you don't have access to the other set of strings. You are now limited to trying to play the note on another string that you might have needed a half a second ago to do whatever.

So I promised that this was going to come back to something that made sense. So the whole point of this is that I was thinking of the cross-strung harp because it's one of these two solutions for the same problem. And not necessarily that one of them is better than the other, but in certain cases, there is definitely a priority. One of them really, truly did better.

EL: Yeah. That's so cool. And I have to ask one more question about the cross-strung harp. I don't want to totally derail this, but I'm going to slightly. So you know, when I watch my sister play like her right hand, you know, as on one side of the strings that are left hand is on the other. So with the cross-strung harp, then, your right hand and left hand — if you were playing, like on the diatonic, would have one would have to be up and one would have to be down, right? Because the strings go…

LM: Yeah. I will admit I've never played one.

KK: You’d switch them, right?

EL: Yeah, like if you move to flats, you move like this.

LM: You kind of want to be able to do both at once. So your technique would change entirely. I don't actually know how you would deal with this. If you want to Google even more things, there’s also a thing called a Welsh triple, and the triple — so what this is is now, this will make your brain hurt if you try to look at this thing — it’s now three rows of perpendicular vertical strings. So instead of being crossed through the middle, they’re, like, up, straight up. And so now you've got white notes on the outside, that are spaced wider than a normal harp, with sharps and flats down the middle. And so now you have to reach through the strings to try to grab.

EL: That seems bad.

LM: Oh, and I’ve played one of those too. And it's like your eyes just can't focus. Like I can't decide which layer of strings to be looking at. It was it was rough,

KK: But I'm sure that's mostly a function of having grown up playing the one that you play.

LM: That’s true.

KK: If you grew up playing the cross-strung or the West triple, that would just feel absolutely, totally natural.

EL: Yeah, but I guess you'd have to do something for visual markers, like so the harp has, is it the F and C strings are a different color to help you? Because on the piano, you can look at where the black keys are, and that tells you what notes you're playing. On the harp if they're all the same color, it's just a nightmare. And I guess on the Welsh triple, maybe you'd have to…

LM: I don’t remember what the coloring was on the Welsh triple. Yeah, on a modern harp, C’s are red and F’s are black. And then if you get older strung harps, they used to be switched, and that also makes my brain hurt.

EL: Oh no!

LM: It’s very much what you're used to.

EL: Yeah. Well, that's so cool. And what a funny coincidence that you’re harpist, just like my wonderful sister.

LM: Hi to your sister.

EL: Hi, Rachel!

KK: Do you still play much?

LM: Oh, not much anymore, unfortunately. I've moved my harp around to multiple cities and houses. Yeah, so I basically, I kind of put myself through grad school playing weddings. So that was nice. And then after, you know, kids and everything else, it's just kind of turned into less so.

KK: Well, you’ll come back.

LM: It might come back.

KK: Yeah. As someone who's now an empty-nester, I can tell you it does happen. So the other thing is, you know, my kid’s about to go off to grad school for music composition. So apparently just like Evelyn’s sister, he’s going to go be a theoretical musician, I guess. He's a percussionist, though, you know, and I can't understand how he plays drumset right. That’s, like, two arms and two legs, doing four different things. But harp sounds pretty bad too.

LM: Yeah, I thought I had to move around a bunch of stuff. They are so much worse. They got, yeah, that was that was always the trick. So when I was in college, the harpists saw got our own practice rooms, so we didn't have to move the harps around. Everybody else had to, like, fight for it. But the the percussionists also got their own practice rooms where they got, like, nobody wants to move around, you know, the xylophone.

KK: Yeah, although I've certainly moved Gus’s drum kit plenty of times.

LM: Oh, yeah. No, if you've gotten to the point that you've bought vehicles based on your instrument…

EL: Yep. I was going to say that's Rachel’s, I think Rachel has taken a harp to a car place to, like, test in the lot to see how easy it is to load.

LM: I have done that. I've purchased vehicles entirely because it could fit my harp in the back and I've confused — oh my goodness, I love showing up to car dealerships like this. It's my new favorite thing.

EL: Yeah. So before we end this episode, I just had — I didn't get a chance to say this earlier. But Max and Min can both be names, and I just feel like max flow and min cut just sound really snappy and sound like either a superhero duo or maybe, like, superhero antagonists or something. I’ve just got to put this idea out there for like, you know, a graphic novel about Max Flo and Min Cut.

LM: I want to read that book. Absolutely.

KK: Or Max Flow sounds like a news reporter from the ‘20s. You know, like His Girl Friday, right? You know, Cary Grant and Rosalind Russell or Max Flow and Min Cut in there. Yeah. So we also give our guests a chance to plug anything, or where can we find you on the line or …

LM: Oh, I definitely should have thought of that in advance. I don't know. Come see MSU. I like her. I like our new department. That's probably the first thing I should plug. I'm on Twitter more than I really should be. So you can always find me there. [Her handle: @elizabethmunch] That's — Yeah. I love Math Twitter. It's a happy place.

KK: Mostly.

LM: Yeah, I think that's mostly it. The other thing I guess I should plug is, so I'm very involved in the women in computational topology network. So if anybody is interested in anything in that general direction, so this is a group for not just women. So we're aiming for a broader community of gender minorities and a place for people to do math in a supportive environment because that, in my opinion, is the reason I am still here, and so I want to create that for other people.

EL: Cool, yeah.

KK: I can verify Liz is well-known for her no assholes rule.

LM: I yes — oh good, we can swear on this podcast.

KK: Sure. Why not?

LM: You just did. I do have a tendency to curse like a sailor, and so my husband before I started this was like, “Remember, Liz, you need to not swear on that.” I was like, “Okay.” I have a very strict “does not collaborate with assholes” rule. And it is gotten me very far in life and made for much better research.

KK: Life’s too short not to have that.

LM: Exactly. Yeah, exactly.

EL: So now are we going to go back and rerecord, like, the swear version of max flow min cut? Just kidding.

LM: Now that I'm allowed to curse for the whole thing?

KK: It’s pretty light swearing.

LM: You’ve got your R rating, don’t worry.

KK: No, no, no, R rating requires f bombs.

LM: See? We’re doing okay.

KK: So PG— so movie ratings for PG-13, you’re allowed one f-bomb. If there's more than one it gets an automatic R.

LM: Good to know. So we’ll hold off on the f-bombs, just in case this turns into a movie.

KK: That’s why you don't hear it as much in PG-13 movies because they they deliberately leave them out to get a PG-13 rating so the teenagers can come. But you can use any other swear you want. Anything. But just one f-bomb.

EL: All right. Well, with that important knowledge…

LM: We’ve gone on so many tangents at this point. I probably got math in here somewhere, right? Somebody did math.

EL: Yes. Our math, hair, harp, and movie ratings podcast can now conclude.

KK: That’s right.

EL: Great to have you, Liz.

LM: Thank you. Thank you both for having me. I really appreciate it.

KK: Take care.

[outro]

On this episode, we were happy to talk with Liz Munch, an applied mathematician at Michigan State University, about the max flow, min cut theorem. Here are some links you might enjoy after you listen to the episode.
Munch's website and Twitter account
The Women in Computational Topology Network
The Max-flow Min-cut theorem at Brilliant.org
The Ford-Fulkerson algorithm on Wikipedia
The cross-strung harp on Wikipedia
Harp.com's history of the harp

Episode 66 - Érika Roldán

15 maj 2021 | 37 min

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb. I am a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you, Evelyn?

EL: I’m all right. It's been raining a little here, which is very good, because we are in a perhaps somewhat historic drought and every bit of moisture we can get is fantastic. Probably not very close to your experience in Florida right now.

KK: It’s been pretty dry. But yeah, it's not really an issue for us. I mean, it’s actually really lovely right now, and I'm looking forward to kayaking some this week.

EL: Oh, fun.

KK: And my son graduates college in two weeks. And yeah, all kinds of fun stuff on the horizon for us. So anyway, let's talk math, though.

EL: That is exciting. Yes. We are very happy today to have Érika Roldan joining us. So yes, Érika, would you like to introduce yourself?

Érika Roldán: Yeah, thank you. Thank you very much for the invitation. I'm super happy to be here. And, well, I'm a postdoc right now. I finished my PhD thesis in 2018. And then I started jumping here and there, from Mexico to the states and now Europe. I’m at Munich, Technical University of Munich, and École polytechnique fédérale de Lausanne in Switzerland is my co-host. And yeah, I have this fellowship, the Marie Curie fellowship, for 11 more months, and then jumping again. Yes.

EL: Yeah. Well, that's very exciting. And what is your field of research?

ÉR: Well, it's stochastic topology and topological and geometric data analysis. I think most of the time, most of the brain time, goes to that. But also, there is something that is related because it gives extremal examples — you will never see them typically when you're using kind of random processes — but these extremal examples allow is to contrast with random ones, so I also do extremal topological combinatorics a bit.

EL: Okay, and I also am familiar with some work that you've done in recreational mathematics, which I guess might have to do with this extremal combinatorics too. And so if I can self-promote and Érika-promote a little bit, a couple of the puzzles that Érika has worked on appeared in the mathematics-themed calendar that I put together a couple years ago, which is still available for purchase through the bookstore of the American Mathematical Society and which is not specific to a year, so you can still enjoy this calendar whatever year it is when you are listening to this episode. So anyway, yeah, you did a couple of fun puzzles. I don't know if you want to talk about any of those. I am actually blanking right now on there's one with like, polyomino things, right?

ÉR: Yes. So there are two. One of them actually has a very special place in my heart because it was my first paper, and I wrote it for the Gathering for Gardner, this meeting that is every two years in Atlanta. It is a wonderful meeting. It is the first mathematical community, research community, that I got in contact with. And yeah, I did a complete analysis and characterization of a type of puzzles with colored cubes, and you have to stack them, and you have to do a tower and have some interesting coloring properties. And the most famous one is called Instant Insanity, just to give you the name. The name has a little bit of a clue of how interesting it is to try to solve them by trial and error. And yeah, I guess I did some computations and everything to characterize all possible different kinds of puzzles like this. And, yes, that's one of the of the entries of the calendar, this this wonderful calendar. Thank you for sending it to me. I enjoy it very much. And yes, and the other one was about maximally many holes with polynomials.

EL: Yes, that’s right.

ÉR: And for sure, we're going to go back to polynomials. Because it's related with the story that I want to talk about today about my favorite theorem.

EL: Yes. And you've provided the perfect segue now. What is your favorite theorem?

ÉR: Yes. Well, first of all, I want to say that I was thinking, and I changed my mind different times during the past two weeks. And I decided that I wanted to talk today about my favorite theorem, choosing it in particular for my mother to be able to follow the podcast. So today's the 10th of May. And, yes, because of the corona crisis, I think a lot of people have not been able to travel, and I haven't seen my mother for more than, almost a year and a half. And this is a way of sending her all my love and appreciation. So my favorite theorem, okay, so one of the things that my mom used to do, or does still very, very well, is shuffling cards. So every time that we gather together with my grandma, and in Mexico, this is very common that you gather with your family very often. It doesn't matter what, you always want to be in a huge crowd, just eating and playing. And yeah, she's a very good shuffler of cards. And my favorite theorem is a theorem by Perci Diaconis, the theorem for today is by Persi Diaconis, where he proves, and actually there is like a set of different papers that have different ways of modeling this shuffling, the usual shuffling that we know, and they prove how many shuffles you need to do to be a sampling from a fairly uniform distribution from all possible ways of having 52 cards of a deck in a certain order.

KK: This is a very famous result, and it's a surprising number, you're gonna tell us the number, right?

ÉR: Actually, it depends. It depends on the distribution that we use. And then this depends on the algorithm. And but yes, it's a very famous theorem. And it's very well-appreciated by by mathemagicians. Persi Diaconis is one of the most well-known mathemagicians.

KK: I have to say, I'm not surprised this is your favorite theorems. I've been at various TDA [topological data analysis] conferences with you. And topologists like to play fast and loose with distributions, and you are always quizzing us about which distribution we're trying to use. It's always “Wait a minute, wait a minute, what's the distribution?”

EL: So,

KK: Go ahead, Evelyn.

EL: Yeah. Um, so for someone who has not thought about card shuffling a whole lot, what do you mean by different distributions?

ÉR: So let me withdraw the word “distributions” for the first part.

EL: Okay.

ÉR: So you want to play poker or any other game, and you want to shuffle the cards in such a way that you feel comfortable betting $100. So, I have the deck of cards. And let's say that I decide to shuffle like this: I take the uppermost card, I take it, and then I put it in one of the possible positions that is not on top, or could be on top actually. And I choose where to put it uniformly randomly. That means I select one of the possible 51 spots, 52 spots, and I put it there. And let's say that I do this five times. Question for both of you: Will you bet $100 with me in this poker game?

KK: Uh, no.

EL: Probably not.

KK: It doesn’t feel very well-shuffled.

ÉR: Exactly. And I think no one will do it. Like, it doesn't matter if you know or not probability. We have a sense of when things are shuffled, even if we haven't heard the word “probability.” And so, let me just tell you that actually, for doing this, if I do this 10 minutes, you will start getting more and more comfortable, right?

EL: Yeah.

ÉR: At some point, you can say, “Okay, let's play.” That's enough after an hour, right? And the solution, or roughly the number from starting from which you feel comfortable and theoretically can be proven that is the kind of right stopping time, is around 200 shuffles.

EL: So this is not a very efficient way to shuffle, in case that wasn't already clear to everyone. One card at a time. Not so good.

ÉR: Yeah, well, in a Mexican family it will be okay, because everyone will be talking everything. No one will get, just, bored. But perhaps if you are there just for playing, it's not the right way to go. Now, in general, I want to say here, the right rate. In general, if you have n cards, with this shuffling you have to wait nlogn time. So n times logn is the rate that with this kind of shuffle will give you a uniform distribution as a convergence, and when you start to feeling that yeah, it is uniformly sampled. Now let's compare these to the actual way that my mom shuffles the cards. And this is commonly referred to as a riffle shuffle. So you take the deck of cards, and then you kind of try to cut it in into two packs, half perhaps. And then what you do is you hold each one of the packets with each one of your hands. And then you make — I’m trying to generate a mental image that everyone knows — and then you start with your hands, like, trying to put one after the other one, alternating from the packets from one and the other one. And there are different things to do here to convince you that this is for sure a different way of shuffling than the first one. That is one thing, and the other thing is, so if I do it one time, perhaps you will not be very happy. But whenever we're playing with friends usually is what? Five times, six times.

EL: Yeah.

KK: It depends on how many times I drop them. You know, when you do it, you do it the down and then you do that little flip where you reverse the cards to get that little “shhhh” sound. It took me for—I was an adult before I could finally ever do that. But then a few cards always go flying, right?

ÉR: And this is one of the things that I love seeing my mom doing because I think she has some secrets that she doesn't tell us because she does it so well. Like amazingly well.

KK: My mom was good at that too, actually. That's interesting. Okay.

ÉR: We have these two different ways of shuffling. And we we kind of feel that, or we're comfortable with saying a riffle shuffle shuffles faster than the other way. And so now I want to use to use this shuffling perspective to go and shuffle other mathematical structures. Because what happens is that I ended up using this way of shuffling objects, or mathematical entities, in my research. And that was a very pleasant outcome of my PhD thesis. Yeah.

EL: So you're saying that like with these polyominoes with holes and stuff, there's some way to think about shuffling polyominoes, or do you shuffle entire configurations of them?

ÉR: Yes, exactly. Yes. It's super fun. So okay, let me start with Tetris.

EL: Okay.

ÉR: I think if we start with Tetris, everyone who's listening to us will go directly to the picture, the image of what is a polyomino.

EL: Oh, that's right. Yeah, we didn't actually say what a polyomino is. So in Tetris, you know that each picture is made up of four blocks. And so that's a tetromino is. So it's like a polyomino, I guess, is something that's made up of any number of these blocks.

ÉR: Yes, yes. Exactly. So. I think my, my favorite definition of a polyomino was the first mathematical definition given by Solomon Golomb when he was a PhD student. He gave a talk, and he said, well, a polyomino is a rook-wise, connected subset of squares of the infinite checkerboard.

EL: Okay.

KK: Rook-wise.

ÉR: So you can imagine you select a finite number of squares, and then you place a rook in any one of the squares, and you have to be able, with only rows and column moves, to go and visit any other square of the structure. So rook-wise connected.

EL: Yup.

ÉR: Yeah. And with Tetris, what happens is that we — let's now come back to to the mindset of sampling. And when we're playing Tetris, there is some entity that is giving us one polyomino after another one. Question: I don't know if people have thought about this. But it's a fun thing to do. I did one morning, and I was like, “I don't know.” Yes. So how is this done? How does the game decide how to give you the next polyomino?

KK: It’s not just random?

EL: Yeah, I guess I assumed it was — there was a however many Tetris pieces there are. I actually can't think off the top of my head, but six or seven, maybe different pieces?

ÉR: Seven.

EL: Okay, then just so a 1/7 chance of each one, is that not the case?

ÉR: Well, you soon will be able to play the game that you're describing, because I'm programming a version with uniform distribution, just to make people really mad and upset.

EL: Okay.

KK: I can see, actually, now that I think about it, if it were just random, uniform, that would be bad. You’d lose pretty quickly, right? I mean, now that I think about it, often you'll get that long, the straight four in a row right when you need it, right? And and if it were uniform, you wouldn't necessarily expect that to happen.

ÉR: It could, you could have bad luck. Yeah. You could have had as long as you want bad luck.

KK: Right? Sure. That's right.

ÉR: Right. So the algorithm is very interesting. What they do is, they take the seven, the possible seven polyominoes with four squares that they give, and then they shuffle these seven in a way that you have a one over seven factorial, each one of the possibilities of the order has one over seven factorial, that means it’s a uniformly distributed sample. Okay, so what what can happen? Well, the worst that you can wait for that “I” shape is that if you have at the beginning of a set of seven, and then in the next set of seven, you have it at the end. That’s the worst that can happen in terms of these wonderful pieces that allow you to kill four rows.

KK: Interesting. Okay, so they're just choosing a random shuffle. Oh, wow. Well, that is better.

ÉR: It is for playing.

EL: I guess with, like, computer games like this. We, as you're saying earlier, Kevin, we don't actually want it completely random because we don't want to get, you know, five squares in a row or something like this, which can definitely happen when you play it randomly. Especially when you're me and my brother when we were, like seven and nine years old, always going in and playing for two hours to try to erase the other ones’ high scores.

KK: I still remember, so when I was in graduate school, I had a Nintendo Gameboy. My mother gave me one of these, and I played Tetris. And I can still hear that song. The music that went along with it, and I would go to sleep at night picturing these damn pieces just falling. I can hear this song over and over and over my head.

ÉR: That is a thing I think that happened when we were doing a PhD because I was always with my Nintendo playing Tetris and procrastinating, beautifully procrastinating. And it turns out.

EL: I have a question. How do you — like, did you find how Tetris is programmed? Or did you just do some — Like, how did you figure out that this is how Tetris works?

KK: Yeah.

ÉR: Well, so I started reading about it. But first of all, I started playing, and I started noticing this pattern, right? This pattern, because you never see — I think, I think what what tells you immediately that it cannot be a random distribution, uniform random period, is that you will never see three times the the same shape.

EL: Okay, you could see two in a row, if it happened, right at the end and then the beginning, but you can't see three in a row.

ÉR: You can’t see three in a row! So this is one of the immediate signs. So whenever you don't see three in a row, and you have a decently small set — and we're going to go back on the size of the set, because it plays a role. So it plays a role, the distribution that you have in the set, and it plays a role also the size of the set. And so in this case, because it's such a small set, not seeing three in a row is like, uh-uh, something fishy is happening here. This cannot be the uniform distribution, period. Yes.

EL: Okay.

ÉR: Yeah, and this is with four squares. And now imagine that we three get a very nice contract by a company that tells us, I want you to build a very nice Tetris, but it has to be for any number of tiles. And then we start discussing, okay, so let's see, one of the things that we need to do is to extend, or just to apply perhaps this algorithm, but it has to work for any n, any number of squares, let's say 57 squares, right? So, first step, go and find all polyforms with 57 squares.

KK: That’s a lot.

EL: Yeah.

ÉR: That’s a lot. Yes. And I kind of, yeah, I kind of, in purpose, choose 57. Because 57 is the first amount of squares that we as humans, don't know how many polyominoes there are with 57 squares.

EL: Okay.

KK: All right.

EL: It’s just too too many for for us to have sorted them out somehow.

ÉR: It’s too many for the patience that people and computers and computer time have. Like, 56, I think was two years.

EL: Wow.

KK: So there's some algorithm but it is not super-efficient.

EL: It hasn't stopped yet.

ÉR: It’s a little bit worse. Because even even if we wanted to say, Okay, let's cut at 56 because we know how many there are, or up to 56 and we deliver the game like this, right? So the problem is that this algorithm can count them. But this algorithm cannot really store the pieces or actually look at them, which is something that is possible to do. This is one of the magics of mathematics and computer science, that we can say something about something that we cannot see. And then we know up to 28 tiles, Tomás Oliveira e Silva in in 2015, was able to look at them and, for example, say the symmetries that they have because it's unknown, really, how many of them are going to have which kind of symmetries and so on and so forth, all the geometric and topological properties that you can have with polyominoes. And yes, so I guess we have to talk to the manager and say to the manager, we have a problem here. I don't think we can create Tetris for any amount of tiles.

EL: Yeah. Plus, I mean, it would be — I bet a lot of these tiles, as they get more pieces in them, have holes in the middle that you can just never plug. Because you can you can have rook-wise moves that will sort of surround this hole. Which I am thinking of because of your puzzle from the calendar. So I guess, do you happen to know like, at what size Tetris stops being fun?

KK: Eight.

ÉR: There is the answer. Well, with eight, perhaps not yet. Because almost all of them will not have holes.

KK: Right. But you’re still going to get that one with a hole every once in a while. And then you're done. Well, not yet. But it's, it's gonna cause problems.

ÉR: Super difficult.

EL: Yeah.

ÉR: Yes. And that is a very, very good observation, because actually, one of the things that is a mathematical truth is that most polyominoes have holds.

KK: Sure.

EL: Okay.

ÉR: Meaning with probability one, as n goes to infinity, if you take a polynomial at random, it will have most likely a linear amount of holes with respect to the number of tiles.

KK: Okay.

ÉR: And this is one of the topics that I started studying in my PhD, is how does the number of holes grow with respect to the number of tiles? And because we're asking a question upon a set that we don't know how many there are, and we don't know how to actually look at all of them. Right?

EL: Yeah.

ÉR: So the only way that we can say something is, for example, to have actual numbers, is sample. Sample and tell me. Sample them. Yeah, but we have another problem. How do you sample from a set that you don’t know?

EL: Yeah, that doesn't seem easy.

ÉR: Yeah. And that is huge. That is humongous. Because it’s — the number of polyforms is growing exponentially fast with respect to the number of tiles.

KK: Right? That's what I was going to say. I mean, even, you know, if you're up to, like, 10, the number is probably so large that you can't construct a reasonable game out of it, right? Yeah. So I mean, so Tetris is — so I have this game with blocks called Katamino, where they’re pentominoes, right? And that's fun, and challenging enough. Yeah. Okay. So now I understand why you're interested in stochastic topology, this feels very, if you're looking for holes in these things, you've set up some rules. And now the question is when do the Betti numbers change and things like that.

ÉR: Yes, yes. And now that you mention polyominoes, I also like like putting here and there one puzzle, so that people, perhaps even build it in their homes or something. So when Matthew Kahle was in Mexico visiting for my candidacy exam, we exchanged puzzles, and one of the puzzles that we exchanged is you take an eight by eight, a chess board, a usual eight by eight chess board. And then you have and then you break it on the head — no. And then the story in Diedonne’s book, it goes like this, that there were two royal persons playing chessboard, one loses, and then breaks the chessboard on the head of the other one, and then you get pieces. And these pieces are the 12 pieces that you were referring to, Kevin, the pentominoes, plus the two by two square. And then one of my favorite puzzles, and now so because Matt did this, built this puzzle with bamboo, with a very, very thin bamboo material, and he gave it to me as an exchange of the puzzles that we had. We actually had a night of puzzles and games back there.

KK: That’s fun.

ÉR: A lot of polynomials in my mathematical life.

KK: Very cool.

EL: Yeah. So, another thing we like to do on this podcast is asks our mathematicians to pair their theorem or example or their mathematics with something else. And so, what would you like to pair with this theorem about shuffles? And it may be even with the applications to all of these polyomino puzzles.

ÉR: I will say that is drawing and 3D modeling.

EL: Okay.

ÉR: One of the things that I like is to run these run some of these algorithms for generating poly-cubes in 3D, and different algorithms. And then to get these 3D crazy structures, and I like doing the modeling. So I guess, is the is a way of I, I think, it’s random art?

KK: Sure. Sure.

EL: And what programs do you use to actually draw those? ÉR: Oh, well, here, and this is one of the things that I have been really into it ,is there is a program called Maya. And it’s for doing 3D modeling and rendering. But it can be combined with Python.

KK: Okay.

EL: Okay. So do some other math with it, and then put it into Maya?

ÉR: Yes. And so I found someone to give me some classes on Maya. And then this person was always like, “But how do you manage to do 1000 cubes and decide where to put them?” And no, like, I just click and play, and I have an algorithm that decides to go all the way to 1000. So I guess it was a very interesting combination, knowing the kind of fast-track things that we do as mathematicians, and then mixing that with art. I like digital art. Yes.

EL: Nice.

KK: Very cool. Okay, so so we also like to give our guests a chance to plug anything, or where can we find you on the internet, or anything else you want to tell us let people know.

EL: Yeah, if people want to play anything that you made, or where they can find this frustrating Tetris game that you'll be coding once you make it.

ÉR: Yes. That is going to be available very soon. But also, because I’ve been developing these apps with a computer science collaborator, and one of the apps that I have, and that I will provide a link for people to play, is for finding polyforms formed by squares and triangles. So you can also play this in the infinite the triangle tessellation of the plane, or you can play this with any tessellation. You could play it even in hyperbolic spaces, you could play it in higher-dimensional spaces, but the app that I’m going to share stays with the square and the triangular grid. So, you will be able to find polyforms with maximally many holes and it will tell you if you are actually reaching the maximum number of holes that you can for that amount of tiles, so it has that property. And I guess another another random way that I have for for producing these random polygons is like a cell growth model. And what you do is you start with a cell, and then you add uniformly random one of the cells that are neighboring, so you can imagine this is a cell colony growing. And there is a huge area of research called first passage percolation that has been analyzing these kind of models. But what I do is I study the topology of this other way of randomly generating polyforms. And I have an app that I will also provide here so that people can think about it and perhaps come up with some conjectures that later, they can see in one of my papers with Benjamin Schweinhart and Fedor Manin.

EL: Okay, great.

KK: That’s fun.

EL: So, you’ll get homework after this episode, but it's not a quiz. So it's still okay.

KK: All right. Well, this has been a lot of fun, Érika. I always like to hear about these sorts of questions, even though I'm no good at them. I always like to talk to people who are good at counting and these kinds of things. So it's been it's been great fun talking to you. Thanks for joining us.

ÉR: Yeah. Thank you so much for the invitation. And yes, looking forward for seeing you in the USA soon.

EL: Yes. Bye.

ÉR: Bye.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Érika Roldán, a Marie Curie fellow at Technical University Munich and École Polytechnique Fédérale de Lausanne, about shuffling cards and Tetris pieces.

To read about the mathematics of riffle shuffles, this article by Persi Diaconis is a good place to start. To get a copy of the American Mathematical Society page-a-day calendar, click here. (If you already have a calendar, check out Dr. Roldan's puzzles on August 28 and October 12.)

Dr. Roldán shared some other links to and explanations of some of the apps and videos she mentioned in the episode:

The COVID crisis has allowed me to start developing digital material for my research, teaching, and outreach on mathematics and its applications. It has also allowed me to collaborate as a developer of digital material (in Germany) with artists whose projects promote gender equality, and diversity & inclusiveness awareness. Here, I share some of the links to explore this digital playground (new digital material created will be posted soonish at my website: http://www.erikaroldan.net/):
1) https://000612693.deployed.codepen.website
Follow the link above to find Extremal Animals, that is, polyforms with maximally many holes. A polyform is built by gluing together squares or triangles (in the case of this app) by their edges. And a hole in a polyform (that mathematicians call the first Betti number in this 2D case) is a finite connected component of the complement of the polyform. To get some intuition, just build a polyform with squares with 7 tiles and one hole, or a polyiamond with 9 triangles and one hole. Could you create one hole with less tiles in any of these two cases?
Have a look at these papers for the maths behind this (Extremal Topological Combinatorics) puzzle of finding polyforms with maximally many holes:
https://arxiv.org/pdf/1807.10231.pdf
https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i2p56/pdf
https://arxiv.org/abs/1906.08447v1
2) https://000612976.deployed.codepen.website
Here a link to an app that has a model to generate a random polyform by a cell growth process called The Eden Model. Pay attention to how the holes are created and destroyed as time (the number of tiles) evolves. Do you have any conjectures about how the number of holes is changing concerning time? Have a look at this link to see if your conjectures are stated and/or proved in this paper:
https://arxiv.org/abs/2005.12349
3) My first proto-game with Unity was developed for the Film “Broken Brecht” directed and produced by Caroline Kapp and Manon Haase, for the Brechtfestival Augsburg, Germany (Mar 2021). This is a project that will be extended during 2021! Here some links to the festival, the proto-game, and an extract from the film that happens within the proto-game.
Brecht Festival
https://brechtfestival.de/brokenbrecht/
Extract from Broken Brecht
https://vimeo.com/542287814
Link to the 3 min Archive Video Game
https://simmer.io/@ErikaRoldanRoa/~56f30f68-048c-c027-7aa0-aeaca82508fc
4) Some 3D models created with Python & Maya to explore (random) cubical complexes.
https://sketchfab.com/erikaroldan

Episode 65 - Howard Masur

8 april 2021 | 29 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast where there's no quiz at the end. I remember we did that tagline, like, I don't know, probably two years ago or something. And I forgot that I wanted to keep doing it. But I did it today. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. I forgot that tagline, too, and it's a pretty good one. Let's, let’s—look.

EL: We’ll see if we remember later.

KK: After our last recording session, we agreed we needed we needed a real tagline. So yeah. We're recording this on February 18, which means that Texas is largely without power and frozen.

EL: Yeah.

KK: And it's 82 degrees in Florida today.

EL: Oh wow. Yeah. Most of my family is in Texas, and it is not great.

KK: Do they have power? Or? No?

EL: Most of them do. All of them do sometimes.

KK: Right. Actually, now I think water is getting to be a problem now. Right?

EL: Yeah. I haven't heard about any problems with that from my family. But yeah, it's not great. I hope that it warms up there soon and everything can come back online. But yeah, today, we're very happy to be talking with Howard Masur, who is in a place that is very used to being cold and snowy. So yeah, Howard, do you want to introduce yourself? Tell us a little bit about yourself?

Howard Masur: Okay, thank you. First of all, thank you very much for inviting me to do this. I’ve been very excited thinking about about it. Yes, I'm on the math faculty at the University of Chicago. And I've, I guess, been working in mathematics for quite a long time and still enjoy it a great deal. It’s a major part, a very big part of my life. And your invitation to talk about my favorite theorem led me to, you know, think about what that would be and why I chose what I did. And and it made me think that, yes, what I really like the most in mathematics, or one of the things, is mathematics that connects different fields of mathematics. And maybe unexpectedly connects different fields. And I personally, have worked on and off in complex analysis and geometry and dynamical systems, another field. And I love the part of mathematics that sort of connects them.

EL: Well that's perfect. Because I mean, you you're a frequent collaborator with my husband, Jon Chaika. But also with my advisor, Mike Wolf, who, you know, isn't quite in the same area of math generally. So yeah, you have worked in a lot of a lot of different fields that I feel like your name pops up, you know, in a very wide range of things related to geometry, analysis, dynamics, but yeah, you’ve got your finger in a lot of pots.

KK: Right. Well, okay, so what is it? What's your favorite theorem?

HM: Okay. It's called the Riemann mapping theorem.

KK: Yes.

HM: So, let me let me give a little bit of background. The first thing, it involves subsets of the plane which are called simply connected. And this is a notion from topology. And let me just say I looked at one of your podcasts and someone else talked about the Jordan curve theorem, where if you have a simple curve in the plane — it could be very, very complicated — a simple closed curve, then it has an inside and an outside, then the inside is simply connected. And a way of thinking about what simply connected means is heuristically it doesn't have any, it has no holes. But as also has been pointed out, they can be very complicated, Jordan curves. Certainly they can be simple looking like a circle. The inside of a circle is simply connected, the inside of a rectangle. But on the other hand, the Jordan curve can be very complicated like a snowflake, a Koch — I never remember how to pronounce that; is it “coke” snowflake?

KK: Let’s go with Koch [pronounced “coke”].

HM: Pardon me?

KK: Let’s go with that.

HM: Okay. And so that's very complicated. It's the boundary — the curve is a fractal. So already simply connected domains can be very complicated, but they don't even have to be just the inside of a Jordan curve. You could take the plane itself, there’s a very simple example. You could take all the positive real numbers, include zero, and take it away from the complex plane. So the plane minus the positive real axis and also subtract the origin, that’s simply connected, it doesn't have any holes. And it's not the inside of a curve. You could also, on the other hand, here's something that isn't simply connected: you could remove the interval [0,1], including zero and one from your plane, just that interval on the real axis. And that is not simply connected because the complement, or the plain minus that, has a hole, which is that interval [0,1], it can be thought of as a hole. So that's the notion of simply connected. I don't know whether I should say more. I mean, that's what I thought to say about what simply connected means.

KK: That’s great. Yeah, yeah, that's a good explanation.

HM: Okay, and so that's a topological notion. And then the other thing that goes into this theorem is a notion from geometry, well, actually a notion from geometry and a notion from complex analysis. But let me let take a basic notion from geometry, which is called conformal. And the idea is that if you suppose you have two domains in the plane, and you have a transformation from one to the other, you say it's conformal if it's angle-preserving. So that means that if in the first domain, you have a pair of arcs — or maybe you prefer to think of them as straight lines, but it's better to think of a couple of arcs — that meet at a point, and then you apply the transformation, and you get a pair of arcs that meet in the image under the transformation. And you could measure the angle that you started with between the pair of arcs and the angle of the images of the pairs of arcs, and if the angles are equal at every point for every pair of arcs at those points, then you say the transformation is conformal, angle-preserving. Now, in some ways, the nicest — so let me give some examples that are and are not. The nicest transformations, certainly of the plane, are linear transformations.

KK: Sure.

HM: Given by two by two matrices, and they turn out not to be typically conformal. There are some that are, for example, a rotation about the origin is conformal. You know, if you have two lines and you rotate them, the angle they make after rotation is the same as the angle they started with. If you — this isn't strictly a linear transformation, it’s called affine — if you take a translation of the plane, if you take every point and you add the same vector, think of them as vectors, that's angle-preserving, that's a conformal transformation. Here's another one that's back to linear. If you take, for example, every point, which has, say, coordinates (x,y), and you multiply x by 2 and y by 2, so you multiply the coordinates by the same number, 2. That's called a scaling. And that's angle-preserving. One can sort of check that out. What that transformation does is, for example, it takes a square with one vertex at the origin, a unit square, and then another vertex on the x-axis at the point (1,0) and another at the point (0,1), last point at (1,1), and it takes a unit square to a two by two square, and that's angle preserving. But that's it — well, and the composition — but typical linear transformations are not angle-preserving. So, for example, if you took (x,y) and the transformation took (x, y) to (2x, y/2), so it multiplies in the x direction by 2 and multiplies in the y direction by a half, it takes a unit square into a rectangle, and that's not angle-preserving. It preserves the right angles, but it doesn't preserve other angles.

EL: Yeah, you can imagine the diagonal is, you know, [demonstrates with arm gestures that are very helpful to podcast listeners].

HM: The diagonal is closer to the x-axis, so the diagonal which made an angle of 45 degrees will be moved with the x-axis. The x axis goes to itself, and the image of the diagonal is moved closer to the x axis. Yeah, exactly.

So there aren’t maybe, there aren't so many linear transformations of the plane to itself, and so let me tell you what the theorem is, and this is a beautiful, beautiful theorem, I think, and it was really a cornerstone of, in the 19th century, of the beginnings of complex analysis. Oh yes, I’m sorry. Before I do that, let me also connect conformal, as I had mentioned, to complex analysis. One also can think of the euclidean plane as the complex plane, where (x,y) becomes x+iy, becomes a complex number z, and then conformal, another way of saying it, is that the map, the transformation from some region in the plane to some other region in the plane has a complex derivative. It’s what you call complex analytic. It has a derivative and the derivative is not zero. Again I looked at your podcast. Someone talked about the Cauchy-Riemann equations, and that's exactly what complex analytic means is that the Cauchy-Riemann equations hold. Where where w is u+iv and z is x+iy, then it's complex analytic if ux=vy and −uy=vx. That’s the Cauchy-Riemann equations, and that's from complex analysis. It has the names Cauchy and Riemann, who where in some sense the founders of complex analysis. And that's equivalent to conformal, so even there just in this, there's already kind of an amazing theorem that relates — I think obviously you had somebody on your podcast maybe talk about this — that relates complex analysis to geometry, conformal meaning angle-preserving and complex analytic meaning, let's say, the Cauchy-Riemann equations hold.

KK: Right.

HM: Okay, so the theorem is that if I take any simply connected set’s domain in the complex plane, other than the complex plane itself, okay? And I take the unit disc — so that's inside the circle of radius one, so that's simply connected — I can find a conformal transformation from the unit disc to this simply connected domain, and maybe thinking about the inverse, it's a conformal transformation from that (maybe crazy) simply connected domain to the unit disc, and so that's the Riemann mapping theorem

EL: Yeah, and it's just amazing. I mean I think there's part of me that still doesn't believe that it's true. I've actually just, I don't know when it was, maybe a month or two ago, I think I was brushing my teeth or something and just thinking, why hasn't someone pick the Riemann mapping theorem yet for My Favorite Theorem?

HM: Okay, all right.

KK: It's a really mind-blowing theorem. So when I teach the undergraduate complex analysis course that we have, I don't get to it until the very end.

HM: Yeah.

KK: And it's kind of hard. You can't even really prove it especially at that level, but students just look at me like, there's no way this is true. This just can’t be true. So it's really remarkable that anything — I mean, you're right. I mean, these simply connected domains can be bizarre. But they're conformally equivalent to the unit desk. That's just blows my mind still. Yeah,

EL: Yeah. It's just hard to imagine, like, this fractal snowflake, you know, how can you straighten that out enough to just be like a circle?

HM: Let me contrast it — and this also goes back kind of to the founding mathematicians of the subject. If I take what's called an annulus, let's say I take the circle of radius 1. And I take the circle of radius R, where R is bigger than 1. And I take the region between them. So the region between two concentric circles, that's not simply connected because it has a hole, namely, the inside of the unit circle is the hole. And so if I take one of radius, the inner is radius 1, the outer radius is R, and I take another one, inner radius 1 and outer radius R’. And let's say R’ is not equal to R. So it's a different outer radius. They are not conformally equivalent, even though they are very simple boundaries, their circles. So there was something very, very special about simply connected. And that's also kind of what makes the theorem amazing. And then the fact that it doesn't work for something not simply connected started a whole field of mathematics that has been going on for close to 200 years.

EL: And so was this kind of a love at first sight theorem for you the first time you saw it?

HM: You know, I guess I'm not 100% sure. I was in college a little while ago, and I don't don't think I had complex analysis in college. And so I may not have run into it then. But certainly, as a first-year graduate student at University of Minnesota, and my professor, who then became my thesis advisor within a year, you know, for my PhD advisor, that was somehow his field. And so I certainly learned it as a graduate student. And that led me — again, I can't exactly say it led me to what I do — but, you know, it certainly had a big influence, and things that I do sort of have grown out of this whole history of this, from the from from the Riemann mapping theorem.

KK: So, is this one of those theorems is actually named correctly? Did Riemann actually prove it?

HM. I don't know, I'm not a historian. You know, I mean, I could ask. For that matter, are the Cauchy-Riemann equations named after the right people? Yeah. I mean, I know the modern proof that one sees in books on the Riemann mapping theorem is not due to Riemann. It’s I think, early 20th century.

EL: Is it Poincaré maybe?

HM: You know, my mind is going blank here for a second.

EL: It’s someone.

HM: I don't know. I'm not a historian, and I did not look it up to say “Does Riemann really deserve credit?”

KK: But wait, I looked at Wikipedia. I’m cheating. The first rigorous proof of the theorem was given by William Fogg Osgood in 1900.

HM: Oh, okay. Okay. Yeah.

KK: So apparently Riemann, this is in his thesis, actually. But there were some issues, it depended on the Dirichlet principle. And Hilbert sort of fixed it enough that it was okay. But Osgood is credited with the first rigorous proof.

HM: Well, isn’t it also somehow the case again, that mathematicians 200 years ago did not quite have the rigor that we have now?

KK: That’s true. Cauchy sort of put limits on the right footing more or less, but I think it still took a little while to get it cleaned up, right? So are there any really interesting applications of this theorem that you like? Or is it just beauty for its own sake?

HM: Gosh, you know, I'm not sure. I think beauty for its own sake, I mean, but also to my mind, it opened up a whole branch of mathematics where you study, well, for example, you study surfaces. Or maybe it's the difference between topologists and geometers. A topologist says that a doughnut, famously, a doughnut is the same as a coffee cup with a, you know, with a handle and so forth. And geometers say, well, we could put different ways of measuring angles, different metrics on a torus that are not conformally equivalent, that there's no transformation from one to the other that preserves angles.

KK: Right.

HM: And this Riemann mapping theorem says, No, you can't do that for simply connected. They are conformally the same. But as soon as we move to topologically more complicated things like a torus or even these annuli, or surfaces with more holes, genus, then there are different ways of putting metrics and measuring angles and so forth. And so it opened up, and again, this actually also has Riemann’s name to it, it’s the Riemann moduli space, is the study of all metrics on a space. And so, yeah, again, I haven't thought of an application so much to other fields, but something that is a beautiful and unexpected theorem that opened up whole vistas of mathematics, I think, in the last whatever. I don't remember when Riemann stated this problem. When did he live, in the 1840s?

KK: Yeah, middle 1800s.

HM: So it’s been 175 years or something that people studying, have been studying? consequences in some sense of, of this or analogs of this?

EL: Yeah. Well, and so this is something: I never wonder it at the right time to check and see, but is there a place where you can go and say, like, this is my domain 1 — and maybe it's a square or maybe it's the flag of Nepal or something — and this is my domain 2, or just the unit circle, and here is the conformal map between them. Is that something that exists?

HM: Typically not. There are certainly examples where you can, but it's very, very rare that you can write down an explicit formula for the map. That's again, maybe why it's such a beautiful theorem, but you cannot, I don't let’s see, I hope I'm not — maybe you can do it for a circle to an ellipse. Um, maybe. I'm not 100% sure. There are people who know much, obviously know much, much more about finding something explicit. But in general, no, if you take some crazy Jordan curve, no way, do you know an explicit formula.

EL: You just know it's there.

HM: You know it's there.

KK: Well, that's important, though, right? If you're going to go looking for a needle in the haystack, you do, in fact, want to know there's a needle in it.

EL: Yeah.

KK: All right. So another fun part of our podcast is we ask our guests to pair their theorem with something. So we're dying to know what pairs well with the Riemann mapping theorem?

HM: Well, I thought about that a lot.

KK: This is the harder part.

HM: I tried desperately to find food, but I couldn't think of really the right thing. So I do love music. And this is maybe crazy far-fetched, but I paired it with Stravinsky's Rite of Spring only because to me, this Riemann mapping theorem revolutionized geometry and complex analysis. And I think of the Rite of Spring of Stravinsky, which was premiered in the early 20th century, revolutionized modern music, contemporary music. That's the best I can do.

EL: I like that.

KK: Well, I do too. And for all we know, there were riots after Riemann published his theorem.

HM: Could have been.

KK: You know, “there’s no way this is true!” Mathematicians stormed out.

HM: Maybe he gave a lecture and people threw tomatoes at him.

EL: Yeah, well, I must say when I was thinking about asking you to be on the podcast, I did think about the many wonderful meals that we have shared together, and I know that Howie is a great appreciator of the finer things in life, including music, too. I think we've gone to a concert together. And yeah, so I thought that this would be an excellent thing. You know, I was I was talking with Jon earlier about, what is Howie going to pair with it? And my first thought was actually pancakes, which I think are a little pedestrian, but that you can make them into so many different shapes. And there's even, there are people who will do these things where, you know, if you pour the batter on, in a certain way, you know, you can get these beautiful things. I mean, part of it is part of the batter cooks longer than the rest of it. And so you've got shading based you know, how they do I've seen, I think, you know, Yoda and like, I don't know, all sorts of different things. There’s this Instagram account. But that was one, all these different shapes you can do. I'll say that Jon actually suggested jigsaw puzzles. Oh, no, sorry, he first suggested jigsaw puzzles, but there’s only one right way to do that. But then he said tangrams, you know those things with all the shapes, you know, there's a square and triangles and stuff. And then you can rearrange them to make all these different shapes, although those are non-continuous maps. So it wouldn't be quite as good. But, I do like the Rite of Spring. And it means that Stravinsky is doing really great on My Favorite Theorem because Eriko Hironaka actually picked Stravinsky also.

KK: Firebird.

HM: She picked the Firebird. I’ll have to look at her podcast. Maybe I'll give her a zoom meeting and we can compare music and the math.

EL: Yes. But I like that. And I am now going to ret-con in some riots following Riemann declaring that you can make these conformal maps.

KK: Well, this has been great fun. I do love the Riemann mapping theorem and Howie, thanks for joining us this

HM: Well, thank you for having me. It was a pleasure.

On this episode of My Favorite Theorem, we were happy to talk with Howard Masur, a math professor at the University of Chicago, about the Riemann mapping theorem. Here are some links you might find interesting as you listen.
Masur's website
Evelyn's article about the Koch snowflake
Jeremy Gray's article about the history of the Riemann mapping theorem (pdf)
A recording of Stravinsky conducting the Rite of Spring
Did the Rite of Spring really cause a riot at its premiere?

Episode 64 - Pamela Harris and Aris Winger

11 mars 2021 | 49 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. We need a better tagline, but I'm not going to come up with one today. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And I think that our guests might be able to help us with that tagline. But we'll get to that in a moment because I have to share with you a big kitchen win I had recently.

KK: Okay.

EL: Which is that that I successfully worked with phyllo dough! It was really exciting. I made these little pie pocket things with a potato and olive filling. It was so good. And the phyllo dough didn't make me want to tear out my hair. It was just like, best day ever.

KK: Did you make it from scratch?

EL: No, I mean, I bought frozen phyllo dough.

KK: Okay, all right.

EL: Yeah, yeah, I’m not at that level.

KK: I’ve never worked with that stuff. Although my son and I made made gyoza last month, which, again, you know, that that's a lot of work to because you start folding up these dumplings, and you know. They’re fantastic. It's much better. So, yeah, enough. Now I'm getting hungry. Okay. It's mid afternoon. It's not time for supper yet. So today we have we have a twofer today. This is this is going to be great, great fun. It's like a battle royale going here. This will be so much fun. So today we are joined by Pamela Harris and Aris winger. And why don't you guys introduce yourself? Let's start with Pamela.

Pamela Harris: Hi, everyone. I like how we're on Zoom, and so I get to wave. But that’s really only to the people on the call. So for those listening, imagine that I waved at you. So I am super excited to be here with you all today. I'm an associate professor of mathematics at Williams College. And I have gotten the pleasure to work with Dr. Aris Winger on a variety of projects, but I'll let him introduce himself too.

Aris Winger: Hey everybody, I’m Aris Winger. I'm assistant professor at Georgia Gwinnett College. I've been here for a few years now. Yeah, no, we, Pamela and I have been all over the place together. I've been the honored one, to just be her sidekick on a lot of things.

PH: Ha, ha, stop that!

EL: So we're very excited to have you here. So you've worked on several things together. The reason that I thought it would be great to have you on is that one of the things is a podcast called Mathematically Uncensored. And it's a really nice podcast. And I think it has a fantastic tagline. I was telling Aris earlier that it just made me very jealous. So we've we've never quite gotten, like, this snappy tagline. So tell us what your podcast tagline is. And a little bit about the podcast.

PH: Maybe I can do the tagline. So our tagline is “Where our talk is real and complex, but never discrete.”

AW: Yeah, that's right. That is the tagline. And yeah, it's a good one. And sometimes I have to come back to it time and again to remember, so that we live up to that during the podcast. We're taping the podcast later today, actually. And so it should be out on Wednesday. So yeah, the show is about really creating a space for people of color in the mathematical sciences and in mathematics in general, I think. And so one of the ways—I think for us the only way that can happen—is we have to start having hard conversations. Right. And so a realization that comfort and staying on the surface level of our discussions doesn't allow for us to have the true visibility that all people in mathematics should have. And so for too long, we've been talking surface-level and saying, “Oh, we have diversity issues. Oh, we should work harder on inclusion.” No, actually, people are suffering. No, actually, here's our opinion. And stop talking about us; start talking to us. So it really is a space where we're just like, you know what, screw it. Let us say what we think needs to be said. Listen to us. Listen to people who look like us. And yeah it’s hard. It's hard to do the podcast sometimes because when you go deeper and start to talk about harder topics, then there are risks that come with that. Pamela and I, week after week, say, “Oh, I don't know if I really should have said that.” But ,you know, it's what needs to be said, because we're not doing it just for us. We're doing it to model what what needs to happen from everybody in this discipline, to really say the things that need to be said.

KK: Have you gotten negative feedback? I hope not.

AW: Yeah, that’s a good question. So I mean, I think that the emails we've gotten are have been great and supportive. But I think, so for me, I'm expecting no one to say — I’m expecting the usual game as it is, right, that people aren't going to say anything, but of course there's going to be backlash when you start saying things that go against white privilege and go against the current power structures. You know, I'm expecting to be fired this year.

PH: Yeah, those are the conversations that we have constantly — that we’re having on the podcast are things that Aris and I are having conversations about privately. And so part of what's been really eye-opening for me in terms of doing a podcast is that I forget people are listening. There are times Aris and I are having just a conversation, and I forget we're recording. And I say things that I normally would censor. If I were in a mixed crowd, if I were in a department meeting, if I were at a committee meeting for, you know, X organization. And I think it's not so much that we would receive an email that says, “Hey, you shouldn't have said X, Y, and Z,” it’s that we are actually getting targeted. For example, I was just virtually visiting Purdue University giving a talk about a book that Aris and I wrote, supporting students of color. And accidentally, the link got shared to the wrong people. And all of a sudden, I'm getting Zoom-bombed at a conversation. That's targeted, right? So those are the kinds of things that we are experiencing as people of color, and we have to have conversations about how are we ensuring that this isn't the experience when you bring a Black or brown mathematician to talk virtually at your colloquium. And if we're not talking about that, then no one is talking about that, because people are trying to hide their dirty laundry. Purdue University is not putting out an email to their alumni saying, “By the way, we invited Pamela Harris to show up and talk about how we best support students of color. And then we got Zoom-bombed, and somebody was writing the N-word and saying f BLM.” Right? Like, that's not happening.

AW: Yes. Wait, they didn't say anything about it?

PH: Well, they're actively doing things about it. But you know they're not putting out the message.

AW: Right. So then it gets sanitized, right? So a traumatic attack gets sanitized to be something else. There are two things about the podcast that Pamela and I, and the Center for Minorities in the Mathematical Sciences, really are trying to work with is making sure that we call out these things, but then not to center it, right, because the the podcast itself is supposed to be about our experiences. But a lot of ways there's a significant part of our experiences that is tied to having to continuously fight against this type of oppression against us.

EL: Yeah. And I think it's really important to have that. And it's so important that it decenters— I think I was listening to an episode recently where you talked about the white gaze and what you have to deal with all the time in trying to present things to a majority white audience. And I think it's really important for us white people to listen to this and realize that not everything is about and for us. And I mean, there are so many things in life where this is true: movies, TV shows, books and stuff. And yeah, I think it's great that that your voices are there and having these conversations, and I think that people should listen to your podcast.

AW: I appreciate that. Yeah. Because it requires a deep interrogation, a self-interrogation by white people to really deal with the feelings. Let me just step back and give the usual disclaimer. Everybody's nice. Everybody's good. Nobody's mean. Nobody is a bad person. Let me just say all that to get that out of the way, right? But what we're talking about is that when I say something on the show, when Pamela says something on the show and you get this feeling like “Wow, that doesn't feel good to me,” then you need to take some time and figure out why it is that you're feeling this way. And it's tied to your privilege, something that you need to interrogate, and it will make you a better person and for everybody.

KK: I don't know. I can't wrap my head around people, like, Zoom-bombing. This is nothing that would ever come to my mind. “You know what, I'm going to go Zoom-bomb this person.” I just…

EL: Well, I mean, it’s just a bad way to spend your time, but not everyone has the same time priorities.

AW: Well, no. So I think that's a great question. And let me just say that it that's how deep and pervasive it is in people, right, that people grow up and have this experience of being raised by other people who have ingrained within them that it is fundamentally, and in some sense, it just burns their soul to have somebody who does not look like them, have someone who is “lesser” than them take the center stage, be deemed the expert. And so again, I'm not calling these people bad. But there is something within some of us that says — and it’s called white supremacy, by the way — that we all have, that we all have to fight, that is so ingrained in some people that they feel compelled to do it. And so they, again, no one's going to fix that for them. And the person who did this to Pamela has it in spades, right? And so when we say that, so I think too often we make it an intellectual exercise, right? We say that it just makes no sense. Right? It doesn't make any sense because white supremacy makes absolutely no sense. But it is a thing. And it's there. And that's what it is, right? So I've been working a lot on calling, naming things so that we don't get confused, because as long as we don't name it, then it just gets to be out there. Like, “Oh, I don't understand.” We understand this exactly. It's called white supremacy. And we need to fight it in our discipline, and across the board.

PH: And it doesn't always just show its face via Zoom-bombing with the N-word in the chat, right? It shows up with who you invite to your podcast. It shows up with who's winning awards from our big national organizations. It shows up with who gets tenure, who even lands into a tenure track position, who even gets to go into graduate school, who actually majors as a mathematician, who actually goes to college, who actually graduates high school, who actually gets told that they're a mathematician. Right? So this is showing its ugly head in very visual ways that we all feel a huge sense of, “Oh, no, this is terrible. I'm sorry, this happened to you.” But the truth is that white supremacy is in everything within the mathematical sciences. And so you know, we got to pull it at its root, my friends. At its root!

AW: Yes.

PH: So this was just one way in which it showed itself, but I want to make it clear that it is pervasive.

KK: Sure. Right.

EL: So what I love about hosting this podcast is that we get to know both people and their math and their relationship to their math. And so we're gonna pivot a little bit now, maybe pivot a lot now, and say, Okay, what are your favorite theorems? And, yeah, I don't know who wants to go first. But, yeah, what's your favorite theorem?

KK: Yeah, let’s hear it.

PH: I’ll do it. I’ll go first. I always like hearing Aris talk. So I'm just like “Aris, go,” right? But no, I’m going to take the lead today. Alright, so I wanted to tell you about this theorem called Zeckendorf’s theorem. I don't know if you know about it.

KK: I do not.

PH: And it goes like this. So start with the Fibonacci numbers without the repeated 1. So 1, 2, and then start adding the previous two, so 3-5-8, and so on. Alright. So if you start with that sequence, his theorem says the following, if you give me any positive integer N, I can write it uniquely as a sum of non-consecutive Fibonacci numbers.

AW: Oh, wow!

EL: Uniquely?

PH: Yes. And this is why you need to get rid of the 1, 1. Because otherwise you have a choice. But yeah. So it's hard to do off the top of my head, because I'm not someone who, like, holds numbers. But say, for example, we wanted to do 20. Maybe we wanted to write the number 20 as a sum of Fibonacci numbers that are not consecutive. So what would you do? You would find the largest Fibonacci number that fits inside of 20. So in this case, it would be 13.

AW: Yeah.

PH: 13 fits in there. Okay, so we subtract 13. We're left with 7. Repeat the pattern.

KK: Ah, five and two.

KK: Five and two! They're non consecutive.

KK: Okay.

PH: Yeah.

AW: Wow!

PH: Three is in between them, and eight is in between the others. And so you can do this uniquely. And so this is using what's known as the greedy algorithm because you just do that process that I just said, and it terminates because you started with a finite number.

KK: Sure.

PH: And so the the proof, of course, there's the, you know, “Can you do it?” but then “Can you do it uniquely?” So the thing that you would do there is assume that you have two different ways of writing it, each of which uses non-consecutive, and then you would argue that they end up being exactly the same thing. So that, in fact, they use the same number of Fibonacci numbers and that those numbers are actually the exact same.

KK: Sure, okay.

EL: Yeah. Like I'm trying to figure out — and I don't, I also am not super great at working with numbers in my head just on the fly. But yeah, I'm trying to figure out, like, what would have gone wrong if I had picked eight instead of 13 to start with, or something? And I feel like that will help me understand, but I probably need to go sit quietly by myself and think about it. Because there’s a little pressure.

PH: Yeah, it's a little subtle. And it might be that you don't get big enough, you end up having to repeat something.

EL: Yeah, I feel like there's not enough left below eight to get me there without being consecutive.

PH: Yeah. Right.

AW: Right. Because you’ve got to get 12. Yeah, yes. Yeah.

KK: Yeah, it makes sense, right? Like, I guess, you know, if you pick the largest one less than your number, then it's more than halfway there. That's sort of the point, right? So that's how you prove it terminates, but also the the non-uniqueness, the non-uniqueness seems like the hard part to me somehow, but also the non-consecutive. Wait a minute, I don't know, which is.

AW: Well it sounds hard, period.

KK: Yeah. I like this theorem. This is good. What attracts you about this theorem? What gets you there?

PH: So I, in part of my dissertation, I found a new place where the Fibonacci numbers showed up. And so once you find Fibonacci numbers somewhere new, I was like, what else is known about these beautiful numbers? And so this was one of those results that I found, you know, just kind of looking at the literature. And then I later on started doing some research generalizing this theorem. So meaning, in what other ways could you create a sequence of numbers that allows you to uniquely write any positive integer in this kind of flavor, right, that you don't use things consecutively, and consecutively, really, in quotes, because you can define that differently. And so it led me to new avenues of research that then I got to do. It was the first few research projects with some of my undergraduate students at the Military Academy. And then I learned through them — they looked him up — that he actually came up with this theorem while he was a prisoner of war.

AW: Oh, wow!

PH: This is when Zeckendorf worked on this theorem. And to me, this was really surprising that, you know, my students found this out. And then I was like, “See, mathematics, you can just take it anywhere.” lLke this poor man was a prisoner of war, and he's proving a theorem in his cell.

KK: Well Jean Leray figured out spectral sequences in a German POW camp.

PH: I did not know that!

AW: Anything to pass the time.

KK: What else are you going to do?

EL: I mean, Messiaen composed the Quartet for the End of Time — I was about to say string quartet, but it's a quartet for a slightly different instrumentation, in a concentration camp, or a work camp. I'm not sure. But yeah, I'm always amazed that people who can do that kind of creative work in those environments, because I feel like, you know, I've been stuck in my house because of a pandemic, and I'm, like, falling apart. And my house is very comfortable. I have a comfortable life. I am not as resilient as people who are doing this. But yeah, that is such a cool theorem. I'm so glad that you said that. And I'm trying to think, like, Lucas numbers are another number sequence that are kind of built this way. And so is there anything that you can tell us about the the sequences that you were looking at, like, I don't know, does this work for Lucas numbers? I don't know if you've looked at that specifically, or did you look at ways to build sequences that would do this?

PH: Yeah. So we started from the construction point of view. So rather than give me a sequence, and then tell me how you can uniquely decompose a number into a sum of elements in that sequence, we worked backwards. So one of the research projects that we started with is what we called — there's a few of them — but one, it was a “Generacci” sequence. And so what we would do is, instead of thinking of the numbers themselves, imagine that you have buckets, an infinite number of buckets, you know, starting out the first bucket all the way to infinity, and you get to put numbers into the sequence in the following way. So you input the number 1 to begin with, because you need a number to start the sequence. And since you want to write all positive integers, well, you’ve got to start with 1 somewhere. So you stick the number 1 in the first bucket. And then you set up some system of rules for which buckets you can use to pull numbers from that then you add together to create new numbers. Well, you only have one bucket, and you only put the number 1 in it. So then you move to the next bucket. Well, okay, you want to build the number two, and you only have the number one, and as soon as you pull it from the bucket, you don't have any other numbers to use. So let's stick the number two in the second bucket. Oh, well, now I could maybe in my rule, grab a number from two buckets, and add them together to get the next number. Oh, that starts looking familiar. The third bucket will have not the number three, because you were able to build it. So what next number could you grab? Well, maybe you can stick in the 4 in there. And so by thinking of buckets, the numbers that you can fill the buckets with, that you couldn't create from grabbing numbers out of previous buckets under certain rules, you now start constructing a sequence. And provided that you very meticulously set up the rules under which you can grab numbers out of the buckets to add together to build new numbers, then you do not need to add that number into the buckets, because you've already built it.

EL: So what rules you have about the buckets will determine what goes in the buckets.

PH: Exactly, exactly. So you might say okay, maybe our buckets can contain three numbers. And you're not allowed to take numbers out of consecutive buckets, or neighboring buckets, or you must give five buckets in between. So what must go into the buckets to guarantee that you can create every single number and you can do so only uniquely? And so these are these bin decompositions of numbers. But you are working backwards. You start with all the numbers, and then you decide how you can place them in the buckets and how you can pull them from the buckets to add together. So I'm being vague on purpose, because it depends on the rules. And actually it's quite an open area of research, how do you build these sequences? You set up some some capacity to your bucket, some rules from where you can pull to add together. And the nice thing is that it's very accessible, and then it leads to really beautiful generalizations of these kinds of results like that of Zeckendorf.

EL: This is very cool.

AW: Fantastic.

EL: All right. So Aris, I feel like the gauntlet has been thrown.

KK: Yeah.

AW: Yeah, well mine is simple. This is not a competition. Yeah, no. I guess mine is influenced — I’ve been thinking about a bunch of different things, but I keep coming back to the same one, which I think is influenced by my identity as a teacher first and foremost when I think about the fundamental theorem of calculus. I just keep coming back to that one. And I don't know how many people have used this one with you on this podcast before, but for me, it hits so many of the check marks of my identity in terms of thinking about myself as a mathematician and a teacher, in the sense that for a lot of students who get to calculus, it's one of the first major, major theorems that will show up in their faces that we actually call out and say this is a theorem. And we call it fundamental, right? We don't often bring up the fundamental theorem of algebra in college algebra, right? Or in other places, or the fundamental theorem of arithmetic, right? But so it's one of these first fundamental theorems. And so it also helps to tell the story of a course, right? And so that really hits the teacher part of me where too often people in the calculus sequence spend all this time talking about derivatives, and all of a sudden, we just switch the anti-derivatives. And we don't really say why. You'll figure out in the next couple of sections, and then we start adding rectangles, and we don't say why. And so it really is, at least the way that the order of calculus has gone and in terms of how to teach it, in my experience, it really is this combination, like, oh, this is why we've been doing this. And this is the genius of relating two things. Sometimes I've gone in, and I've talked about, like I put up a sine curve, and a cosine curve, and we talk about how one of them measures the area under the curve. And then I pretend to bump my head and get amnesia. And then I'll come and say, “Oh, look, looks like we've been talking about derivatives. Right?” And they’re like, “Wait, what do you mean we're talking about derivatives?” “This is the derivative of this one.” And they’ll go, “What? We were measuring the area under the curve.” “Well, we’re also measuring the derivative, right?” This is the derivative, but this is measuring the area. And it's like, Oh, right, and so it's just one of these “aha” moments, where if people have been paying attention, it's like, oh, that's actually pretty cool, right? And then also in terms of the subject itself in relationship to high school, just really thinking about — because I get a lot of students who know all the rules, right? And they look at the anti-derivative with the integral sign and say “That's the integral.” Well, that's an anti-derivative, right?

EL: We’re not there yet.

AW: Yeah, that the anti-derivative and the integral are actually different. And so just having that conversation. And it also is a place to talk about the history of the subject and stuff like this.

EL: Yeah, I love it. And, at least for me, I feel like it's a slow burn kind of theorem. The first time you see it, you're like, “Okay, it's called the fundamental theorem of calculus. I guess some people think it's really important.” So that might be your Calculus I class. And then you see it again, maybe in an introductory real analysis class. And you're like, “Oh, there's more here.” And then you teach calculus, and you’re like, “Ohhhh!”

AW: Oh right, yes!

EL: Your brain explodes. You're like, “This is so cool!” And then your students are where you were several steps ago. And they’re like, “Okay, I guess it’s all right.”

AW: If I get the success rate of like, I've had three or four people go, “Whoa!” And it's like, okay, yeah, you're with me. And so this is out of hundreds of people.

EL: If you can get a few people that do that the first time they see it, that’s awesome.

AW: Yes. Yeah. No, it's been fulfilling for sure. And so then the proof itself, you know, it's also great, because then it culminates all of the theorems that you've been talking about beforehand. Depending on the proof, of course, but like, there's the intermediate value thereorem. There's the mean value theorem for integrals. There's unique continuity, at least in this version of it, in order for it to work. So yeah, it's great.

KK: So when you teach calculus, there's always two parts to the fundamental theorem. And so I like the one where the derivative of the integral is the function back, right? That's the fun, like for the mathematician in me, this is the fun part. Your students never remember that. Right? They always remember the other one, where we evaluate definite integrals by finding it the anti-derivative. So I was going to ask, if you had to pick one of the two, which one is your favorite?

AW: I mean, part of it is because at least the way that I've taught it, we're coming out of the mire of Riemann sums.

KK: Right.

AW: And so people have suffered through doing rectangles so much. And then I just get to say, “Oh, you don't have to do this anymore.” I mean, I've had a few students go, you know, now that we do — I always use the antiderivative of x squared on zero to 10, or the area under the curve of x squared from zero to 10. And like, sometimes I'll say, “Oh, that's 1000 over 3, right?” And then it was like, “Well, how did you do that so quickly?” We'll see. Right? But then, at the end, when I'll say, okay, and then we do another one again. And then I show how to apply the theorem, and people say, “Well, why didn't you just say that?” And then we have a great conversation there about how this isn't about the answer, that this is about a process and understanding the impact of mathematical ideas, that the theorem, as with all theorems, but this one is my favorite, is an expression of deep human intellect. And that if we reframe what theorems are, we get a chance to rehumanize mathematics. And so I think that too often in our math classes, and our math discourse, we remove the theorems from the humanity of the people who created them. And so people get deified, like Newton and Leibniz, but you know, these same people had to sit down and work hard at it and figure it out.

KK: No, it’s certainly a classic, but it is surprising how little it has come up on our podcast. It was the very first episode.

AW: Oh, okay.

KK: Yeah. Amie Wilkinson chose it. And then this will be episode 60-something.

AW: Okay. Yeah.

PH: Wow.

EL: We've talked, we've mentioned it in some other episodes. But it isn’t — I mean, there are just — I love this podcast, obviously, I keep doing it. And there are just so many types of theorems. And I love that you two picked different types. Yours, Aris, is one of these classics. Everyone who gets to a certain point in math has seen it, hopefully has appreciated it also. And Pamela, you picked one that none of us had ever heard of and made us say, “Whoa, that's so cool!” And people just have so many relationships with yours. And that's what this podcast is really about. Actually it's not about theorems. It's about human relationships with theorems and what makes humans enjoy these theorems. And so you picked two different ways that we enjoy theorems. And I just love that. So yeah.

KK: Yeah, that is what we're about here, actually. I mean, I mean, yeah, the theorem. But actually what I like most about our podcast, so let's toot our own horn here. We’re trying to humanize mathematics. I think everybody has this idea that mathematicians are a very monolithic bunch of weird people who just — well, in movies we’re always portrayed as either being insane, or just completely antisocial. And I mean, there’s some truth and every stereotype, I suppose, but we are people, and we love this thing. We think it's so cool. And sharing that with everyone is really what's so much fun.

AW: Yeah. And I think also that, for me, the theorem itself, and what it reveals, touches something that’s inside of us. There’s something about it, right? There’s the “Whoa” part that is that is indescribable and that I think really touches to our humanity. There is a eureka moment where you're just like, “Oh, I understand this now.” Or this connection is amazing, right? Yeah, it's indescribable.

KK: So we all agree these things are beautiful. So here's a question. Where do people lose this? I mean, I have a theory, but — because we've all had this experience, right? You're at a cocktail party and someone says, they find out you're a mathematician and like, oh, record scratch. I hate math. Okay.

AW: Yes, yes, yes.

PH: But I don’t think they hate math, though, Kevin.

KK: No, they don’t. Nobody hates math. Nobody hates math when they're a kid. That's exactly right. So I think when they say that they mean that the algebra caused them trouble. When x’s started showing up.

PH: I don't even think that's it.

KK: Okay. Good. Enlighten me because I want an answer to this that I can’t find.

PH: I don't think it's that people hate math or that they hate that the alphabet showed up all of a sudden in math that they hate how people have made them feel when they struggle with math. Math is an inanimate object. Math is not going out there and, like, punching people in the face. It's the way that people react to other people's math. Right? The second that you don't use the language in the way that somebody expects you to use it and you're trying to communicate properly and somebody says, “That’s not how you say it. It's not FOILing. It's called distributing!” Right? But you knew what I meant when I said FOIL the binomial!

KK: Of course I did.

PH: FOILing this gives you the middle term, blah, blah, right? So it’s again about human interactions. And if you make someone feel dumb, they'll never like what it is that they're trying to learn

AW: Amen to that. And they will conflate the two, which is what always happens.

PH: That’s exactly it!

AW: They will replace the experience with the subject itself, when in fact, they're talking about the experience. Yeah. So yeah, we've been working a lot about this in the last few years, Pamela and I and Dr. Michael Young, about when people say they hate mathematics, they’re really talking about their mathematical experience. So my immediate response to your question is just bad teaching. Let's just call it what it is.

PH: Right.

AW: I don't want to get on my podcast too early. We're recording later.

PH: We’re recording in a bit, yeah.

AW: But yeah, we're talking about people. And I say this as a loving critique of the greatest discipline in the history of people. I truly believe that, but I believe that the way we teach it, and the cultural norms we take with it, devalues people, and so I want every person who's listening to this now to then the next time they hear somebody say they hate it, look at them as an innocent person who had a bad mathematical experience. And then, because I see too often amongst my people in the community who say they hate having these conversations with people who say they hate it. And I think we need to return innocence back to that person. And say that this is not a person who hates you or even hates the subject. This is a hurt person. Yes, this is a person who has been damaged in our subject. And by the way, I go much farther than that. It's our responsibility to try and help repair that because this person is going to impact their cousin, their child, their relative, by bringing this hate of the subject, when in fact, it doesn't have anything to do with the subject.

EL: Yeah. It’s about the traumatic experiences. And actually, I think mathematicians often have a bit of a persecution complex and think this is the only place where people have this reaction. But one of my hobbies is singing, and in particular, singing with large groups of untrained people who are just singing because we love singing. And the baggage that people bring to singing is similar. I’m not saying it's entirely the same, but people have been made to feel like their voice isn't good enough.

AW: Yes.

EL: They have this trauma associated with trying to go out and do this sometimes. Obviously a lot of people love to sing and will do it in public. A lot of people love to sing at home and are scared of doing it in public because they're worried about, you know, their fourth grade music teacher, who told them to sing quieter, or whatever happened.

PH: Yes,

AW: That’s right. That's right. And the connection is similar, because what are we saying? We're saying that if you don't hit this right note, then it doesn't count. As opposed to if you don't get the answer seven, then we're not going to value you because the answer is seven, right? Because we have this obsession with the correct answer in mathematics. Right.

PH: And not only that, but also doing it fast.

AW: Yes.

PH: You and I have talked about this before, that — maybe in singing, this is different. I'm not sure. I definitely can relate to the trauma of never singing out loud in public. But is there this same sentiment that you must get it perfect the first time and pretend that it doesn't actually take you hours of training?

EL: I mean, it comes up. There’s definitely, people can feel more valued if they're quicker at picking things up than others, although, you know, it's not the same. There's no isomorphism between these two, I don't know, to bring a little silly math lingo in. But there definitely, there are a lot of similarities, and I think about this a lot, because two things I love in my life are math and singing with my friends. And, you know, I just see these relationships. But yeah, I could go on a whole rant, and I want to not do that.

AW: No, no, no, I appreciate you bringing it up.

EL: But I think it's a really interesting correspondence.

AW: And then the final one is that, you know, in the music space, what is it that we really should be trying to do, value everybody's voice? And in mathematics, we should be valuing everybody's contribution. Right? This is all we're saying. And what does each discipline look like when we value people's voices, no matter where they are on the keys? And we value everyone's contribution to trying to solve a problem.

EL: Yeah, yeah. And how can we help people, you know, grow in the way they want to? You can say, like, “Oh, I like I am not as good a sight reader as I want to be. How can I get better?” How can we help people grow in that way without feeling cut down?

AW: Yeah.

EL: Yeah, it is true for math, too. Yeah. It's just, everything is connected. Woo.

AW: Yes. But you know, we've been talking about, you know, these human relationships we all have with math. And so another part of our podcast that we love is forcing you to do make one more human connection between math and something else with the pairing. So what goes well, Pamela, with this theorem about uniquely writing the numbers in terms of the Fibonacci sequence?

PH: So I was trying to think about my favorite food, and when it was the epitome of perfection, and I came up with, okay, so if we're going to pair it with something to drink, I was like, I want to think about happy moments. Because this feels like a happy theorem. And so I want to go with some champagne.

KK: Okay.

PH: Okay, I was like, “We're gonna go fancy with it!” But then for food, I'm thinking about, oh, this is hilarious. So I went to a conference in Colombia, we visited Tayrona which is a beach in Colombia. And on the side of the beach, I paid to have ceviche, fresh ceviche. And I've never been happier eating anything in my life. And so I imagine myself learning Zeckendorf’s theorem at the beach in Tayrona in Colombia, with some champagne and the ceviche.

EL: Oh man.

AW: Wow.

PH: Beat that, Aris! Beat. That.

AW: There’s no way. So wait, so I want to make sure I understand. So is this while you're reading the proof? Or is this while you’re—

PH: This is like the gold standard. If I were to put all the, like, uniqueness of my favorite food, my favorite drink and my favorite theorem, I would put them in a location which is Tayrona in Colombia, at the beach, eating ceviche sipping on some champagne, learning Zeckendorf’s theorem.

AW: Okay.

KK: Is this the Pacific coast or the Caribbean?

PH: You’re asking questions I should know the answer to, and I believe it’s the Caribbean.

KK: Okay.

PH: Nobody Google that. [Editor’s note: I Googled that. It is the Caribbean.] I have no idea where they took me in Colombia. I just went.

KK: Sure.

EL: Yeah, that sounds so lovely as I look out of my window where there's snow and mud from some melted snow.

PH: Ditto.

AW: So I yeah, I think for the fundamental theorem of calculus, I think this is something that's just classic. Like you're just having a nice pizza and some ginger ale. You're just sitting down and you're enjoying something hopefully that everybody likes and that connects with everybody, that everybody hopefully sees that they get to get that far. So yeah, I mean, my daughter recently — I didn't realize this. She's 9. And we were talking. We visited my aunt in DC. My aunt raised me. And my daughter was much younger at that time, but then every time she thinks about going to visit, she thinks about the ginger ale that my aunt got her because that was the only time she ever got ginger ale. So she’s like, “Oh, I like your aunt, Daddy, because you know, I had ginger ale there.” And I was like, Oh, I should have ginger ale more often. So that made me think of that.

PH: That’s adorable.

EL: I can really relate to that feeling of, like, when you're a kid, something that is totally normal for someone else isn't what's normal for your family. So you think it's a super special thing.

AW: It’s amazing.

EL: I think I had this with, like, Rice-a-Roni or something at my aunt's house, and my mom didn't use Rice-a-Roni, and I was like, “Whoa, Mom, you should see if you can find Rice-a-Roni.”

PH: Amazing.

EL: She was like, “Yeah, they have Rice-a-Roni here.”

AW: Rice-a-Roni’s the best.

KK: I haven't had that in years. I should go get some.

AW: Me either. All right.

PH: That’s how you know you made it.

KK: You know what? You know, single mom and all that, and I lived on Kraft macaroni and cheese when I was a kid. And yeah, you would think I don't like it any more. But, aw man.

PH: Listen, that thing is delicious. So good.

AW: I was about to say.

EL: They know what they’re doing. Yeah. Well, that's great. And I mean, pizza is my favorite food. As great as ceviche on the beach sounds, pizza, just, when you come down to it, it's my favorite food. And so I love that you paired the fundamental theorem of calculus with my favorite food.

KK: So I'm curious, there must be a human who doesn't like pizza, but have you ever met one? I've never met one.

PH: No.

EL: I know people who don't like cheese. And cheese is not — I mean, to me cheese is essential to the pizza experience, but you can definitely do a pizza without cheese.

AW: Yeah. No, my wife also always says that for her it's about the sauce. So I think she might be a person who can get rid of the cheese if the sauce is right. Yeah.

KK: But the crust better be good too.

AW: Of course, of course. It's a full package here.

EL: But okay, so you say that, but on the other hand, I would say that bad pizza is still really good.

KK: Sure.

EL: I mean, you can have pizza that you're like, “I wish I didn't eat that.” But I have very rarely in my life encountered a slice of pizza that was like, “Oh, I wish I wish I had done something else other than eat that pizza.”

AW: It’s actually a pretty unbeatable combination, right? Tomato sauce, cheese and bread.

PH: Yeah. It kind of can't go wrong. Yeah.

KK: When I was when I was in college, there was a place in town. It was called Crusty’s Pizza, and I don't think it exists anymore. And it was decidedly awful. But we still got it because it was cheap. So we would occasionally splurge on the good pizza. But you could get a Crusty’s pie for like five bucks.

AW: Absolutely.

KK: This is dating myself. But yeah, absolutely. Always. All right, so we've got we've got theorems, we’ve got pairings. You've plugged your podcast pretty well, although you can talk about it more if you'd like. Anything else that either of you want to plug, websites, the Twitter?

EL: Yeah, but can you say a little more about the book that you mentioned?

AW: Yeah, the book is a series of dialogues that was an extension of an AMS webinar series that we gave about advocating for students of color mathematics. And so we had just decided, you know, there was so much momentum, we had hundreds of people coming every time to the four-part series. And so we were like, you know, we've gotten to a place where we've given all these talks, and then you give talks, create momentum, and then it just ends. And we're just like, you know, what, not this time. Let's create a product out of this. And so, we decided quickly to get the book together, just answering some of the unanswered questions from the webinar series. So we had the motivation, in terms of answering their questions. And yeah, we got it together. And it was an honor. So it really is just a list of our dialogues, a transcription of our dialogues, answering some of the unanswered questions from that webinar series. And so it's gotten some really good reviews, and people are using it in their departments. And so it's been fantastic so far.

PH: Yeah, I think that's that's the part that I'm really enjoying, getting the emails from people who have purchased the book. And so maybe I should say the full title, so it is Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics. And the things that I hear from folks who have purchased the book — so thank you all so much for the support — is that they didn't expect that there is part of a workbook involved in the book. So it isn't just Aris and I going back and forth at telling you things. I mean, a lot of that there is, that is part of the content. But there's also a piece about doing some pre-reflection before we start hearing some of the dialogue that we have, and then also the post part of it. So how are you going to change? And how are you going to be a better advocate for students of color in mathematics? And so it leaves the reader with really a set of tools to come back to time and time again. That's really what I see as a benefit of the book. And people are purchasing it as a department to actually hold some kind of book club and really think about what of the things that we suggest that professors implement in their department, in their classrooms, in their institutions, what they can actually do. And so the reception has been really wonderful. And I'm just super thankful that people purchase the book, and we're supporting our future work.

EL: Yeah. And can you also mention, is it minoritymath.org, the website that hosts Mathematically Uncensored?

AW: That’s correct. That's right. So yeah, that's the home of the podcast. And that's a place where we're trying to create voices for underrepresented minorities in the mathematical sciences. And so you can go there not just for the podcast, but for other content as well that centers around that experience.

KK: Okay.

EL: Fantastic. Thank you so much for joining us.

KK: Yeah.

EL: I had a blast.

PH: Thank you.

KK: This was a really good time.

EL: Yeah. Over lunch today, I'm going to be writing down numbers and writing them in terms of Fibonacci numbers. It’s great.

AW: It will be fantastic.

PH: Awesome.

AW: Thanks.

PH: Bye, everyone.

KK: Thanks, guys.

On this very special episode, we had not one but two guests, Pamela Harris from Williams College and Aris Winger from Georgia Gwinnett College, to talk about their podcast, Mathematically Uncensored, and of course their favorite theorems. Here are some links you might be interested in as you listen to the episode.

Harris's website

Winger's profile on Mathematically Gifted and Black

Mathematically Uncensored, the podcast they cohost
Minoritymath.org, the Center for Minorities in the Mathematical Science, a website with information and resources for people of color in mathematics
Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics, their book
Zeckendorf's theorem and a biography of Edouard Zeckendorf

Jean Leray, a French mathematician who worked on spectral sequences as a prisoner of war
Olivier Messiaen's Quartet for the End of Time, composed when he was a prisoner of war
A paper generalizing the Zeckendorf theorem by Harris and coauthors
Our episode with Amie Wilkinson, who also chose the Fundamental Theorem of Calculus, making it 2 for 2 among mathematicians with the initials AW.

Episode 63 - Lily Khadjavi

11 februari 2021 | 51 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the podcast from 2021. I don't know why I said that, just, it's a math podcast, and it is currently being taped in 2021. I'm your host Evelyn Lamb. I'm a freelance math writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. No, look, it's important to say it's 2021 because 2020 lasted for about six years. It was—I couldn't wait for 2020 to be over. I don't think 2021 is much better yet. It's January 5. I'll leave our listeners to figure out what's going on right now that might be disturbing. And, and yeah, but anyway, no, happy new year. And I had a very nice holiday. My son has been home for nine months now. He's going to go back to school finally next month to finish up his senior year in college. And I did nothing for a week. I mean, like when I say nothing, I mean nothing. Just get up, watch some TV, like we’re watching old reruns of Frasier, like this is the nothing levels I saw. It was fantastic.

EL: Very nice.

KK: How about you guys? Did you have a nice holiday?

EL: Um, I had a bad bike accident right before Christmas. So I had some enforced rest. But I'm mostly better now. I have gotten on my bike a couple times, and nothing terrible has happened. So still a little more anxious than usual on the bike. We were taking a ride yesterday and I could tell I was just like, not angry, but just, you know, nervous and worried. And it's just like, Okay, I'm just at the scene of the trauma, which is my bike seat, and getting over it. But I hope I will continue to not fall off my bike and keep going.

KK: That’s the only thing to do. Back in my competitive cycling days when I was a postdoc, I had some pretty nasty crashes. But yeah, you just get back on. What else are you going to do? So anyway, enough of that. Let’s talk math.

EL: Yes. And today, we are very happy to welcome Lily Khadjavi to the show. Hi, will you introduce yourself and tell us a little bit about yourself?

Lily Khadjavi: Hi. Oh, thanks, Evelyn. It's so great to be here. I'm Lily Khadjavi, as you said. I'm a professor of mathematics at Loyola Marymount University, which is in Los Angeles, California. I'm a number theorist by training, but I'd say that I'm lucky to have taken some other mathematical journeys, especially since graduate school, and I don't know, for example, this past year, maybe my biggest excitement is I was lucky to be appointed to a state board in California. So by the Attorney General, Xavier Becerra, to be appointed to an advisory board looking at policing and law enforcement and the issue of profiling. And so that's an issue that's very important to me. And it was an unexpected mathematical journey.

EL: Yeah.

LK: If you’d asked me 20 years ago, what would I be up to, I might not have thought of that. And I've taken many a bike spill in my day, so I could feel some nice affinity being here today. You’ve just got to get back on and be careful, of course.

EL: Yeah. And that that must be an especially important issue in LA, because I know the LAPD has been the subject of some, I guess, investigations and inquiries into their practices and things like that.

LK: That's exactly right. And over the years, it was under a consent decree, so an agreement between the US Department of Justice and the City of Los Angeles, with many aspects monitoring police practice. And actually, some of that included data collection efforts looking at traffic stops. And that, combined with teaching a statistics course, is what really gave me a window more into policing practice, into problems that where I wanted authentic engagement for my students with the real world and took me on, maybe I'll say unexpected journeys to law conferences and elsewhere, as I started to learn more about the issues, the ways that as mathematicians, we can bring tools to bear on on these social questions too.

EL: Yeah, very cool.

KK: Yeah.

EL: So what is your favorite theorem? And I know that's an unfair question, but I will ask it anyway. And then, you know, you can run away with it.

LK: Yeah. I know this podcast is not visual, but I'm already kind of smiling in a terrified way because I found this question so difficult, really an impossible task, because I thought it's like asking me when my favorite song—I don't know, do you have a favorite song?

EL: That is hard to say. If you asked me, I would start listing things. I would not, probably, be able to tell you one thing.

LK: What do you think, Kevin?

KK: I, uh, Taxman?

LK. Okay, I thought you would name the opening the music for the podcast as a favorite too.

EL: Oh, yeah.

LK: You know, shout out to that.

KK: I do like that. But now, you know, maybe What Is Life by George Harrison? Single?

LK: Oh, yeah. Okay, well, maybe I'll count that as listing, which is what Evelyn started to do. Because it's very, difficult.

KK: It is.

LK: You know, I was really wrestling with this. And it got me kind of thinking about why do we like certain theorems. I think I pivoted to what Evelyn said. I started wanting to make lists. And of course, it's fun to talk about things that are new to everyone. And, you know, it's been a remarkable podcast, and lots of people have staked out, I mean, they've grabbed those beautiful favorite theorems. But I started thinking, could you have a taxonomy? I really saw a taxonomy of theorems. Not by discipline. So not a topological statement or an analytic proof, but by how mathematicians feel about them, or the aesthetic of them. And so my first you know, category had to be sort of the great workhorses, like those theorems that get so much done, but they also they never cease to amaze you. And I mean, it’s hard not to point right away to the fundamental theorem of calculus, and I think maybe in your very first episode. That's right, that might be what?

KK: Yeah, Amie Wilkinson.

EL: Yes, Amie Wilkinson just came in and snatched that one. Although as everyone knows, we do double theorems, you know, we don't have a rule that you can't use the same theorem again.

LK: No, because that's one we use again and again and again. You know, even this past semester, I was teaching multivariable calculus. And you know, we have this march through line integrals, double, triple integrals, and we build, of course, to Green’s theorem, Stokes’ theorem, the divergence theorem. So these main theorems in calculus that the machinery is heavy enough for the students that even if I'm trying to put them in a context where, “Oh, this is really all the fundamental theorem of calculus,” I think that gets obscured obscured for students first trying to get their head around these theorems. Even though you relate them, you say, Oh, but they've got the boundary of this—maybe endpoints of a curve or some other surface boundary, and you're relating it as the relationship between differentiation integration, and it's so it's beautiful stuff. But I think I'm not convinced my students thought of it as the same theorem, even if I tried to emphasize this perspective. But still, they, all of us can be blown away by how powerful the theorem is in all of its incarnations. And so that's a great workhorse. So we don't have to talk at length about that one. It's been here before, but you know, you just have to tip your hat to that one. But I was wondering, are there other great workhorses something you put in that in that category?

KK: So I argue—I mean, so you mentioned the fundamental theorem—the workhorse there is actually the mean value theorem.

LK: Hmm.

KK: Because the fundamental theorem, at least for one variable, is almost a trivial corollary of the mean value theorem. And I didn't appreciate that until I taught that sort of undergraduate analysis course for the first time. And I said, “Wait a minute.” And then I sort of came up with this joke, I'm actually going to write a book. It's like a “Where's Waldo” style thing: Where's the mean value theorem? Because in every proof, it seemed like, Well, wait a minute, by the mean value theorem, I can pull this point out. Or there is one, I don't know where it is, but it's in there somewhere. So I really like that one.

LK: That’s a really great perspective. I also will say that I did not happen on that feeling until teaching analysis for the first time, of course, versus, you know, for seeing these theorems or learning about them, and even learning them in analysis, not just using them in calculus. Know, that reminds me that it wasn't till grad school, maybe taking a differentiable manifolds class, and that's not really my area. But seeing, Oh, you can define a wedge product, you can define these things in a certain way. Oh, they really are literally all the same theorem. But I like this perspective, maybe that would have been a way to convince my students a little bit more, to kind of point to the mean value theorem, because it would put them on more familiar turf too. I really like that. Yeah. Are there other workhorses?

EL: So the first one that came to my mind was classification of surfaces, in topology, of like, you know, the fact that you can do that—I feel like I it's like so internalized to me now. And yeah, I don't know, that for some reason that came to mind, but it's been a long time since I did research and was keeping up with, you know, proving things. So yeah, it’s—but yeah, I think I would say that anyway.

KK: Yeah. And I would sort of think anything with fundamental in its name right.

LK: Yeah, I was thinking that.

KK: So the fundamental theorem of arithmetic, okay, so that you can factor integers as products of primes, or the fundamental theorem of algebra, that every polynomial with complex coefficients has a root. But then more obscure things like the fundamental theorem of algebraic K-theory. You guys know that one?

LK: That one, I'm afraid does not trip off my tongue.

KK: All it is, is it's a little bit weird. It just says that the K-theory of if you have a ring, maybe it needs to be regular, that if you look at the K-theory of the ring, and the K-theory of a polynomial ring in one variable over it, they're the same. And the topological idea of that is that, you know, it's a contractibility argument somehow. And so it's fundamental in that way.

LK: These are great workhorses. Yeah. And also, Evelyn, you mentioned the classification, like these results are just so fundamental. So in whether they have fundamental in the name or not, they are.

EL: Like, naming it fundamental, it's almost like cheating that point. Or, maybe not cheating, maybe stealing everyone else's thunder. It’s like, “No, I already told you that this is the fundamental theorem of this.”

LK: My poor students, whenever I want them to conjure up the name and think of something that way, I make the same corny joke. I'm like, “It's time to put the fun back into…” and they’re like, “Ugh, now she's saying fundamental again.” So yeah, I was thinking, too, that in different fields, we reach back, even as we're doing different things in our own work, back to those disciplines that we were sort of steeped in. And I think for topologists, there are so many great theorems to reach to.

KK: Sure.

LK: But I was thinking even like the central limit theorem in statistics and probability, so this idea that you could have any kind of probability distribution—start with any distribution at all—but then when you start to look at samples, when the samples are large enough, that the mean is approximated by a normal distribution. That somehow never ceases to amaze me in the way that the fundamental theorem of calculus, too. Like, “Oh, this is a really beautiful result!” But it's also a workhorse. There are so many questions in statistics and probability that you can get at by gleaning information from the standard normal distribution. So maybe I’d put that into a workhorse category.

KK: Sure.

EL: Actually, Heine-Borel theorem, maybe could be kind of a workhorse, although I'm sort of waiting for for you to say that it's actually the mean value theorem too.

KK: No, it's just, it's just that, you know, compact sets are closed and bounded. That's it. Right?

EL: Yeah. Yeah, actually, yeah, that, once again, is such a workhorse that it's often the definition that people learn of compactness.

LK: That’s right.

EL: Like the first time they see it. Or, like such an important theorem that it it almost becomes a definition. Actually the Pythagorean theorem, in that case, is almost a definition.

KK: Sure.

EL: Slash how to measure distance in the Euclidean plane.

LK: Yeah, that's a good example. So maybe now we have so many workhorses, well, another category I was thinking of — it's beautiful stuff. I was thinking of those theorems where the subtlety of the situation kind of sneaks up on you. So maybe you hear the statement, and you kind of even think, “Oh yeah, I believe that,” like the Jordan curve theorem, I think you had a guest speak about this, too. So this, you know, idea of a simple closed curve. So you just draw it in the plane, there's an inside, and it divides the plane into an inside and outside. And I kind of really remember—I can't tell you what day of the week it was—but I remember the first time this came up in a class, and I thought, “Yeah.” But then we started thinking about how would you go about proving something like this, or even just being shown, someone drawing, a wild enough crazy curve, where suddenly you can't just eyeball it and immediately see what's inside and what's outside. So I don't know what this category or set of theorems should be, but the subtlety sneaks up on you even though statement seems reasonable.

EL: “I can't believe I have to prove this.” Maybe that’s slightly different. Well, what I mean is like, I can't believe this is a—It seems so intuitive that understanding that there is something to prove is a challenge, in addition to then proving it.

LK: Yeah. And maybe you can't even prove it—Well, how about the four color theorem? So this map coloring theorem, this idea that the four colors suffice, so if you have states or counties or whatever regions, you want to make your map of, that if they share a common edge boundary, then use different colors, that four colors is enough. I don’t know, has a human being ever proven that? My understanding is that it took computing power.

KK: It’s been verified.

EL: I think they’ve reduced the number of cases, also, that have to be done from the initial proof, but I still think it's not a human-producible proof.

KK: That’s right. But I think Tom Hales actually verified the proof using one of these proving software things. So I mean, yeah, but that was controversial.

LK: That brings up a neat question about what constitutes proof in this day and age. I've seen interesting talks about statements where, or journals where something's given as this: “Okay, here's a theorem. And here's the paper that's been refereed.” And then later, oh, here's something that contradicts it. And people are left in a sort of limbo. Well, that's another discussion, things unproven, un-theorems, I don't know. Well, anyway, in this category, that's going to help the subtlety of the situation sneaks up on you. If I start coloring maps, testing things out, after a while, I’d say, “Oh, there's a lot to this.” But the statement itself has an elegant simplicity.

KK: Well, it's not easy. So I curated a math and art exhibition at our local art museum, in the Before Times, and one of the pieces I chose was by a Mexican artist, and it's called Figuras Constructivas. And it was just two people standing there talking to each other, but it was sort of done in this—we’ve all done, you probably when you were a kid—you took a black crayon and scribbled all over a page, and then you fill in the various regions with different colors, right? It reminded me of that. And the artist used five colors. And so when I was talking about this to the to the docents, I said, “Well, why don't we create an activity for patrons to four-color this map?” So they did, they created it, because it was just a map. And they did it, and the docents were just blown away by how difficult it was to do a four-coloring. You know, five colors is fairly easy. But four was a real challenge.

LK: That sounds really fun. And what a great example of math and art coming coming together. And my understanding of the history of this, too, is that the five-color theorem was proved not just before four colors, but was kind of doable in the sense that

EL: I think it’s just not that hard.

LK: Certainly not that hard in the sense of firing up the computers and whatever else has done.

KK: Needing a supercomputer in 1976.

LK: Which is basically my phone, maybe. Well, I had another category mind, which is, theorems where the proofs are just so darn cute.

KK: Okay.

LK: And so what I was thinking of—I tried to have an example for each of these—which was the reals being uncountable.

EL: Yeah.

LK: And I think you've had guests talk about this. And you know, like a diagonalization argument, like say, just look at the reals only from 0 to 1. And suppose you claim that that is a countable set. Okay, go ahead and list them in order, in whatever ordering you've got for countability. And then you can construct a new element by whatever was in the first place of your first element, do something different in your first place, whatever was in the second place of the second element, do something different in your second place of your new element, and so on down the line. So you go along the diagonal, if you had listed these and so this, I don't know my crude description of a diagonalization argument, that you can construct a new element that wasn't in your original set and so contradict the countability. I don't know, I thought that's really cute.

EL: Yeah. And that was probably the first theorem that really knocked my socks off.

KK: Mm hmm. It's definitely a greatest hit on our show.

EL: Yeah.

LK: So I guess that’s right. We've had a Greatest Hits show, so I don't know, this taxonomies kind of disintegrating, like “Workhorses,” “Just so darn cute,” “Situation sneaks up on you.” But yeah, I don't know if there are others that fit into the “Just so darn cute.” That was the one that came to mind because I kind of wanted it on my favorites, and then I was like, “Oh, someone's already talked about this on the show.”

KK: Well, I really like—so I'm a topologist. And I really like the theorem that there are only four division algebras over the reals. So the reals, the complexes, the quaternions and the octonians. And it's a topological proof. Well, I mean, there's probably an algebraic proof. But my favorite proof is topological. So I don't know if it's cute.

EL: That isn't what you'd expect the proof of that to be, for sure.

KK: No. And it's it's sort of—I'm looking through it. So I taught this course last year, and I'm trying to remember the exact way the proof goes, not that our listeners really want to hear it. But it involves cohomology. And it's really pretty remarkable how this actually works. Oh, here it is. Oh, yeah. So it involves, it involves the cohomology rings of real projective spaces. And so if you had one of these division algebras, you look at some certain maps on cohomology, and you sort of realize that things can't happen. So I think that's very, well, I don’t know if it’s cute, but it's a pretty awesome application of something that we spend a lot of time on.

LK: Yeah, it’s so neat when a different field. So you know, we have these silos, historically: algebra, topology, and so on. So the idea that a topological proof gives you this algebraic result is already a delight, but then that's heavy machinery. That's sounds like a really neat.

KK: Or fundamental theorem of algebra, right?

LK: Well, that's when I was thinking when you started saying saying, “Oh, there's a topological proof.” I started thinking, “Oh, fundamental theorem of algebra.” You know, fire up your complex analysis. And yeah, neat stuff. Yeah.

EL: Well, and there's this proof of the Pythagorean theorem that I have seen attributed to Albert Einstein, I think, that has to do—Steve Strogatz wrote, I think, an article for The New Yorker about it. So Oh, yeah, listening to my bad explanation of it semi-remembered from several years ago, you can go read it. But it has to do basically with scaling. And it's a kind of a surprising way to approach that statement.

KK: I think it was in the New York Times [editor’s note: Evelyn was right, it’s the New Yorker! [note to the editor’s note: Evelyn is the editor of this transcript]], or it's also in his book, The Joy of X, I think it's in there too. And yeah, I do sort of vaguely remember this, it is very clever.

it's a nice one to record.

LK: Yeah, this makes me want to swing back to many things. It's also reminding me, so here we are in pandemic times. And so at the university I'm at, we're not spending time in the department, but you reminded me that when I wander around the department, sometimes we have students’ projects, or work from previous semesters, up here and there, along with other posters. And I'll look at something and say, “Oh, I haven’t thought about Pythagorean Theorem from that context, or in that way.” So just different representations of these. So maybe there should be a category where there are so many proofs that you can reach to, and they're each delightful in their own way, or people could you could start to ask people what's your favorite proof instead of a favorite theorem, maybe.

KK: I think we did that with Ken Ribet because he did the infinitude of primes. He gave us at least three proofs.

LK: And I think three pairings to boot. Yeah. Nice. I'm wondering if another, so there was the “so darn cute,” how about something where the simplicity of the statement draws you in, but then the method of the proof may just open up all kinds of other problems or techniques. So in other words, I guess what I'm saying is some theorems, we really love the result of the theorem. Maybe the Fundamental Theorem of Calculus. That result itself is so useful. But on the other hand, Fermat’s Last Theorem, I don't know if anyone's even pointed to that on the show, but something in number theory where the statement was—I mean, this is how I got suckered into number theory. That's what I would say. So you have this statement. You mentioned the Pythagorean theorem, so this idea that, that you could find numbers where the sum of two squares is itself a square, like three squared plus four squared equals five squared, but what if you had cubes instead, could you find a cubed plus b cubed equals c cubed, or any a to the n plus b to the n equals c to the n. And, you know, that's a statement that, although the machinery of number theory that's developed to ultimately prove this is so technical, and involves elliptic curves and modularity, all kinds of neat stuff, but that the statement was very simple. And of course, at some level, then it wasn't even just proving that statement. It was the tools and techniques we can develop from that. But I remember telling a roommate in college about, “Oh, there's this theorem, it's not even proven.” So that was a question too. Why are we calling this a theorem? So back in the day, that was not a theorem, but it was still called Fermat’s Last Theorem. And in telling, you know, relating the story that Fermat was writing in the margin of his I don't know Arithmetica or something in the 1600s. And that he said, “I had the most delightful proof for this, but the margin is too small to contain it.” And my roommate’s first reaction actually was “Has anyone looked through all of his papers to find the proof?” And that was nice, because, you know, coming from a different discipline, studying English and history and so on. Because to me that wasn't the first reaction. It was like, oh, if Fermat had a proof, can we figure it out too? Or can we figure out what he—maybe he had something, but what mistake might he have made? Because there's more to this one perhaps. But anyway, the category was “statements that draw you in with their simplicity.” Maybe the four-color theorem should have landed here.

EL: Yeah.

LK: I don’t know.

EL: Yeah, draw you in. It's kind of—I don't know if this is maybe a bad analogy to draw, but kind of catfishing. Yeah. There’s just this nice, well-behaved statement. And oh, yeah, now it's a giant mess to prove. Actually, maybe like the Jordan curve theorem.

LK: Yeah, maybe a lot of these end up there. Then there's that way, though, if something's finally— sometimes when you finally prove something, you're like, “Oh, why didn't I think of that earlier?” I don't know that Fermat will ever land there for me, but maybe the Jordan curve, maybe there are aspects of some of these that you just come to a different understanding on the other side of the hill.

EL: Yeah. So I think if I were doing this taxonomy, one of my categories—which is probably not a good category, but I think I would have a sentimental attachment to it and be unable to get rid of it—would be like, theorems with weird numbers in them or, or really big numbers in them, like the one that we talked about with Laura Taalman, where there’s this absurd bound for the number of Reidemeister moves you have to do for knots. Like there are some theorems where like, you've got some weirdness, it's like, oh, yeah, this theorem is, works for everything except the number 128. And it's just like, theorems with weird numbers in them, or weird numbers in their proofs, I think would be one of mine. Or, like the proof of the ternary Goldbach conjecture several years ago, which I only remember because I wrote about it, is basically proving that it works up through a certain very large number of just individual cases, and then having some argument that works above 10 to the some large number, and like, that's just a little funny. It's like, “Oh, yeah, we checked the first 12 quadrillion. And then once we did that, we were made in the shade.” And I don't know, I think I think that goes a long way with me.

KK: How about theorems with silly names? Like, like the ham sandwich theorem.

LK: I think the topologists corner the market on this, right? Yeah? No? Maybe?

KK: We really do.

LK: Yeah, the ham sandwich. No, I like so we need to find one that's like, unusual cases, or a funny number comes up and it has a funny name to boot. I love these categories. Well, how about how about something where the statement might surprise the casual listener. So in other words, like, the Brouwer fixed-point theorem, so when I’m I chatting with my students, I say, “Oh, you toss a map of California onto the table (because I'm in California) and there's some point on the map that's lying above its point in the real world.” And then oh, I can do it all over again, toss it again, it doesn't land the same way. And then, and they start to realize, oh, there's something going on here. But I don't know if that's surprising. Maybe my students are a captive audience. I say surprising to the casual listener. Maybe it's surprising to the captive audience. I don't know.

EL: Yeah, well, that's definitely like a one where the theorem doesn't seem surprising, or, you know, the theorem doesn't seem that strange. And then it has these applications or examples that it gives you that you're like, oh, wow, like that makes you think like, for me, it's always the weather. What is it? That there are two antipodal points on the earth with the same, you know, wind speed, or at any given time or temperature, whatever the thing is you want to measure?

KK: The Borsuk-Ulam theorem.

EL: Maybe the same of both? I don't remember how many dimensions you get.

KK: Well, you could do it in every dimension. So yeah, it's the Borsuk-Ulam theorem, which is that a map from the n-sphere into R^n has to send a pair of antipodal points at the same point. Right.

EL: So the theorem, when you read it, it doesn’t seem like it has anything weird going on. And then when you actually do it, you're like, “Whoa, that's a little weird.”

LK: Oh yeah, I like that. Maybe that's true, so many of the things we we look at. So I guess I realized, as I was thinking about these, I was tipping towards theorems where there's also some kind of analogy or way to convey it without the technical details. Certainly, if the category is to draw in the casual listener, or to sucker someone in without the technical machinery. Yeah, so I don't know what would be next in the taxonomy of theorems. Do you have other ideas?

EL: I’m not sure. Yeah, I feel like I’d need to sit down for a little bit. Actually first go through our archives and like look at the theorems that people have picked, and see where I think they would land.

LK: I had a funny taxonomy category that's very narrow, but it could be “guess that theorem.” But I was thinking theorems with cute names or interesting funny names that have also been proven in popular films.

KK: Oh, the snake lemma.

LK: Ding-ding-ding, we have a winner.

KK: You know, don’t pin me down on what the movie is. I can't remember.

EL: I think t's called It’s My Turn.

KK: That’s it.

LK: Wow, the dynamic duo here has exactly. And I have to admit, when I was thinking of it, I was like, “I don’t remember the movie.” And I had to look it up. But anyway, algebra comes to the rescue.

EL: Yeah, I’ve seen that scene from it, but I've never seen the rest of the movie for sure.

KK: Has anybody?

LK: As mathematicians, maybe we should.

EL: I don’t even know if it’s on DVD. It might might never have been popular enough to get to the new format.

KK: And isn’t that the last time that there's any math in the movie? Like it's this opening scene, and she proves the theorem, and then that's it? Never any more?

LK: So it's really a tragedy, that film. But no, they say this is the year that people said, Oh, they watched all of Netflix. I don't know if that's possible. So this is the year, then, to reach out to expand. Or maybe if we rise up and request more streaming options for the movie. I would like to show my students students that. Yeah, but I also admit, I haven’t seen the film.

Maybe a big core category we're missing is those theorems that really bridge different areas or topics. So Kevin, you give an example of a statement that could be algebraic, but it's proven topologically. But then I was thinking, are there theorems that kind of point to a dictionary between areas? And I only had one little example in mind, but maybe I'll call it my little unsung hero, a theorem that won't be as familiar to folks, but I was thinking of something called Belyi’s theorem, so not as well known as the others, perhaps, but that number theorists and arithmetic geometers are really interested in. And then actually, I went ahead and printed out ahead of time, these quotes of Grothendieck, who was so struck when this theorem was announced or proven because he'd been thinking along these lines, but was surprised at the simplicity of their proof. But my French is not very good, so I'm not going to read anything in French. But I don't know if you want to take a moment to talk about this theorem.

KK: Sure.

EL: Yeah.

KK: So what's the statement?

LK: Yeah, so maybe I'll say en route to the statement that number theorists and arithmetic geometers are interested in ramification, but I'm maybe I'm going to describe things in terms of covering maps, and whether you have branching over a covering so. So like, if you had a Riemann surface, you're mapping to Riemann surface, and you had a covering map, you might expect, okay, for every point down below, you'd expect the same number of preimages, or for every neighborhood down below, the same number of neighborhoods, if it's a degree D map, maybe a D-fold cover. And in fact, I remember my advisor first describing this to me by saying, if you had a pancake down below, you'd have D pancakes up above. And it really stuck in my head, frankly, because he was so precise and mathematical in his language at every moment, this was one of the most informal things I ever heard him say. Maybe he was hungry at the moment, he was thinking about pancakes. So as a concrete example where something different could happen, suppose I was mapping to the Riemann sphere, and I suppose I had a map, like I don't know, take a number and cube it, like x cubed, and started asking what kind of preimages points have. For example, x cubed equals 1, there are three roots of unity that map to 1, but something different is happening at zero, so only zero maps to zero. There's no other value that when you cube it, gives you zero. So now we no longer have, instead of a cover, maybe I'll say we have a cover, except at finitely many points. So somehow zero, and in that case, infinity, there's some point at infinity that behaves differently, but everything else has three distinct preimages. And maybe just to make a picture, let's take the interval from 0 to 1. So a little line segment, the real interval, and we could ask what its preimage looks like. And so above 1, there are three points up above. There are three roots of unity that map to 1, and on the other hand 0 was the only point that mapped to zero. And for the rest of the interval, all of those points have three preimages. So you could draw, maybe I'm picturing now a little graph on my original surface that's got a single vertex, say, at zero, and then three segments going out for each of the preimages of the real line, and ending at these three roots of unity, ending at the preimages of 1. And so now I'm not even thinking very precisely about what it looks like. I'm just picturing a graph. So I’m not worrying about how beautiful my drawing is. I just have one vertex over zero and then three branches. So what number theorists describe in terms of ramification, in this setting we might think of as branching. So these branch points. So I'm interested in saying when I have a map, say to the Riemann sphere, or number theorists might say to the projective line, I'm interested in what kind of branching is happening. And it turns out that — so now Belyi’s theorem — he realized that in the situation where you're branched over at most three points, so in the picture, we had over 0 and also infinity. I was kind of vague about what's happening at infinity. So that was two points. But if there are at most three points where branching happens, something very special is going on. So he was looking at maps from curves to the projective line. So in a nutshell, really what he proved was that a curve is algebraic if and only if there's one of these coverings that's branched at at most three points. So what is that saying? So saying a curve is algebraic? That's an algebraic statement. You're kind of saying, Well, if you had an equation for the curve — suppose I could write down an equation and then the solutions to that equation are the points of the curve — he’s saying that the coefficients have to be algebraic numbers. So they don't just have to be integers. I could have coefficients, like the square root of two could be a coefficient, or i, or your favorite algebraic number, but not pi, or e or any non-algebraic number. So that's an algebraic statement. But saying that that can happen if and only if, and now he has a map actually, from the curve, well I'm going to say from some Riemann surface to the Riemann sphere, that's branched over at most three points, that second statement is very topological. And it's actually sort of combinatorial too, because that graph I was describing earlier, people use those to kind of describe what's happening with these maps. And so the number of edges, the number of vertices, there's a lot of combinatorial information embedded in that picture. And so I don't know how much of the theorem really comes through in this oral description. But the point is, people were really surprised, including Grothendieck was surprised. He was so surprised and agitated, but excited, that he wrote a letter to the editor, and it's been published. Leila Schneps has done these amazing volumes about a topic called dessins d’enfants, or children's drawings, but I have to read a piece of this because he wrote something like “Oh, Belyi announced this very result.” So this idea, he says actually, “Deligne when consulted found it crazy indeed, but without having a counterexample at hand. Less than a year later, at the International Congress in Helsinki, the Soviet mathematician Belyi announced this very result, with a proof of disconcerting simplicity contained in two little pages of a letter of Deligne. Never was such a profound and disconcerting result proved in so few lines.” So Belyi had actually figured out not only a way to show that these maps exist, but he had a construction. And it reminds me of something you were saying earlier, Evelyn, where the construction exists, maybe it's an unwieldy construction, in the sense that if you really wanted to work with these maps, you might want to do better, and if you try to bound, something I tried to do earlier, you get these really huge degree bounds on maps that are not so practical, in a sense, but the fact that you could do it, so it was the fact not only of the existence, but also there was a constructive proof, opened the door to lots of other work that folks have done.

And maybe I just want to say I was looking — so my French is not good enough to read and translate on the fly. But this “disconcerting result” the word that was used déroutant, can also mean strange and mysterious and unsettling. So even our taxonomy could include unsettling proofs or unsettling results. But I really wanted to put this in the category of something that that bridges different areas, because this picture I was describing earlier really was just a graph with three edges and four vertices. It’s an example of what Grothendieck called, he nicknamed them dessins d’enfants, or children's drawings, the preimagesof this interval. And yeah, so this is really a topic that's caught people's imagination, and Frothendieck was thinking “Are there ways to get at the absolute Galois group?” Because these curves I mentioned were algebraic, so something behind the scenes here is purely algebraic. You can look at Galois actions on the coefficients, for example. But meanwhile, you have this topological combinatorial object. And when you apply this action, we preserve features of the graph, we preserve the number of vertices and edges and so on. Can you start to look at conjugate drawings? And so these doors opened up to these fanciful routes, but it also pointed to these bridges between areas. Maybe algebraic topology is full of these, where you have some algebraic tools, but you're looking at something topological, just things that bridge or create dictionaries between between areas of mathematics, I think are really neat. Yeah. So in the end, you could even bring a stick figure to life this way. So I described this funny-looking graph with just three edges, but you could actually draw a stick figure in this setting, labeling vertices and edges. So I'm picturing, I don't know, literally a little stick figure.

EL: Yeah.

LK: And give some mathematical meaning to it. And then through these through Belyi’s theorem, and through this dictionary, is actually related to curves and so on. And then you can do all kinds of fun things. Like I mentioned some Galois action, although I wasn't specific about it. You could start to ask, are there little mutant figures in the same family as a stick figure? Maybe there's a stick figure with both arms on one side? And is that conjugate somehow to your original, and so somehow there was something elusive about this. The proof had eluded Grothendieck. But it opened this door to very fanciful mathematics. And there's really been kind of an explosion of work over the years looking at these dessins d’enfants. It's a podcast, but I saw you nodding when I mentioned these children’s drawings.

EL: Well, that's a term I've definitely seen. And then not really learned anything about it. Because I must admit, algebraic geometry is not something that my mind naturally wants to go and think about a whole lot.

LK: There’s a lot of machinery, and actually one direction of Belyi — I said this theorem as and if and only if — but one direction was sort of known and takes much more machinery. And it was this disconcerting direction, as Grothendieck said, that actually took less somehow. Some composition of maps and keeping track of ramification, or using calculus to see where you have multiple images of points, or preimages. Yeah, in fact, Grothendieck, there was one last sentence I found, I culled from this great translation by Leila Schneps, who said, “This deep result, together with the algebraic geometric interpretation of maps, opens the door to a new unexplored world, within reach of all, who pass without seeing it.” And you know, we really don't usually see mathematicians speaking in these terms about their work. So that's something I loved. I also loved that Belyi’s proof was constructive too, because even if it creates bounds, I might not want to use, it becomes a lynchpin in other work that connects — the fact that it could be made effective, like not just that this map exists, but you can actually have some degree bound on a certain map, is a lynchpin. And maybe the funniest example takes me to a last category, which is how about theorems that may not be theorems? Like what counts as a theorem? And there's this statement called ABC conjecture. Which is—

EL: A can of worms.

LK: Yeah, so is it proven or not?

KK: It depends on who you ask.

LK: Yeah, so there’s this volume of work by Shinichi Mochizuki, it’s 500-plus pages, and he's created this, I think it was called inter-universal Teichmüller theory. And I, you know, I can't speak to it, but experts are chipping away, chipping away. And maybe it's — I don't know if it's too political to say it's in kind of a limbo. There may be stuff there. There's a lot of machinery there. And yet, do lots of people understand and sort of verify this proof? I'm not sure we're there.

KK: I mean, he’s certainly a respected mathematician. So that's what people taking it seriously. But that's right. But didn't Scholze point to one particular lemma that he thought wasn't true? And the explanations from Kyoto have not been satisfying?

LK: Yeah, I don't have my finger on the pulse. But it’s this funny thing where if you unravel a thread, does the whole thing come apart? And on the other hand, when Wiles proved Fermat’s last theorem, well, some people realized that it would need to do a little something more here. But then it happened. And it kind of was consistent with the theory to be able to sure to fill that in. Yeah. So this is — I don't know, it's exciting to me, but it's also daunting. But this ABC conjecture, so I mentioned Belyi’s theorem. So there's a paper that assuming the ABC conjecture — we don't know if we have a proof, but going back when we've still just called it a conjecture — you can imply or from that, you get so many other results in number theory that people believe to be true. And Noam Elkies has this paper ABC implies Mordell, so Faltings’ theorem, so this theorem about numbers of points on curves. And there's this, I thought this is funny. So I’ll mention this last thing, but this paper has been nicknamed by Don Zagier: Mordell is as easy as ABC. And it's kind of funny, because they're quite difficult no matter how you slice it. You've got something that's still an open problem. And then something that had a very difficult proof. So to say one thing is as easy as the other is sort of perfect. Yeah, there's much more to say about the ABC conjecture, but maybe that's a topic for My Favorite Conjecture.

EL: Yeah. Or My Favorite Mathematical Can of Worms.

KK: Yeah, yeah. Okay, so.

EL: I like this.

KK: Yeah. Well, I was going to say it might be time for the pairing.

EL: I think it is.

KK: So I think I think maybe you're going to pair something with Belyi’s theorem, but maybe not. Maybe there’s something else.

LK: Yeah, I wanted to. I feel like I didn't do justice to Belyi’s theorem, and originally, I'll admit it, I was going to say a gingerbread man because I mentioned stick figures. And so I was like, okay, pairing, well, I love food, made me think of food, made me think of a gingerbread man because of this theory of dessins, or drawings, of Grothendieck. So you can attach a meaning to this little stick figure. And maybe when you're baking, you start making funny-looking figures and those are your Galois conjugates, I don't know. But actually, you know, I was so long on this list of theorems, I'll be short. I think I just have to go with coffee too. Maybe a gingerbread man and coffee because, you know, I wanted to be clever and delicious. But instead I’m just going with coffee because, well, I drink a lot of coffee. They say mathematicians turn coffee into theorems. So can't go wrong. And during the pandemic working at home, I would say I've consumed a lot of coffee in all its incarnations. And maybe it takes me back, too. When I was first hearing about Belyi’s theorem and elsewhere, I was very lucky to have the chance to spend some time in the Netherlands because my advisor Hendrik Lenstra was spending time there, and so as students, we got to go for periods of time. It was very influential to me to be there. But there's a coffee you can get in the Netherlands, which is probably sort of cafe au lait meets latte. But it's called something like koffie verkeerd, and I'm going to mispronounce it, but it basically means messed up coffee. And that's one of my favorite coffees, coffee with, it has too much milk in it. I guess that's what messes it up. So maybe that will be my pairing, just to stick with coffee.

KK: All right. Yeah.

EL: Well, I thought you might go like a pairing for this whole taxonomy and just go with, like, the taxonomy of animals, which, you know, I feel like we didn't do a great job of like, getting theorems exactly into one category or another. And historically, that has also been true for our understanding of biology and like, how many kingdoms there are, you know, in terms of, like, animals, plants, and then a bunch of other stuff.

LK: That’s right, I'm counting on someone to hopefully listen enough to this sprawling, fanciful discussion and say, “Oh, no, no, no, here's how we should do it,” and actually come up with a decent but entertaining, I hope, taxonomy.

EL: Well, we also like to give our guests a chance to plug anything. You know, if you have a website, books or projects that you're working on, that you want people to be able to find online, feel free to share those.

LK: Yeah, that's such a gracious door that you open to everyone. And I mean, maybe I do want to say, in honor of work with collaborators, that math sent me on sort of an unusual journey, as I mentioned in the beginning. So now, for example, looking at the issue of racial profiling and statistics and policy and law. And I do think that there are ways that mathematicians are very creative and can carry that creativity to all of their endeavors, including many of us are spending a lot of time in the classroom. And so that interest has led to a collaboration with Gizem Karaali. She's at Pomona College. And so we do have some books that we've been lucky to co-edit, so many creative people have contributed to. So these are books around mathematics for social justice. There are some essays. There are contributed materials of all sorts. The first volume came out in 2019, in the Before Times. The second volume is due out in 2021. But these represent the work of so many people. And actually, many of the theorems that have come up in your beautiful podcast have come up there, like Arrow’s impossibility theorem around voting theory. Kevin, I think you've been in talks about gerrymandering. And that’s, you can imagine, a topic of great interest. And these materials are more introductory, for folks to bring into the classroom. But as I said, I think mathematicians are very creative, and so it's neat to see what other people have done. And so I hope others will be inspired by those examples as they're creating authentic engagement and cultivating critical thinking for ourselves and all the students we work with.

EL: Yeah, well we’ll make sure to put links to that in the show notes.

KK: Sure.

LK: Yeah. Well, thank you for a sprawling conversation today.

KK: This has been a sprawl, but it has been a lot of fun, actually. I kind of felt like you were interviewing us a little more.

LK: Oh, I that sounds fun to me.

KK: Yeah. This is a great one. I'm going to look forward to editing this one. This will be a good time.

LK: Well, maybe a lot will end up on the editing floor.

KK: I hardly ever cut anything out. I really don't.

LK: There’s always a first time.

EL: You’re on the hot seat!

KK: Lily, thanks so much for joining us. It's been a lot of fun.

LK Thank you for your time.

On this episode, we talked with Lily Khadjavi, a mathematician at Loyola Marymount University in Los Angeles. Instead of choosing one favorite theorem, she led us through a parade of mathematical greatest hits and talked through a taxonomy of great theorems. Here are some links you might enjoy as you listen.

Khadjavi's academic website
Her website about mathematics and social justice, which includes the books she mentioned with co-editor Gizem Karaali
Leila Shneps's book The Grothendieck Theory of Dessins d'Enfants
Steve Strogatz's article about Einstein's proof of the Pythagorean theorem

Try your hand at four-coloring Joaquin Torres-Garcia’s Figuras Constructivas
And some past episodes of My Favorite Theorem about some of the theorems in this episodes:
Adriana Salerno and Yoon Ha Lee on Cantor's diagonalization argument
Henry Fowler and Fawn Nguyen on the Pythagorean theorem
Susan D'Agostino on the Jordan curve theorem
Belin Tsinnajinnie on Arrow's impossibility theorem
Ruthi Hortsch on Faltings' theorem
Ken Ribet on the infinitude of primes
Francis Su and Holly Krieger on Brouwer's fixed point theorem

Episode 62 - Tai-Danae Bradley

15 januari 2021 | 32 min

Evelyn Lamb: Welcome to my favorite theorem, a math podcast. I'm Evelyn Lamb, one of your hosts. And here's your other host.

Kevin: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. It's been a while. I haven't seen your smiling face in a while.

EL: Yeah. I've started experimenting more with home haircuts. I don't know if you can see.

KK: I can. It's a little a little longer on top.

EL: Yeah.

KK: And it's more of more of a high and tight thing going here. This is Yeah. All right. It looks good.

EL: Yeah, it's been kind of fun. And, like, depending on how long ago between washing it, it has different properties. So it's very, it's like materials science over here, too. So a lot of fun.

KK: Well, you probably can't tell, but I've gone from a goatee to a plague beard. And also, I've let my hair grow a good bit longer. I mean, now that I'm in my 50s, there's less of it than there used to be. But I am letting it grow longer, you know, because it's winter, right?

EL: Oh yeah. Your Florida winter. It's probably like, what? 73 degrees there?

KK: It is 66 today. It's chilly.

EL: Oh, wow. Yeah, gosh! Well, today we are very happy to invite Tai-Danae Bradley to the podcast. Hi, Tai-Danae. Will you tell us a little bit about yourself?

Tai-Danae Bradley: Yeah. Hi, Evelyn. Hi, Kevin. Thank you so much for having me here. So I am currently a postdoc at X. People may be more familiar with his former name, Google X. Prior to that, I recently finished my PhD at the CUNY Graduate Center earlier this year. And I also enjoy writing about math on a website called math3ma.

EL: Yes, and the E of that is a 3 if you're trying to spell it.

TDB: Yeah, m-a-t-h-3-m-a. That's right. I pronounce it mathema. Some people say math-three-ma, but you know.

EL: Yeah, I kind of like saying math-three-ma my head. So, I guess, not to not to sound rude. But what does X want with a category theorist?

TDB: Oh, that's a great question. So yeah, first, I might say for all of the real category theorists listening, I may humbly not refer to myself as a category theorist. I'm more of, like, an avid fan of category theory.

KK: But you wrote a book!

TDB: Yeah, I did. I did. No, I really enjoy category theory, I guess I'll say. So at X, I work on a team of folks who are using ideas from—now this may sound left field—but they're using ideas from physics to tackle problems in machine learning. And when I was in graduate school at CUNY, my research was using ideas in mathematics, including category theory, to sort of tackle similar problems. And so you can see how those could kind of go hand in hand. And so now that I'm at X, I'm really just kind of continuing the same research interest I had, but, you know, in this new environment.

EL: Okay, cool.

KK: Very cool.

EL: Yeah, mostly, we've had academics on the podcast. We’ve had a few people who work in other industries, but it's nice to see what's out there, like, even a very abstract field can get you an applied job somewhere.

TDB: Yeah, that's right.

EL: Yeah, well, of course, we did invite you here to talk about your job. But we also invited you here to ask what your favorite theorem is.

TDB: Okay. Thank you for this question. I'm so excited to talk about this. But I will say, I tend to be very enthusiastic about lots of ideas in mathematics at lots of different times. And so my favorite theorem or result usually depends on the hour of the day. Like, whatever I’m reading at the time, like, this is so awesome! But today, I thought it'd be really fun to talk about the singular value decomposition in linear algebra.

KK: Awesome!

TDB: Yeah. So I will say, when I was an undergrad, I did not learn about SVD. So I think my undergrad class stopped just before that. And so I had to wait to learn about all of its wonders. So for people who are listening, maybe I could just say it's a fundamental result that says the following, simply put. Any matrix whatsoever can be written as a product of three matrices. And these three matrices have nice properties. Two of them, the ones on the left and the right, are unitary matrices, or orthogonal if your matrix is real. And then the middle matrix is a diagonal matrix. And the terminology is if you look at the columns of the two unitary matrices, these are called the singular vectors of your original matrix. And then the entries of the diagonal matrix, those are called the singular values of that matrix. So unlike something like an eigen decomposition, you don't have to make any assumptions about the matrix you started with. It doesn't have to have some special properties for this to work. It's just a blanket statement. Any matrix can be factored in this way.

EL: Yeah, and I, as we were saying, before we started recording, I also did not actually encounter this in any classes.

KK: Nor did I.

EL: And yeah, it’s something I've heard of, but not never really looked into because I didn't ever do linear algebra, you know, as part of my thesis or something like that. But yeah, okay, so it seems a little surprising that there aren't any extra restrictions on what kind of matrices can do this. So why is that? I don't know if that question is too far from left field.

TDB: Maybe that's one of the, you know, many amazing things about SVD is that you don't have to make any assumptions. So number one, in mathematics, we usually say multiplying things is pretty easy, but factorizing is hard. Like, it's hard to factor something. But here in linear algebra, it's like, oh, things are really nice. You just have this matrix, and you get a factorization. That's pretty amazing. I think, maybe to connect why is that—to connect this with maybe something that's more familiar, we could ask, what are those singular vectors? Where do they come from? Or, you know, what's the proof sketch of this?

EL: Yeah.

TDB: And essentially, what you do is you take your matrix, you multiply it by its transpose. And that thing is going to be this nice real symmetric matrix, and that has eigenvectors. And so the eigenvectors of that matrix are actually the singular vectors of your original one. Now, depending on like, if you multiply them the transpose of the matrix on the left or right, that will determine whether, you know, you get the left or right singular vectors. So, you might think that SVD is, like, second best: “Oh, not every matrix is square, so, we can't talk about eigenvectors, oh, I guess singular vectors will have to do.” But actually, it's like picking up on this nice spectral decomposition theorem that we like. And I think when one looks out into the mathematical/scientific/engineering landscape, and you see SVD sort of popping up all over the place, it's pretty ubiquitous. And so that sort of suggests it’s not a second-class citizen. It's really a first-class result.

EL: Yeah. Well, that's funny, because I did, when I was reading it, I was like, “Oh, I guess this is a nice consolation prize for not being an invertible square matrix, is that you can do this thing.” But you're telling me that that was—that’s not a good attitude to have about this?

TDB: Well, yeah, I think SVD, I wouldn't think of it as a consolation prize, I think it is quite something really fundamental. You know, if you were to invite linear algebra onto this podcast and ask linear algebra, what its favorite theorem is, just based on the ubiquity and prevalence of SVD in nature, I'd probably bet linear algebra would say singular value decomposition.

EL: Yeah, can can we get them next?

KK: Can we get linear algebra on? We’ll see. Okay, so I don't know if this question has—it must have an answer. So say your matrix is square in the first place. So you could talk about the eigenvalues, and you do this, I assume the singular values are different from the eigenvalues. So what would be the advantage of choosing the singular values over the eigenvalues, for example?

TDB: So I think if your matrix is square, and symmetric, or Hermitian, then the eigenvectors correspond to the singular vectors.

KK: Okay, that makes sense.

TDB: But, that's a good question, Kevin. And I don't have a good answer that I could confidently go on record with.

KK: That’s cool. Sorry. I threw a curveball.

TDB: That’s a great question.

KK: Because then singular values are important. The way I've always sort of heard it was that they sort of act like eigenvalues in the sense that you can line them up and that the biggest one matters the most.

TDB: Exactly, exactly. Right. And in fact, I mean, that sort of goes back to the proof that we were talking about. I was saying, oh, the singular vectors are the eigenvectors of this matrix multiplied by its transpose. And the singular vectors turn out to be the square roots of the eigenvalues of that square matrix that you got. So they're definitely related.

KK: Okay. All right. Very cool. So what drew you to this theorem? Why this theorem in particular?

TDB: Yeah, why this theorem? So this kind of goes back to what we were talking about earlier. I really like this theorem because it's very parallel to a construction in category theory.

KK: Yes.

TDB: Maybe people find that very surprising. We're talking about SVD. And all of a sudden, here's this category theory, curveball.

EL: Yeah, because I really do feel like linear algebra almost feels like some of the most tangible math., and category theory, to me, feels like some of the least tangible.

KK: So wait, wait, are you going to tell us this is the Yoneda lemma for linear algebra?

TDB: No. Although that was going to be my other favorite theorem. Okay, so I'm excited to share this with you. I think this is a really nice story. So I'm going to try my best because it can get heavy, but I'm going to try to keep it really light. But I might omit details, but you know, people can maybe look further into this.

So to make the connection, and to keep things relatively understandable, let's forget for a second that I even mentioned category theory. So let’s empty our brains of linear algebra and category theory. I just want to think about sets for a second. So let me just give a really simple, simple construction. Suppose we have two sets. Let's say they're finite, for simplicity. And I'll call them a set X and a set Y. And suppose I have a relation between these two sets, so a subset of the cartesian product. And just for simplicity, or fun, let’s think of the elements of the set X as objects. So maybe animals: cat, dog, fish, turtle, blah, blah. And let's also think of elements in the set Y as features or attributes, like, “has four legs,” “is furry,” “eats bugs,” blah, blah, blah. Okay. Now, given any relation—any subset of a Cartesian product of sets—you can always ask the following simple question. Suppose I have a subset of objects. You can ask, “Hey, what are all the features that are common to all of those objects in my subset?” So you can imagine in your subset, you have an object, that object corresponds to a set of features, only the ones possessed by that object. And now just take the intersection over all objects in your subset? That's a totally natural question you could ask. And you can also imagine going in the other direction, and asking you the same question. Suppose you have a subset of features. And you want to know, “Hey, what are all of the objects that share all of those features in that subset I started with?” A totally natural question you could ask anytime you have a relation.

Now, this leads to a really interesting construction. Namely, if someone were to give me any subset of objects and any subset of features, you could ask, “Does this pair satisfy the property that these two sets are the answers to those two questions that I asked?” Like, I had my set of objects and, Oh, is this set of features that you gave me only the ones corresponding to this set of objects and vice versa? Pairs of subsets for which the answer is yes, that satisfy that property, they have a special name. They're called formal concepts. So you can imagine like, oh, the concept of, you know, “house pet” is like the set of all {rabbits, cats, dogs}, and, like, the features that they share is “furry,” “sits in your lap,” blah, blah, blah. So this is not a definition I made up, you know, you can go on Wikipedia and look at formal concept analysis. This is part of that. Or you can usually find this in books on lattice theory and order theory. So formal concepts are these nice things you get from a relation between two sets.

Now, what in the world does this have to do with linear algebra or category theory, blah, blah, blah? So here's the connection. Probably you can see it already. Anytime you have a relation, that’s basically a matrix. It's a matrix whose entries are 0 and 1. You can imagine a matrix where the rows are indexed by objects and the columns are indexed by your features. And there's a 1 and the little x little y entry if that object has that feature and 0 otherwise.

KK: Sure.

TDB: And it turns out that these formal concepts that you get are very much like the eigenvectors of that 0-1 matrix multiplied by its transpose. AKA, they're like the singular vectors of your relation. So I'm saying it turns out—so I'm kind of asking you to believe me, and I'm not giving you any reason to see why that should be true—But it's sort of, when you put pen to paper paper and you work out all of the details, you can sort of see this. But I say it's like because if you just do the naive thing, and think of your, your 0-1 matrix as a linear map, like as a linear transformation, you could say, okay, you know, should I view this as a matrix over the reals? Or maybe I want to think of 0 and 1 as you know, the finite field with two elements. But if you try to work out the linear algebra and say, oh, formal concepts are eigenvectors, it doesn't work. And you can sort of see why that is. we started the conversation with sets, not vector spaces. So this formal concept story is not a story about linear algebra, i.e., the conversation is not occurring in the world of linear algebra. And so if you have mappings—you know, from sets of objects to sets of features—the kind of structure you want that to preserve is not linearity, because we started with sets. So we weren't talking about linear algebra.

So what is it? It turns out it's a different structure. Maybe for the sake of time, it's not really important what it is, or if you ask me, I'll be happy to tell you. But just knowing there's another kind of structure that you'd like this map to preserve, and under that right sort of context, when you're in the right context, you really do see, oh, wow, these formal concepts are really like eigenvectors or singular vectors in this new context.

Now, anytime you have a recipe, or a template, or a context, but you can just sort of substitute out the ingredients for something else, I mean, there's a bet that category theory is involved. And indeed, that's the case. So it turns out that this mapping, this sort of dual mapping from objects to features, and then going back features to objects, that, it turns out, is an example of adjunction in category theory. So there's a way to view sets as categories. And there's a way to view mappings between them as functors. And an adjunction in category theory is like a linear map and its adjoint, or like a matrix and its transpose. So in category theory, an adjunction is — let me say it this way, in linear algebra, an adjoint is defined by an equation involving an inner product. Linear adjoint, there's a special equation that your map and its adjoint must satisfy. And in category theory, it's very analogous. It's a functor that satisfies an “equation” that looks a lot like the adjoint equation in linear algebra. And so when you unravel all of this, it's almost like Mad Libs, you have, like, this Mad Lib template. And if you erase, you know, the word “matrix” and substitute in the whatever categorical version of that should be, you get the thing in category theory, but if you stick in “matrix,” oh, you get linear algebra. If you erase, you know, eigenvectors, you get formal concepts, or whatever the categorical version of that is, but if you if you have eigenvectors, then that's linear algebra. So it's almost like this mirror world between the linear algebra that we all know and love, and like, Evelyn, you were saying, it's totally concrete. But then if you just swap out some of the words, like you just substitute some of the ingredients in this recipe, then you recover a construction in category theory, and I am not sure if it's well known — I think among the experts in category theory it is — but it's something that I really enjoy thinking about. And so that's why I like SVD.

EL: So I think you may have had the unfortunate effect of me now thinking of category theory as the Mad Libs of math. Category theorists are just going and erasing whatever mathematical structure you had and replacing it with some other one.

KK: That’s what a category is supposed to do, right? I mean, it's this big structure that just captures some big idea that is lurking everywhere. That's really the beautiful thing, and the power, of the whole subject.

TDB: Yeah, and I really like this little Mad Lib exercise in particular, because it's kind of fun to think of singular vectors as analogous to concepts, which could sort of maybe explain why it's so ubiquitous throughout the scientific landscape. Because you have this matrix, and it’s sort of telling you what goes with what. I have these correlations, maybe I organize them into a matrix matrix, I have data and organize it into a matrix. And SVD sort of nicely collects the patterns, or correlations, or concepts in the data that's represented by our matrix. And, I think, Kevin, earlier you were saying how singular values sort of convey the importance of things based on how big they are. And those things, I think, are a little bit like the concepts, maybe. That’s sort of reaching far, but I think it's kind of a funny heuristic that I have in mind.

KK: I mean, the company you work for is very famous for exploiting singular values, right?

TDB: Exactly. Exactly.

KK: Yep. So another fun part of this podcast is we ask our guests to pair their favorite theorem with something. So what pairs well with SVD?

TDB: Okay, great question. I thought a lot about this. But I, like, had this idea and then scratched it off, then I had another idea and scratched it off. So here's what I came up with. Before I tell you what, I want to pair it pair this with, I should say, for background reasons, this, Mad Libs or ingredients-swapping recipe-type thing is a little bit mysterious to me. Because while the linear algebra is analogous to the category theory, the category theory doesn't really subsume the linear algebra. So usually, when you see the same phenomena occurring a bunch of places throughout mathematics, you think, “Oh, there must be some unifying thread. Clearly something is going on. We need some language to tell us why do I keep seeing the same construction reappearing?” And usually category theory lends a hand in that. But in this case, it doesn't. There's no—in other words, it's like I have two identical twins, and yet they don’t, I don’t know, come from the same parents or something.

KK: Separated at the birth or something?

TDB: Yeah. Something like that. Yeah, exactly. They’re, like, separated to birth, but you're like, “Oh, where are their parents? Where were they initially together?” But I don't know that, that hasn't been worked out yet. So it's a little bit mysterious to me. So here it is: I'm going to pair SVD with, okay. You know, those dum-dum lollipops?

KK: Yeah, at the bank.

TDB: Okay. Yeah, exactly. Exactly. Just for listeners, that’s d-u-m, not d-u-m-b. I feel a little bit—anyway. Okay, so the dum-dum lollipops, they have this mystery flavor.

KK: They do.

TDB: Right, which is like, I can't remember, but I think it's wrapped up with a white wrapper with question marks all over it.

EL: Yeah.

TDB: And you're letting it dissolve in your mouth. You're like, well, I don't really know what this is. I think it’s, like, blueberry and watermelon? Or I don't know. Who knows what this is? Okay. So this mystery that I'm struggling to explain is a little bit like my mathematical dum-dum lollipop mystery flavor. So, you know, I like to think of this as a really nice, tasty mathematical treat. But it's shrouded in this wrapper with question marks over it. And I'm not quite really sure what's going on, but boy, is it cool and fun to think about!

EL: I like that. Yeah, it's been a while since I went to the bank with my mom, which was my main source of dum-dum lollipops.

TDB: Same, exactly. That's funny, with my mom as well.

EL: Yeah. That that's just how children obtain dum-dums.

KK: Can you even buy them anywhere? I mean, that’s the only place that they actually exist.

EL: I mean, wherever, bank supply stores, you know, get a big safe, you can get those panic buttons for if there's a bank robber, and you can get dum-dum lollipops. This is what they sell.

TDB: That’s right.

KK: No, it must be possible to get them somewhere else, though. When I was a kid trick-or-treating back in the 70s, you know, there would always be that cheap family on the on the block that would either hand out bubblegum, or dum-dums. Or even worse, candy corn.

EL: I must admit I do enjoy candy corn. It's not unlike eating flavored crayons, but I’m into it. Barely flavored. Basically just “sweet” is the flavor.

KK: That’s right.

EL: Yeah, well, so actually, this raises a question. I have not had a dum-dum in a very long time. And so is the mystery flavor always the same? Or do they just wrap up some normal flavor?

KK: Oh, that’s a good question.

EL: Like, it falls off the assembly line and they wrap it in some other thing. I never paid enough attention. I also targeted the root beers, mostly. So I didn't eat a whole lot of mystery ones because root beer is the best dum-dum.

KK: You and me! I was always for the root beer. Absolutely.

EL: And butterscotch. Yeah.

TDB: Oh, yeah. The butterscotch are good. So Evelyn, I was asking that same question to myself just before we started recording. I did a quick google search. And I think what happens, at least in some cases, like maybe in the past—and also don't quote me on this because I don't work at a dum-dum factory—but I think it was like, oh, when we're making the, I don't know, cherry or butterscotch flavored ones, but then the next in line are going to be root beer or whatever, we’re not going to clean out all of the, you know, whatever. So if people get the transition flavor from one recipe into the other, we’ll just slap on the “mystery.” I don't know, someone should figure this out.

KK: Interesting.

EL: I don't want to find out the answer because I love that answer.

KK: I like that answer too.

EL: I don't want the possibility that it's wrong, I just want to believe in that. That is my Santa Claus.

KK: And of course, now I’m thinking of those standard problems in the differential equations course where you’re, like, you're doing those mixing problems, right? So you've got, you know, cherry or whatever, and then you start to infuse it with the next flavor. And so for a while, there's going to be this stretch of, you know, varying amounts of the two, and then finally, it becomes the next flavor.

TDB: Exactly.

EL: Well, can you quantify, like, what amount and which flavor dominates and some kind of eigenflavor? I'm really reaching here.

TDB: I love that idea.

EL: Yeah. Oh, man. I kind of want to eat dum-dums now. That’s not one of my normal candies that I go to.

TDB: I know, I haven't had them for years, I think.

KK: Yeah, well, we still have the leftover Halloween candy. So this is, we can tell our listeners—What is this today? It's November 19?

EL: 19th, yeah.

KK: Right. So yeah, we bought one bag of candy because we never get very many trick-or-treaters anyway. And this year, we had one small group. And so we bought a bag of mini chocolate bars or whatever. And it's fun. We have a two-story house. We have a balcony on the front of our house. So this group of kids came up and we lowered candy from our balcony down. When I say “we” I mean my wife. I was cooking dinner. But we still have this bag. We're not candy-eaters. But you're right. I'm jonesing for for a dum-dum now. I do need to go to the bank. But I feel a little cheap asking for one.

EL: Yeah. I feel like, you know, maybe 15, 16, is where you kind of start aging out of bank dum-dums.

KK: Yep, yeah. Sort of like trick-or-treating.

EL: Well, anyway, getting back to math. Have we allowed you to say what you wanted to say about the singular value decomposition?

TDB: Yeah. I mean, I could talk for hours about SVD and all the things, but I think for the sake of listeners’ brains, I don't want to cause anyone to implode. I think I shared a lot. Category theory can be tough. So I mean, it appears in lots and lots of places. I originally started thinking of this because it cropped up in my thesis work, my PhD work, which not only involved a mixture of category theory, but linear algebra for, essentially, things in quantum mechanics. And so you actually see these ideas appear in sort of, you know, “real-world” physical scenarios as well. Which is why, again, it was kind of drawing me to this mystery. Like, wow, why does it keep appearing in all of these cool places? What's going on? Maybe category theory has something to say about it. So just a treat for me to think about.

EL: Yeah. And if our listeners want to find out more about you and follow you online or anything, where can they look?

TDB: Yeah, so they can look in a few places. Primarily, my blog mathema. com. I'm also on Twitter, @mathema as well, Facebook and Instagram too.

EL: And what is your book? Please plug your book.

TDB: Thank you. Thank you so much. Right. So I recently co-authored a book. It’s a graduate-level book on point-set topology from the perspective of category theory. So the title of the book is Topology: A Categorical Approach. And so this is really—we had in mind, sorry about this with John Terilla, who was my PhD thesis advisor, and Tyler Bryson, who is also a student of John at CUNY. And we really wrote this for, you know, if you're in a first-semester topology course in your first year of graduate school. So basic topology, but we were kind of thinking, oh, what's a way to introduce category theory that’s sort of gentler than just: “Blah. Here’s a book. Read all about category theory!” We wanted to take something that people were probably already familiar with, like basic point-set. Maybe they learned that in undergrad or maybe from a real analysis course, and saying, “Hey, here's things you already know. Now, we're just going to reframe the thing you already know in sort of a different perspective. And oh, by the way, that perspective is called category theory. Look how great this is.” So giving folks new ways to think and contemplate things they already know, and sort of welcoming them or inviting them into the world of category theory in that way.

KK: Nice.

EL: Yeah. So definitely check that out if you're interested in—the way you said like “Blah, category theory” —he other day, for some reason, I was thinking about the Ice Bucket Challenge from, like, I don't know, five or six years ago, where people poured the ice on their head for ALS research. (You’re also supposed to give money because pouring ice on your head doesn't actually help ALS research.)

TDB: Right.

EL: But yeah, it's like this is an alternative to the Ice Bucket Challenge of category theory.

TDB: That’s right. That's a great way to put it. Exactly.

EL: Yeah. Well, thank you so much for joining us. It was fun.

KK: This was great fun. Yeah.

On this episode, we had the pleasure of talking with Tai-Danae Bradley, a postdoc at X, about the singular value decomposition. Here are some links you might find relevant:

Bradley's website, math3ma.com
Her Twitter, Facebook, and Instagram accounts

The book she co-wrote, Topology: A Categorical Approach

Episode 61 - Yoon Ha Lee

10 december 2020 | 22 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, coming at you from the double hurricane part of 2020 today. I mean, I'm not near the Gulf Coast so it's it's not quite as relevant for my life, but that is the portion of the year we are in right now. I am one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And here's your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. It's just hot here. But you know, there have been, like, fire tornadoes, right, in California? This is all very on-brand for 2020. This year can’t end soon enough.

EL: Yeah, we say that. I feel like I've said that at the end of many previous years, and then it's not great.

Yoon Ha Lee: As a science fiction writer, I have to say never assume it's the worst. It can always get worse.

EL: Yes.

KK: Right, right, right.

EL: Yes. And that is our guest, Yoon Ha Lee. So yeah, would you like to introduce yourself, tell us a little bit about yourself, and maybe talk about your writing a little bit, how you got to writing from the degrees that you have in math.

YHL: So my name is Yoon Ha Lee. I'm from Houston, and I'm a science fiction and fantasy writer. I actually went to Cornell to get a degree in history, and then I realized that history majors starve on the street. So I switched to math, so that I could have an income and ended up not becoming a mathematician. My best-known books are probably the Machineries of Empire trilogy, which is Ninefox Gambit, Raven Stratagem and Revenant Gun. It's space opera, lots of ships blowing up everywhere. And then a kid's book, Dragon Pearl, which is out from Disney Hyperion in the Rick Riordan Presents series. And that one is also a space opera, because ships blowing up is just fun.

EL: Yeah, well, and that's funny. I think I just put together—I had seen the Rick Riordan publishing imprint before, and I just started reading Percy Jackson the other day. And so it's like, oh, that's who that guy is.

KK: And I think I might be the only one among us who is old enough to have seen the biggest space opera, Star Wars, in the theater in its first release.

YHL: Yeah, my parents let me see it on the television when I was six years old, and I was terrified at the point where Luke gets his hand cut off.

KK: That’s Empire.

YHL: I think the second one? I forget which movie it was, but he gets his hand cut off and I had nightmares for weeks. And I'm like, Mom and Dad, Why? Why? Why did you think this was an appropriate movie for a six-year-old? And then I got all the storybooks and I wanted the lightsaber and everything, so I guess it worked out.

KK: Of course, yeah. Well, my movie story—we’re getting off track, but it's it's a good movie story. So when I was six years old, in 1975, my parents thought it would be a good idea to take me to the drive-in to see Jaws. And I had nightmares for months that there was a shark living under my bed, a huge shart that was going to get me.

EL: My husband was born I think right around the time one of them was released. I don't remember which one now. But we were talking with one of his colleagues one time and figured out that on the day he was born, that colleague was going to see that movie, like, the day it came out.

KK: I’m going to guess it was I'm gonna guess it was Jedi. I don't know exactly how old you guys are. But that's that's my guess.

EL: That sounds right. Yeah, I'm not a big star wars person. But yeah, I guess I've always not been sure, like, “space opera.” The term is something that I feel like I know it when I see it. But I don't really know, like, how to describe it. Is it just—do you feel like a categorization of space opera is, like, ships blowing up?

YHL: Ships blowing up, generally bigger, larger-than-life characters, larger-than-life stakes, big galactic civilization types of things. It's basically the Star Wars genre.

EL: Yeah

KK: It works.

EL: yeah. And the Machinery of Empire—the reason that I invited you on here is because I just read Ninefox Gambit a few weeks ago and just thought, you know, this person sure uses a lot of math terms for a novel! So mathematicians might be especially interested in reading this one, it has shenanigans with calendar systems that are based on math and arithmetic and stuff. So yeah, that was fun. So you, in addition to getting a bachelor's degree in math, you got a master's in math education, right?

YHL: Yes, at Stanford. And I ended up not using it for very long. I was a teacher for, like, half a year before I left the profession.

EL: Okay, and was it just that your writing was taking off and you wanted to do that more? Were there other reasons?

YHL: A kid came along. That was the big reason. Yeah.

EL: Oh. Yeah. That definitely can take a lot of time.

KK: Ah yeah, just a little bit.

EL: Well, that's great. So what is your favorite theorem?

YHL: My favorite theorem is Cantor's diagonalization proof. And I discovered it actually in high school as a footnote in Roger Penrose’s The Emperor's New Mind. It was really just sort of a sidelight to the extremely complicated and hard-to-follow argument that he was making in that book on the nature of consciousness and quantum physics, which, as a high schooler, you know, it basically went over my head. But I was sitting there staring at this footnote and going “I don't understand this at all.” He said in the footnote that Cantor had proven that the real numbers, the set of real numbers, has a cardinality greater than the set of natural numbers. And of course, I was a high schooler. I hadn't had a lot of math background. So my understanding of these concepts was very, very shaky. But he said if you make a list of, you know—pretend that you have a list of all the real numbers and you put them, you know, 1, 2, 3, 4, you put them in correspondence with the natural numbers, and then you go down diagonally, first digit of the first number, second digit of the second number, third digit of the third number, and so on. And then you shift it by one. So if the numeral in that place is two, it becomes three, if it's nine, it becomes zero, and so on. So you can construct a number that is not on the list, even though your premise is that you have everything on the list. And I think this was the first time that I really understood what a proof by contradiction was. My math teachers had attempted very hard to get this concept into my head. And it just did not go through until I read that proof and meditated upon it. And it's funny, because I spent most of my life as a kid thinking that I hated math. And yet there I was in the library reading books about math, so I guess I didn't hate it as much as I thought I did.

EL: Yeah, I was thinking a high schooler reading that Penrose book is definitely—yeah, you had some natural curiosity about math, it sounds like.

KK: Yeah, I'm sort of sort of surprised that your high school teachers were trying to teach you proofs by contradiction. That's kind of interesting. I don't remember seeing any of that until I got to university.

YHL: I don't know that they got into depth about it. But this was at Seoul Foreign School, which was a private, international school in South Korea. And they tried to make the curriculum more advanced, with mixed results.

KK: Sure. It’s worth a shot.

EL: Yeah, and this, this is really one of those Greatest Hits. Like if you're putting together the like, record that you're going to send out or something, like, Math’s Greatest Hits with would include this diagonalization argument. It's so appealing. And we've had another guest select that too, Adriana Salerno a few months ago and yeah, just people. I think a lot of people who eventually do become mathematicians, this is one of those first moments where they feel like they really understand some some pretty high-concept math kind of stuff. So did you see this this proof later in school?

YHL: No. Ironically, most of what I was interested in doing when I did my undergraduate degree was abstract algebra. So I didn't even take a set theory course at all. But I knew it was sort of out there in the water, and I don't know, one of the things I loved about math and that led me to switch my major to math was the idea that there were these beautiful ideas and these beautiful arguments, and just sort of the elegance of it, which was very different from history, where—I love history, and I love all the battles and things, like the defenestration of Prague and all the exciting things happening. But you can't really prove things in history. Like you can't go back and run the siege of Stalingrad again, and see what happens differently.

KK: Maybe we could though, right? We have the computing power now. Maybe we could do that. This sounds like your next novel, right? So simulation of Stalingrad, and this time, the Nazis win or something? I don't know.

YHL: Oh no. I mean, science fiction writers totally do that. There's this whole strand of alternate history, science fiction or fantasy. Harry Turtledove is one author who, he likes to have the story where aliens invade during World War II and then the Nazis and the allies have to have to team up against the aliens kind of stories there. There is a set there is a readership for these things. Sure.

EL: So you use a lot of math concepts in your writing, your fiction writing. So have you ever tried to work in diagonalization, or this kind of idea, into any of your stories?

YHL: This one? No. I mean, occasionally, I remember writing a story in college, actually, called Counting the Shapes. And it was just everything in the kitchen sink, because I was taking point-set topology, and so I used it as a metaphor for a kind of magic that worked that way, and other ideas, like, I don't know, I had recently read James Gleick’s Chaos. So I was really interested in chaos theory and fractals. And I don't know that I was super systematic about it, and I sort of suspect that a real mathematician would look at it and poke holes. You know, I'm using this as a magic system, not as rigorous math, more as a metaphor, I guess, or flavor.

KK: Oh, but I mean, writers do that all the time, right? So I, I taught math and lit class with a friend of mine in languages a few years ago. And, you know, Borges, for example, you know, this sort of stuff is all over his work, these ideas of infinity and, and it's even embedded in Kafka and all this stuff, and it can be a wonderful way to to get your readers to think about something from a point of view they might not have thought of before.

YHL: Well, the interesting thing about Ninefox Gambit and the math terminology that I used for flavor is that 20 publishers turned the book down because they said it had too much math. And I my joke about this is that they saw the word diagonalization in the linear algebra matrix context, and they didn't know what that meant, and they ran away from it. Which was extremely discouraging when my agent at the time, Jennifer Jackson, and I were going out on submission with this book. And it's like, it's basically a space opera adventure where people blow each other up. You don't have to worry about the occasional math term. It's just there as flavor for the magic system. But a lot of people—I’m sure you have encountered the fact that a lot of people in the US have math phobia, and this really does affect the readership as well.

KK: Really?

EL: Yeah, that’s funny, because in some way, I mean, you definitely use the the math language to give a certain flavor to the system that this universe is in, but you could sub it out for, like, any Star Trek term,

YHL: Exactly.

EL: t’s just like, oh, yeah, you could put tricorders and dilithium crystals, or, you know, anything in to serve that that because you know, you're it's not a math textbook, no one's learning linear algebra from reading Ninefox Gambit.

YHL: No, exactly. I actually, when I was originally writing the book, like the rough draft, I had my abstract algebra textbooks out and ready to go. And I was going to construct sort of a game engine, a combat engine of how these battles were going to work in an abstract algebra sense. And my husband who, he's not afraid of math, he's actually a gravitational astrophysicist, and he's arguably better at math than I am. But he sat me down and said, “Yoon Ha, you can't do this. You're not going to have any readers because science fiction readers who want to read about big spaceships blowing each other up do not want to have to wade through a math textbook to get to the action.” And I mean, it turned out that he was absolutely correct. So I ended up not doing that and just using it as, you know, “the force,” except with math flavor.

KK: Linear algebra is the force. All right!

EL: That’s so interesting. I noticed on your website that you have a section for games. So do you also like to design games?

YHL: I do design games. And by design games, I mean tiny little interactive, interactive fiction text adventures or really small tabletop RPGs in the indy sense. You know, three page games for five people, no GM, that kind of thing. So I do enjoy doing that. And it is related to math, I think, but it's certainly not something that we learn to do in any of our math classes.

EL: Yeah, well, I mean, personally, I think it would be very cool. Have you have you written up this potential game, the abstract algebra game thing into an actual game? Or was that kind of abandoned on the editing floor while you were putting the book together?

YHL: It got abandoned on the editing floor. Also because it would have been a tremendous time suck. And, you know, it would have been a fun idea. But if I wasn't going to use it in a book, and it certainly wasn't going to be used in like a computer game or some something like that, there just didn't seem to be enough incentive to go ahead and do it.

EL: Yeah, probably the market of math mathematicians who read sci fi is, you know, not a tiny market but maybe not quite the demographic you're looking for. But I'm just imagining, like, hauling out the Sylow theorems to, like, explode someone’s battle cruiser or something. Just saying that, you know, if you were bored some time and wanted to sink a bunch of time into that.

YHL: if somebody else wrote it, I would definitely buy it and read it, I have to say.

KK: All right. The challenge is out there, everybody. Everybody should get on this.

EL: Yeah, very cool. Yep.

KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. So what pairs well, with Cantor's diagonalization argument?

YHL: Waffles.

KK: Waffles? Oh, well, yeah.

YHL: Because sort of that grid shape. I know, this is super visual. But the waffles I'm thinking of, my husband did his postdoc at Caltech, so we lived in Pasadena. And when we were there, there was this delightful Colombian hotdog place. And they also made the best waffles with berries and fruit and syrup and whipped cream. And those are the waffles I think of when I think of the diagonal slash proof.

KK: Right. And so the grid is actually fairly small. Is it one of those waffle makers?

YHL: Yeah.

KK: Yeah. Okay, so I have a Belgian waffle maker, and it's fine. It makes four at a time, but those holes are pretty big. Right? I'm thinking of, like, the small, Eggo style, right? You can put a lot of digits.

EL: You could also, like, I guess, maybe a berry is too big to fit in them, but I'm just thinking you can put different things in all of them, make sure no two waffles have the same arrangement of syrup and berries and cream.

KK: This is a good pairing. I'm into this one a lot.

YHL: I’m hungry now.

KK: Yeah.

EL: Yeah. I just had lunch, so for once I don't leave this ravenous. So would you like to let people know where they can find you online?

YHL: Online I’m at yoonhalee.com. I'm also on Twitter as @deuceofgears and also on Instagram as @deuceofgears.

KK: Deuce of gears. Is there a story there?

YHL: It’s the symbol of the crazy general in Ninefox Gambit. Okay. And also, because I'm Korean, there are five zillion other Yoon Ha Lees. So by the time I joined Twitter, all the obvious permutations of Yoon Ha Lee had already been taken, so I had to pick a different name.

EL: Yeah, and if I'm remembering correctly, there are sometimes cat pictures on your Twitter feed. Is that right?

YHL: Yes. So the thing that I post periodically to Twitter is that my Twitter feed is 90% cat pics by volume. There are people who, you know, they tweet about serious things, or politics, or so on, and these are very important, but I personally get stressed out really easily so I figure people could use an oasis of cheerful cat pictures.

EL: Yes, I just wanted to make sure our listeners have this vital information that if they are running low on cat pictures, this is a place they can go. It's definitely been an important part of my mental health to make sure to look at plenty of cat pictures during this—these stressful times as they say.

KK: Yeah, on Instagram, I follow a lot of bird watching accounts. So I just get a feed of birds all day. It's better for my mental health.

EL: Well maybe Yoon’s cat would like that,

KK: I suspect yes, that's right. That's right. Yep.

EL: Yeah, we were talking to a friend who said that they have some bird feeders outside, they just have indoor cats. And the cats will meow to get them to open the windows in the morning so they could watch the birds outside. It’s like, “Mom, turn on the TV.”

YHL: I tried putting on a YouTube video of birds, and my cat was just completely indifferent to the visuals. But she kept looking at the speaker where the bird sounds were coming from.

KK: Hmm.

EL: Interesting. I guess maybe hearing is like more of a dominant sense or something? Cats have pretty good vision, though, I think.

YHL: Yeah, I think she's just internalized that nothing interesting comes out of the moving pictures.

EL: Yeah. Well, thanks for joining us. I really enjoyed talking with you.

KK: This has been good.

YHL: It’s been an honor.

On this episode of My Favorite Theorem, we were happy to talk with Yoon Ha Lee, a sci-fi and fantasy writer with a math background, about his favorite theorem, Cantor's proof of the uncountability of the real numbers. Here are a few links to things we mentioned in the episode:
Yoon Ha Lee's website, Twitter account, and Instagram account

Our episode with Adriana Salerno, who also loves this theorem
Roger Penrose's book The Emperor's New Mind

James Gleick's book Chaos
Harry Turtledove

Episode 60 - Michael Barany

12 november 2020 | 41 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast for your quarantine life. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah.

KK: Yeah.

EL: How are you, Kevin?

KK: I'm okay. I had my—speaking of quarantines, I had my COVID swab test this morning.

EL: How was it?

KK: Well, you know, about as pleasant as it sounds. But yeah, I'm sure you've been to the pool and gotten water up your nose. That's what it feels like.

EL: Yeah.

KK: And then it's over. And it's no big deal. I should have the results within 48 hours. It’s part of the university's move to get everybody back to campus, although I don't expect to go back to the office in any serious way before August. But this is late May now for our listeners, who will probably be hearing this in December or something, right?

EL: Yeah. Who even knows? Time has no meaning.

KK: Hopefully this will all be irrelevant by the time our listeners hear this. [Editor’s note: lolsob.] We'll we'll have a vaccine and everything. It will be a brave new world and everything be fine.

EL: It’ll be a memory of that weird time early in year.

KK: That’s right. The before times. So anyway, today, we are pleased to welcome Michael Barany. Michael, why don’t you introduce yourself and let us know who you are and what's up.

Michael Barany: Hi. So I'm a historian of mathematics. I'm super excited to be on this podcast. I feel like I've been listening long enough that the Gainesville percussionists must be in grad school by now.

KK: No. One of them is my son, and he just finished his third year of college.

MB: Okay, yeah. So older than he was anyway.

EL: Yeah.

MB: Yeah, so I’m a historian of mathematics. I'm based at the University of Edinburgh, where I'm in a kind of interdisciplinary social science of science and technology department. So I get to teach students from all over the university how to think about what science means when you step back and look at the people involved and how they relate to society, how ideas matter, how technology's changed the world, all that fun stuff that gets people to really rethink their place in the world and the kind of things they do with their science.

KK: That’s very cool.

EL: And I know some people who are historians of math will get a degree through a math department and some get it through a history department, I assume. And which are you? I always wonder what the benefits are of each approach.

MB: Yeah, that's great. History of mathematics is a really strange field. It’s actually, as a field, a lot older than history of science as a field, and even older than history as a profession.

EL: Huh.

MB: So history of mathematics started as a branch of mathematics in the early modern period. So we're talking like the 1500s, 1600s. There are always debates about what you classify as this or that. And it started as a way of trying to understand how mathematical theories came about, how they naturally fit together. The idea was that if you understood how mathematical theories emerged, you could come up with better mathematical theories, and you could understand the sort of natural order of numbers and the universe and everything else that you want to understand with mathematics. And then more toward the 19th and the 20th century, there are all these different variations of history of mathematics that branched out of fields like history and philosophy, and philosophy of science and history of science. So my undergrad training was in mathematics. My PhD is from a history department, but from a history of science program in that department. But it's possible to get a PhD in history of mathematics from a mathematics department, it's possible to sort of straddle between different departments. And it makes it a really rich and interesting field. Mathematics education departments or groups sometimes give PhDs in history of mathematics. And they really use the history for different purposes. So if your goal is to make mathematics better, you're taking the perspective of someone doing it from a mathematics department. If your goal is to become a better educator, then you can use history for that in a math education context. I tend to do history as a way of understanding how things fit together in the past and trying to make sense of social values and social structures and ideologies and ideas and how those fit together. And that's the approach that that you come at from a history or history of science perspective.

KK: Very cool. And How did you end up in Edinburgh of all places?

MB: Well, so the academic job market is bad enough in mathematics, right, but in history of mathematics, in a good year, there may be two to three openings in history of science jobs in general. So that's the cynical answer. The more idealistic answer is Edinburgh has this really important place in the sociology of science. In the 1970 s and 80s especially, there was this group of kind of radical sociologists in at the University of Edinburgh who sat down. It was called the Edinburgh School of the sociology of scientific knowledge, which is known for this sort of extreme relativism and constructivism view of how politics and ideology shape scientific knowledge. And I did a master's degree in that department many years later, in 2009-2010, sort of getting my feet wet and starting to learn that discipline. And that approach has been really formative for me and my scholarship. And so it was an incredible stroke of luck that they just happened to have an opening in my field while I was on the market. And I was even even more lucky to have the chance to go there.

KK: Wow, that's great. I’ve always wanted to go there. I've never been to Edinburgh,

MB: It’s the most beautiful city in the world.

KK: Yeah, it looks great. All right, well, being a historian of math, you must know a lot of theorems. So the question is, do you actually have a favorite one? And if so, what is it?

MB: So my favorite theorem is more of a definition. But I guess the theorem is that the definition works.

KK: Okay, great.

EL: That works.

MB: Which, actually—saying what it means for a definition to work is actually a really hard problem, both historically and mathematically. So it's interesting in that regard. Ao the definition is the definition of the derivative of a distribution.

KK: Okay.

MB: So distributions, as you’ll recall from, from analysis—I guess, grad analysis I is usually when you meet them.

EL: Yeah, I think it wasn't until grad school for me at least.

KK: I don't know if I've ever met them, really.

MB: So distributions were invented in 1945, more or less. And in the early years, actually, people were saying you could teach this as a replacement for your basic calculus. So the idea was, this would be something that even beginning college students or even high school students would be learning. So it's interesting to see how they have people pitched that the level of a theory or the the relevant audience, and that's part of the story, too. But in earlier stages of one's calculus education, you learn that there are functions that are integrable but not continuous; continuous but not differentiable; differentiable but not continuously differentiable, and so on. And so a big problem is how do you know something's differentiable when you're studying a differential equation or trying to prove some theorem that involves derivatives. And distributions were the kind of magic wand that was invented in the middle of the 20th century to say that's not actually a problem. Basically, if you pretend everything's differentiable, then all the math works out. And when it really is differentiable, you get the correct differentiable answer, and when it's not, then you get another answer that's still mathematically meaningful. But it's sort of your magic passphrase to be able to ignore all of those problems.

So a distribution is this replacement for a function. Where functions have these sort of different degrees of differentiability, distributions are always differentiable and they always have antiderivatives, just like functions do, but every distribution can be differentiated ad nauseam for whatever differential equation you want to do. And the way you do that is through this definition—my favorite definition/theorem—which is you use integration by parts. So that's a technique you use in calculus class, too, as a sort of trick for resolving complicated integrals. And distributions actually don't tend to look at the things that make the calculus problems challenging or interesting, depending on what kind of student you are, or what kind of teacher you are. So you set them up in a way where you don't have to worry about boundary conditions, you don't have to worry about what the antiderivative things are, because you're working with things where you already know what the antiderivative is. And the definition of distribution uses this fact from integration by parts that you essentially move the derivative from one function to another. So we don't have an exact way of saying functionally what the derivative of a distribution is. You can still say if you multiply it by a function that's super-smooth and over a bounded domain—so you don't have any boundary conditions to worry about, and so you always know how to differentiate that—if you multiply that by a distribution, and take the integral, then if you want to take the derivative of that distribution, integration by parts says you can instead throw in a minus sign and take the derivative of that smooth function instead. And so using that kind of trick, of moving the derivative onto something that is always differentiable, you can calculate the effect of differentiating a distribution without ever having to worry about, say, what the values of of that distribution are after you’ve taken the derivative, because distributions are often things that don't have sort of concrete values in the way that we expect functions to have.

EL: And I hope this question isn't very silly. But when you think about integration by parts—you know, if you took calculus at some point and learned this, there's the UV, and then there's the minus the integral of something else. And so for this, we just choose a function that would be zero on the boundary, and that would get rid of that UV term. Is that right?

MB: Exactly? Yeah. So the definition of distribution sets up this whole space of really nice smooth functions. All of them eventually go to zero, and because you're always integrating over the entire domain, and it's always zero when you go far enough out into the domain, those boundary terms with that UV in the beginning just completely disappear, and you're just left with the negative integral, and then with the derivative flopped over.

EL: All right, great. So if anyone was worried about where their UV went, that's where it went. It was zero. Don't worry. Everything's okay. Yeah. Okay. So what is good about this? Or what do you like about this?

MB: Yeah. So I think this is a really interesting definition from a lot of different perspectives. One thing that I've been trying to understand in my research about the history of mathematics is what it means for mathematics to become a global discipline in the 20th century, so to have people around the world working on the same mathematical theory and contributing to the same research program. And this definition is really helped me understand what that even means and how to understand and analyze that historically. So we think, well, you know, a mathematical theory or a mathematical idea is the same wherever you look at it, and whoever's doing it. As long as they can manipulate the definitions or prove the theorem, it shouldn't matter where they are. But if you look historically, at actual mathematicians doing actual mathematics, where they are makes a huge difference in terms of what methods they're comfortable with, how they understand concepts, how they explain things to each other, how they make sense of new techniques. I mean, learning a new mathematics technique is actually really hard in a lot of cases. And so the question is, how do you form enough of an understanding to be able to work with someone who you can't go and have a conversation with over tea the next day to sort of work out your problems? And the answer is, basically, you use things like this definition and take something you're really comfortable with—integration by parts—and give it a new meaning. And by taking old meanings and reconfiguring them and relating them to other meanings, you make it possible for everyone to have their own sorts of mathematical universes where they're building up theories, but to interact in a way where they can all sensibly talk to each other and develop new ideas and share new ideas. So that's one of the things that that's really exciting about that the definition to me.

One of the other things is sort of how do you know what the significance of the definition is? I mean, a lot of people early on said, isn't this just like a pun? Isn't this just wordplay? Quite early on, when Schwartz was sharing this definition, and some people were getting really excited about it. Some people said, well, you know, it's a cool idea. But isn't this just basically integration by parts? What's new? What's interesting about this? And the history really shows this debate, almost, between people with different kinds of values and philosophies and goals for mathematics, for mathematics education, for the relationship between pure and applied mathematics, where they take different ideas of what's really going on with this definition. Is it something that's complex and difficult and profound and important in that way, or is it something that is utterly trivial and simple, and therefore really useful to people who may be, say, electrical engineers who are trying to work with the Heaviside calculus, and need some sort of magic way to make that all add up? And what made distributions and this definition really powerful is it could be these multiple things to multiple people. So you can have mathematicians in Poland, or in Manchester, or in or in Argentina come to these very, almost diametrically opposed views of what it is that's significant or challenging or easy about distributions, and they can all agree to talk to each other and agree that it's worth sharing their theories and inviting them to conferences, and reading their publications, and they can somehow all make a community out of these different understandings.

KK: I’ve never thought about the sociological aspects in that way. That's really interesting. So the theorem that basically says that this definition is a good one. Is that a difficult theorem to prove?

MB: So there are a lot of different parts. It’s not—I guess it doesn't even boil down to one statement.

KK: Yeah, sure. Yeah, that makes sense. Yeah.

MB: So there's the aspect that when you're dealing with a function, but dealing with it using the distributions definition, that anything you do is not going to ruin what's good about it being a function. So anything you do with a distribution, if you could have done it as though it were a regular function, you get the same answer. So that's one aspect of the theorem that sort of establishes this definition. Another aspect is that distributions are, in some sense, the smallest class of objects that includes functions where everything that is a normal function can be indefinitely differentiated. So that's one way of arguing that distributions are sort of the best generalization of functions, and this competition—I mean, there are a lot of different competing notions, or competing ideas for how you can solve this problem of differentiating functions that were circulating in the 1930s and 1940s. And distributions won this competing scene, in part by the aspects of the theorems about the definition that show it’s sort of the most economical, the simplest, smallest, the best in that sense. And then you have all the usual theorems of functional analysis, like everything converges as you expect it to; if you start with something that's integrable, you're not going to lose interpretability, in some sense.

EL: So this might be a little bit of a tangent, and we can definitely decide not to go down this path. But to make this really concrete—so when I think of a distribution, the example I think of—it’s been a while since I've thought of distributions actually, is the Dirac delta function. I naturally just call it a function, but it is really a distribution. And so this is a thing that, I always think of it, it's something that you can't really define what its value is, but it has a convenient property that if you integrate it, you get 1. Like, its area is 1 even though it's supported on only one point, and it is infinitely tall. And so zero times infinity, we want it to be 1 right here.

MB: And magically it turns out to be 1.

EL: Yeah. And basically, if you decide that this function, this distribution, has this property, then things work out, and it's great. Was that before or after Schwartz? Did this definition—was this kind of grandfathered into being a distribution? Or was it the inspiration?

MB: I love how you put that. Yeah. So this, this phrase that you said at the beginning, we call it a function, but it's really a distribution. I mean, that's evidence of Schwartz’s success, right? The idea that what it really is, what it fundamentally is, is a distribution rather than a function, that's the result of this really sort of deliberate—I mean, it's not it's not an exaggeration to call it propaganda in the second half of the 1940s by people like Laurent Schwartz and Marston Morse and Marshall Stone and Harald Bohr and all of these far-traveling advocates for the theory—to say, you think you've been working with functions, you think you've been working with measures, you think you've been working with operator calculus if you're an electrical engineer, for instance. Or you think you've been working with bra and ket, with Dirac calculus for quantum mechanics, but what you've really been doing ultimately, deep down without even knowing it, is working with distributions. And their ability to make that argument was part of their way of justifying why distributions were important. So people who had no problem just doing the math they were doing with whatever kind of language they were doing, all of a sudden, these advocates for distribution theory were able to make it a problem that they were doing this without having the kind of conceptual apparatus that distributions provided them. And so they were both creating a problem for old methods and then simultaneously solving it by giving them this distribution framework.

So, they did this to the Heaviside calculus, which is about 50 years older than distributions. They did this to the Dirac calculus, where the Dirac function comes from, which comes out of the 1920s and 30s. They did this to principal value calculus, which is also an interwar concept in analysis. Even among Schwartz's contemporaries, there were things like de Rham currents, which were—had Schwartz not come along, we would all be saying the Dirac function is really a de Rham current rather than a Schwartz distribution. But then there were even things that came after distributions, or sort of simultaneously and after, that Schwartz was able to successfully claim. Like there was this whole school of functional analysis and operator theory coming out of Poland associated with Jan Mikusiński. Where Schwartz was—because he was able to get this international profile so much more quickly and effectively—he was able to say all of this really clever research and theorems that Mikusiński is coming up with, that's a nice example of distribution theory, even though Mikusiński would have never put that in those terms. So a huge part of this history is how they're able to use these different views of what a distribution really is to sort of claim territory and grandfather things in and also sort of grandchild things, or adopt things into the theory and make this thing seem much bigger than the actual body of research that people who considered themselves distribution theorists themselves were doing.

EL: Okay. And so I think we also wanted to talk a little bit about—you mentioned in your email to us, I hope I'm getting this I'm not getting this confused with anything—how this theory goes with the history of the Fields Medal.

MB: Oh, exactly. Yeah. So this was a really surprising discovery, actually, in my research. I didn't set out—the Fields Medal kind of became one thing, one little bit of evidence that Schwartz was a big deal. I never expected in my research to come across some evidence that really changed how I understood what the Fields Medal historically meant. And this was just a case of stumbling into these really shocking documents, and then having built up all of this historical context to see what their historical implications were. So Schwartz was part of the second ever class of Fields Medalists in 1950. The first class was in 1936, then there's World War II, and then they sort of restart the International Congresses of Mathematics after the war. And Schwartz is selected as part of that second class. The main reason he's part of that class is because the chair of that committee is Harald Bohr, who is the younger brother of Niels Bohr. Actually, in the early 1900s, Harald Bohr was the more famous Bohr because he was a star of the Danish Olympic soccer team.

KK: Oh!

EL: Wow!

MB: He was a striker. His PhD defense had many, many, many more soccer fans that mathematicians. He was this minor Danish celebrity. And he went on to be a quite respectable mathematician. He had his mathematics institute alongside his brother's physics institute in Copenhagen. And during the interwar period especially, he established himself as this safe haven for internationally-minded mathematics in this period of immensely divisive conflict among different national communities. And because he kind of had that role as this respected figure known for internationalism, he was selected by the Americans who organized the 1950 Congress at Harvard to chair the Fields Medal committee. And Bohr, shortly before being appointed to that committee, had encountered Schwartz in a conference that was sponsored by the Rockefeller Foundation and took place in Nancy in France, and he was just totally blown away by this charming, charismatic young Frenchman with this cool-sounding new theory that seemed like it could unite pure and applied mathematicians, that could be attractive to mathematicians all over the world. And so Bohr basically makes it his mission between 1947 and 1950 to tell the whole world about distributions. So he goes to the US and to Canada, and he writes letters all around the world, he shares it with all his friends. And when he gets selected to chair this committee, what you see him constantly doing in the committee correspondence is telling all of his colleagues on the committee what an exciting future of mathematics Schwartz was going to be.

So the problem is, then sort of the question is, what is the Fields Medal supposed to be for? And they didn't really have a very clear definition of what are the qualifications for the medal. There was a kind of vague guidance that Fields left before he died. The medal was created after John Charles Fields’ death. And there was a lot of ambiguity over how to interpret that. So the committee basically had to decide, is this an award for the top mathematicians? Is this award an award for an up-and-coming mathematician? How should age play a factor? Should we only do it for work that was done since the last medal was awarded? A long time to consider there, so that didn't really narrow it down very much in in their case. And they go through this whole debate over what kinds of values they should apply to making this selection. And ultimately, what I was able to see in these letters, which were not saved by the International Mathematical Union, which hadn't even been formed at the time, they were kind of accidentally set aside by a secretary in the Harvard mathematics department. So they weren't meant to be saved. They just were in this unmarked file. And what those letters show is that Bohr basically constructs this idea of what the metal is supposed to be for in a strategic way to allow Schwartz to win. So there's this question, there's this kind of obvious pool of candidates, of outstanding early- to mid-career mathematicians, including people like Oscar Zariski and André Weil, and Schwartz's eventual co-medalist, Atle Selberg. And they are debating the merits of all of these different candidates, and basically, Bohr selects an idea of what the Fields Medal is for, to be prestigious enough to justify giving it to this exciting young French mathematician, but not so prestigious that he would have to give it to André Weil instead, who everyone agreed was a much better mathematician than Schwartz, and much more accomplished and much more successful and very close in age. He was about five years older than Schwartz.

KK: He never won the Fields Medal.

MB: And he never won the Fields Medal, right. And so what you see in the letters from the early years of the Fields Medal is actually this deliberate decision, not just by the 1950 committee, but I was also able to uncover letters for the 1958 Committee, where they consider whether the award should be the very best young mathematicians, and they deliberately decide in both cases that it shouldn't be, that that would be a mistake, that that would be a misuse of the award. Instead, they should give it to a young mathematician, but not a young mathematician that was already so accomplished that they didn't need a leg up.

EL: Right.

MB: And that was my really surprising discovery in the archives, that it was never meant to crown someone who was already accomplished, and in fact, being accomplished could disqualify you. So Friedrich Hirzebruch in 1958, everyone agreed was the most exciting mathematician. He was in his early 30s, sort of a very close comparison to like someone like Peter Scholze today. So already a full professor at a very young age, with a widely-recognized major breakthrough. And they considered Hirzebruch, and they said, No, he’s too accomplished. He doesn't need this medal. We should give it to René Thom or someone like that.

EL: Yeah. And, of course, people like me, who only were aware of the Fields Medal once they started grad school in math—I wasn't particularly aware of anything before that—Think of it as the very best mathematicians under 40 because it has sort of morphed into that over the intervening decades.

MB: Yeah. And one of the cool side effects is now you can now put an asterisk next to—Jean-Pierre Serre is known to brag about being the youngest-ever fields medalist. But the asterisk is that he won in a period when it was still a disqualification to be too accomplished at a young age.

KK: Yeah, but he still won.

MB: He did still win. He’s still a very important mathematician.

KK: You sort of couldn’t deny Serre, right?

MB: Well, they denied Weil, right?

KK: They did. But I think Serre is probably still—Anyway, we can argue about— we should have a ranking of best mathematicians of the ‘50s, right?

EL: I mean, yeah, because ranking mathematicians is so possible to do because it’s a well-ordered set.

KK: That’s right.

EL: Obviously in any field of life, there's no way to well-order people. I shouldn't say any field. I guess you can know how fast people can run some number of meters under certain conditions or something. But in general, especially in creative fields, it's sort of impossible to do. And so how do you choose?

MB: That’s what I love about studying the sociology of science and technology, is that you get these tools for saying—you know, even in fields like running, we think of sprinting as this thing where everyone has a time and that's how fast they are. But look at all of the stuff the International Olympic Committee has to do for anti-doping and regulating what shoes you can wear, like there are all of these different things that affect how fast you are that have to be really debated and controlled. They're kind of ultimately arbitrary. So even in cases like that, you know, it seems sort of more rankable than mathematics or art or something, and you can tell a great sprinter from someone like me who can barely run 100 meters. But at the same time, there are all of these different social and technical decisions that are so interrelated that even things that seem super objective and contestable end up being much more socially determined.

EL: Yeah.

KK: Yeah. All right. So part two of this podcast is you have to pair your theorem with something, or your definition or whatever we're going to call it your distribution, whatever it is.

EL: Yeah. If you treat it as a distribution, it’ll work fine.

KK: That’s right.

MB: Exactly.

KK: So what have you chosen to pair with distributions?

MB: So what I thought I would pair distributions with is a knock-knock jokes.

KK: Okay.

MB: So I did a little bit of research before coming on here, and I basically found there are no good math knock-knock jokes. I mean, someone please prove me wrong, like tweet at me. And yeah, tell me tell me.

KK: Are there good knock knock jokes, period?

EL: Oh, definitely.

MB: Yeah. So I did come up with one that sort of at least picks up on some of the historical themes. So Knock, knock.

KK and EL: Who’s there?

MB: Harold.

KK and EL: Harold who?

MB: Harold is the concept of a function anyway?

That's the best I could do.

EL: Okay.

MB: So why knock-knock jokes? They involve puns. So you're talking about shifting the meaning of something to come up with something new. They're dialogical: there’s a sort of fundamental interactive element. They sort of make communities. So sharing a knock-knock joke, getting a knock-knock joke, finding it funny or groan-inducing, tells you who your friends are, and who shares your sense of humor. And yeah, they fundamentally use this aspect of wordplay to to make something new and to make something social. And that's exactly what the theory of distributions does and what that definition does, just sort of expand your thinking. And they're also sort of seen as kind of elementary, or basic. It's kind of like a kid's joke.

EL: Right.

MB: It’s this question of distributions as this fundamental theory, your basic underlying theory. So I think it sort of brings together all of those aspects that I like about the definition.

KK: You thought hard about this. This is a really thoughtful, excellent pairing. I like this.

EL: Yeah, I like it. I'm trying to figure out what is the analogy to my favorite knock-knock joke, which is the banana and orange one, right, which is classic.

MB: It’s the only one I use in real life.

KK: Sure.

EL: It’s a great one!

KK: Yeah.

EL: Fantastic. But, like, what distribution is this knock-knock joke?

KK: The Dirac function, right? Excuse me, the Dirac distribution.

MB: Yeah. Aren't you glad I didn't say the Dirac distribution? Yeah, no, it's the only one you actually use all the time. Yeah, the Dirac distribution, or there's that theorem that any partial differential equation can be resolved as the sum of derivatives of these elementary distributions. That's your go-to ubiquitous, uses a pun, but uses in a way that kind of makes sense and is kind of groan-inducing, but also you just love to go back and to use it over and over and over again.

KK: Right.

EL: Nice.

KK: I think back in the 70s—dating myself here—I had a book of knock-knock jokes, and it actually had the banana and orange one in it. I mean, it's like, this is how basic of a book this was. So I might be ragging on knock-knock jokes, but of course, I had a whole book of them. So anyway.

EL: Oh, they're great. And especially when a child tells you one.

KK: That’s right. That’s what they’re there for.

MB: The best is when you have a child who hasn't heard the knock-knock joke you’ve heard 10 million times, and you get to be the person to share the groan-inducing pun with the child. I mean, that's how I imagine Schwartz going to Montevideo and explaining distribution theory, like the experience of sharing this pun and having them go “Ohhh” and slapping their forehead. There's this cultural resonance, to introduce something that you immediately grasp. And yeah, that's a really special experience.

KK: Yeah.

EL: So at the end of the show, we like to invite our guests to plug things, and I'll actually plug a couple of your things because we've sort of mentioned them already. You had a really nice article in Nature. I don't remember, it was a couple years ago—

MB: 2018.

EL: —about this history of the Fields Medal, focusing on Olga Ladyzhenskaya, who was on the short list in ’58 and would have been the first woman to get the Fields Medal if she had gotten it, but it was really interesting because it touches on these things about how the Fields Medal became what it is thought of now and how they made that decision at that time. So go read that. And you also have an article about this distribution stuff that I am completely now blanking on the title of, but it has the word “wordplay” in it, and you probably know the title.

MB: There’s “Integration by Parts” as the title.

EL: Okay.

MB: And then there's a long subtitle. So this is the thing any historian does, is they have some kind of punny title and then this long subtitle. I think one of the reasons I empathize with the theory of distributions is, like, this is how I think as a historian. I come up with a pun, and then I work out how all of the things connect together afterwards. You see that in all of my titles, basically, and papers, That's not that's on my website, mbarany.com, and the show notes.

EL: Yeah, we'll put those in the show notes. We'll link to your website and Twitter in the show notes. And yeah, anything else you want to mention?

MB: Yeah, so if you want all of this math and sociology and politics and stuff about academia and the values of mathematics, then my main Twitter account at @mbarany is the one to follow. If you just want sort of parodies and irreverent observations about math history, then @mathhistfacts is my parody account that I started in August, but the key to that is that behind every thing that looks like it's just a silly joke is actually something quite subtle about historical interpretation. And I always leave that as an exercise to the reader. But I do try to—this was my response to, you know, St. Andrews has this MacTutor archive of biographies of mathematicians that has hundreds and hundreds of mathematicians, these sort of capsule biographies. And they have these little examples, or these little summaries, like so-and-so died on this day and contributed to this theory, and it’s just kind of morbid to celebrate them for when they died. But then even the one that makes the rounds every year on Galileo's birthday, so Galileo is actually one of the—not Galileo, Galois. Galois is one of the few people who actually has an interesting death date, whose death is historically significant, and there's a Twitter account that tweets based on on these little biographical snippets, and does it for his birthday rather than his death day and then says, like, “Galois made fundamental contributions to Galois theory.” So this was my response to that account, those tweeting from these biographical snippets saying there's there's more to history than just when people died and what theory named after them they contributed to, and tried to do something a bit more creative with that.

EL: Yeah, that is fun. I felt slightly personally attacked because I did just publish a math calendar that has a bunch of mathematician’s birthdays on it, but I did choose to only do like a page about a mathematician on their birthday rather than their death day because it just seemed a lot less morbid.

MB: Very sensible. There are some mathematicians with interesting death days. So Galois, Cardano. Cardano used mathematics to predict his death day, so it's speculated that he also used some poison to make sure he got his answer right.

EL: Yikes! That’s a bit rough.

MB: But yeah, there are a few mathematically interesting death days. But yeah, I mean, birthdays are okay, I guess. I'm not super into mathematical birthdays anyway, but better than death days.

EL: Yeah. I mean, when you make a calendar, you've got to put it on some day. And it's weird to put it on not-their-birthday. But yeah, that's a fun account. So yeah, this was great. Thanks for joining us, Michael.

MB: Thanks. This was super fun.

On this episode of My Favorite Theorem, we were happy to talk with University of Edinburgh math historian Michael Barany. He told us about his favorite definition in mathematics: distributions. Here are some links you might find interesting.

Barany’s website and Twitter account
His article “Integration by Parts: Wordplay, Abuses of Language, and Modern Mathematical Theory on the Move” about the notion of the distribution

His Nature article about the history of the Fields Medal
Distributions in mathematics

The Dirac delta function (er, distribution?)

The Danish national team profile page of mathematician and footballer Harald Bohr

Episode 59 - Daniel Litt

8 oktober 2020 | 27 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. I have left the county two times since this all happened. We don't have a car, so when I leave my home, it is either on feet or bicycle, which is your feet moving in a different way. But I have biked out of our county now into two different other counties. So it's very exciting.

KK: Fantastic. Well, I do have a car. I bought gas yesterday for the first time since May 26, I think. And yesterday was June 30.

EL: Yes.

KK: And I've gotten two haircuts, but it looks like you've gotten none.

EL: Yes. That’s correct. I’m probably the shaggiest. I've been in a while. My I normally this time of year is buzzcut city, which I do at home anyway. But I don't know.

KK: I will say I’m letting it get a little longer actually. I know I said I got a haircut, but you know, Ellen likes it longer somehow. So here we go. This is where we are. My son's been home for three months, and we haven't killed each other. It's all right.

EL: Great. Yeah, everything's doing as well as can be expected, I suppose. If you're listening to this in the future, and somehow, everything is under control by the time we publish this, which seems unlikely, we are recording this during the 2020 COVID-19 pandemic, right, which—I guess it still stays COVID-19 even though it's 2020 now, to represent the way time has not moved forward.

KK: Right. Time has no meaning. And you know, Florida now is of course becoming a real hotspot, and cases are spiking. And I'm just staying home and, and I have four brands of gin, so I'm okay.

EL: Yeah. Anyway!

KK: Anyway, let's talk math. So we're pleased today to welcome Daniel Litt. Daniel, would you please introduce yourself?

Daniel Litt: Hey, thank you so much. It's really nice to be here. I'm Daniel Litt. I'm an assistant professor at the University of Georgia in Athens, Georgia, likewise, a COVID-19 hotspot. I also have not gotten gas, but I think I've beat your record, Kevin. I haven't gotten gas since the pandemic began.

KK: Wow. That’s pretty remarkable.

DL: I’ve driven, maybe the farthest away I've driven from home is about a 15-minute drive, but those are few and far between.

KK: Sure.

DL: So yeah, I'm really excited to be here and talk about math with both of you.

KK: Cool. All right. So I mean, this podcast is—actually, let’s talk about you first. So you just moved to Athens, correct?

DL: I started a year ago.

KK: A year ago, okay. But you just bought your house.

DL: That’s right. Yeah. So I actually live in northeast Atlanta, because my wife works at the CDC, which is a pretty cool place to work right now.

KK: Oh!

EL: Oh wow.

KK: All right. Is she an epidemiologist?

DL: She does evaluation science, so at least part of what she was doing was seeing how the CDC’s interventions and deployers, how effective they were being help them to understand that.

KK: Very cool. Well, now it would be an interesting time to work there. I'm sure it's always interesting, but especially now. Yeah. All right. Cool. All right. So this podcast is called my favorite theorem. And you've told us what it is, but we can't wait for you to tell our listeners. So what is your favorite theorem?

DL: Yeah, so my favorite theorem is Dirichlet’s theorem on primes in arithmetic progressions. So maybe let me explain what that says.

KK: Please do.

EL: Yes, that would be great.

DL: Yeah. So a prime number is a positive integer, like 1, 2, 3, 4, etc, which is only divisible by one and by itself. So 2 is a prime, 3 is a prime, 5 is a prime, 7, 11, etc. Twelve is not a prime because it's 3 times 4. And part of what Dirichlet’s theorem on primes in arithmetic progressions tries to answer, part of the question it answered, is how are primes distributed? So there is a general principle of mathematics that says that if you have a bunch of objects, they're usually distributed in as random a way as possible. And Dirichlet’s theorem is one way of capturing that for primes. So it says if you look at an arithmetic progressions—that’s, like 2, 5, 8, 11, 14, etc. So there I started at 2 and I increased by 3 every time. Another example would be 3, 6,9, 12, 15, etc—there I started at 3 and increased by 3 every time. So Dirichlet’s theorem says that if you have one of those arithmetic progressions, and it's possible for infinitely many primes to show up in it, then they do. So let me give you an example. So for 3, 6, 9, 12, etc, all of those numbers are divisible by 3. So it's only possible for one prime to show up there, namely 3.

EL: Right.

DL: But if you have an arithmetic progression, so a bunch of numbers which differ by all the same amount, and they're not all divisible by some single number, then Dirichlet’s theorem tells you that there are infinitely many primes in that sequence. So for example, in the sequence 2, 5, 8, 11, etc, there are infinitely many primes, 5 and 11 being the first two [editor’s note: the first primes after 2. But it’s just odd for an even number to be prime]. And it tells you something about the distribution of those primes, which maybe I won't get into, but just their bare existence is really an amazing theorem and incredible feat of mathematics.

EL: So this theorem, I guess, for some of our listeners, and for me, it probably sort of reminds them in some ways of like twin primes or something, these other questions about distributions of primes. Of course, twin primes, you don't need a whole arithmetic progression, you just need two of them. That would be primes that are separated by two, which other than 2 and 3 is the smallest gap that primes can have. And, of course, twin primes is not solved yet.

DL: Yeah, we don’t know that there are infinitely many.

EL: Yeah, people think there are but you know, who knows? We might have found the last one already. I guess that's unlikely. But Dirichlet was proved a long time ago. So can you give me a sense for why this is a lot easier than twin primes?

DL: Yeah, so part of the reason, I think, is that twin primes are much sparser than primes in any given arithmetic progression. So just to give you an example, if you have a bunch of numbers, one way of measuring how big they are is you could take the sum of 1 over those numbers. So for example, the sum of 1/n, where n ranges over all positive integers, diverges; that sum goes to infinity. And the same is actually true for the primes in any fixed arithmetic progression. So if you take all the primes in the sequence 2, 5, 8, 11, etc, and take the sum of one over them, that goes to infinity, since there's a lot of them. On the other hand, we know that if you do the same thing for twin primes, that sum converges to a finite number. And that number is pretty small, actually. We know, up to quite a lot of accuracy, what it looks like. And that already tells you that they're sort of hard to find. And if you have things that are hard to find, it's going to be harder to show that there are infinitely many of them. I mention this sum of reciprocals point of view because it's actually crucial to the way Dirichlet’s theorem is proven. So when you prove Dirichlet’s theorem, it's one of the these really amazing examples where you have a theorem that's about pure algebra. And you end up proving it using analysis. So in this case, the theory of Dirichlet L-functions. And understanding that sum of reciprocals is kind of key to understanding the analytic behavior of some of these L-functions, or at least it’s very closely related.

KK: So I didn't know that result about the reciprocals of the twin primes converging. So even though we don't know that there are infinitely many, somehow…

DL: Yeah, in fact, if there are finitely many then definitely that sum would converge, right?

KK: Yeah, right. That’s—and we even know an estimate of what the answer is? Okay. That’s fascinating.

DL: Yeah, and what you have to do to prove that is show that these primes are sufficiently sparse. And then and then you win. EL: So once again, I am super not a number theorist. So I'm just going to bumble my way in here. But to me, if I'm trying to show that something diverges, I show that it's sort of like 1/n, and if it converges, it's sort of like 1/n2 or, or worse, or better, or however, you want to morally rank these things. So I guess I could imagine it not being that hard to show that twin primes are sort of bounded by n2, or you're like bounded by 1/n2 squared, the reciprocals of that, would that be a way to do this? Or am I totally off?

DL: It’s something like that. You want to show they're very spread out. Yeah, with primes, I do want to mention, so you mentioned like you want to say something like between 1/n or 1/n2. So primes are much, much rarer than integers, right? So it's really somewhere between those two.

EL: Yeah.

DL: So for example, understanding the growth rate of those numbers—the growth rate of the primes and the growth rate of the primes in a given arithmetic progression—is pretty hard. Like that's the prime number theorem, it’s one of the biggest accomplishments of 19th-century mathematics.

KK: Right. Does that help you prove that, though? Maybe it does, right? Maybe not?

DL: Yeah, so proving that the sum of the reciprocals of the primes diverges is much, much easier than the prime number theorem. And as you can prove that in, like, a page or page and a half or something. But it's very closely related to the key input of the prime number theorem, which is that the Riemann zeta function, the subject of the Riemann hypothesis, has a pole at s=1.

KK: All right. Okay. So what's so compelling about this theorem for you?

DL: Yeah, so what I love about it is that it's maybe one of the earliest places, aside from the prime number theorem itself, where you see some really deep interactions between algebra and complex analysis. So the tools you bring in are these Dirichlet L-functions, which are kind of generalizations of the Riemann zeta function. And they're really mysterious and awesome objects. But for me, what I find really exciting about it is that it's like the classic oldie. And people have been kind of remaking it over and over again for the last, like, century. So there's now tons of different versions of the Dirichlet theorem on primes in arithmetic progressions in all kinds of different settings. So here's an example. In geometry, you have a Riemannian manifold, which is kind of a manifold with a notion of distance on it. There's a version of Dirichlet’s theorem for loops in a Riemannian manifold, the first cases of which are maybe do that Peter Sarnak in his thesis. There are versions for over function fields. So I'm not going to be precise about what that means, but if you have some kind of geometric object that's kind of like the integers, you can understand it well and understand the behavior of primes and that kind of object, and how they behave in something analogous to an arithmetic progression. There's something called the Chebotarev density theorem, which tells you if you have a polynomial, and you take the remainder of that polynomial when you divide by a prime, how does its factorization behave as you vary the prime? So there's all kinds of versions of it, and it's a really exciting and cool sort of theme in mathematics.

EL: So kind of getting back to the the more tangible number theory thing—which I guess it's kind of funny that we think of numbers as more tangible when they're sort of the first example of an incredibly abstract concept. But anyway, we'll pretend numbers are tangible. So how does this relate, I remember, and I don't even remember now, I must have been writing some article that related to this, but looking at your primes that are your 1 more than a multiple of 6 versus 1 less and looking at whether there are more or fewer of these. So these are two different arithmetic progressions. The one that's like, you know, 7, 13, let's see if I can add by 6, 19, this, that progression, versus the 5, 11, etc, progression. So is this related to looking at whether there are more of the ones that are one more one less or things like that?

DL: For sure.

EL: I feel like there are all these interesting results about these biases and the distributions.

DL: Yeah, so people call this prime number races.

EL: Yeah.

DL: So what you might do is you might take two different arithmetic progressions and ask are there more prime numbers, like, less than a billion, say, in one of those progressions as opposed to the other? And there are actually pretty surprising properties of those races that I think are not totally well understood. So like even even this recent work of Kannan Soundararajan and Robert Lemke Oliver on this kind of thing.

EL: Oh, yeah, that’s what I was writing about!

DL: Which, yeah, shows some sort of surprising biases. And so that's the reason people think those are cool, is exactly this principle I mentioned before, this general principle of math that things should be as random as they can be. And there are maybe some ways in which our random models of the primes are not always totally accurate. And so understanding the ways in which they're inaccurate and how to fix that inaccuracy, like how to come up with a better model of the primes, is a really big part of modern number theory.

EL: But I guess, the Dirichlet theorem is what you need before you start looking at any of these other things, is you need to know that you can even look at these sequences.

DL: Right. Exactly. Yeah. I mean, how do you study the statistics of a sequence you don't know is infinite? Yeah.

EL: Right.

DL: One thing I’ll mentioned, one cool thing about it is it lets you—it’s not just an abstract existence result. Like, sometimes you just need a prime which is, like, 7 mod 23 to do some mathematical computation. Okay, and if it's 7 mod 23, then it's pretty easy to find one. You can take 7. But if you need a prime, that's a mod b, its remainder upon division by b is a, it's sort of hard to make one in general. And the fact that Dirichlet’s theorem gives them to you is actually really useful. So at least for a mathematician who cares about primes, it's something that just comes up a lot in daily life.

KK: But it's not constructive, though.

DL: Yeah, that's, that's right. It does kind of guarantee that there will be one less than some explicit constant, so in some sense, it's constructive, but it doesn’t, like, hand one to you.

EL: But still, I guess a lot of the time, you probably don't actually need a particular one. You just kind of need to know that there is one.

DL: Yeah.

EL: And where did you first encounter this theorem?

DL: I guess it was, I was probably reading Apostol’s number theory book when I was in college. But I think for me, I didn't really grok it until some other more modern version of it, like one of these remakes showed up for me in my own work. So I wanted to make a certain construction of algebraic curves. So that's some kind of geometric objects defined by some polynomial equations, which have some special properties. And it turned out that for me, the easiest way to do that was to use some version of Dirichlet’s theorem in some kind of geometric context.

KK: Very cool.

DL: So that was really exciting.

KK: Yeah. Well, it's it's nice when, like you say, when the oldies come up on your jukebox. They're useful.

DL: Yeah, exactly.

KK: So another fun thing about this podcast is that we ask our guests to pair their theorem with something. And I mean, I think Evelyn and I are just dying to know what pairs well with Dirichlet’s theorem on primes in arithmetic progressions.

DL: So for me, it's the Arthur Conan Doyle stories about Sherlock Holmes.

KK: Okay.

DL: For a couple different reasons. So first of all, because he's all about making connections between these sort of seemingly unrelated things, just like Dirichlet’s theorem is about making connections between, somehow for the proof, it's about connecting these things in algebra, primes, to things in complex analysis, these L-functions, but then also because it's an oldie that's been remade over and over again. It's still constantly being remade, like with the new BBC Sherlock show.

KK: It’s the best. Yeah, I remember when that was coming out. My wife and I were just so excited every time a new season come out, you know, just “Sherlock! Yes!”

DL: Yeah, just like I'm so excited every time a new version of Dirichlet’s theorem on primes in arithmetic progression comes out.

EL: Yeah, I haven't watched any of the Sherlock TV or movies yet. But we're watching a little more TV these days, and that might be a good one for us to go look at.

KK: It is so good. I mean, the first episode…

EL: Is that the one with Benedict Cumberbatch?

KK: Yeah, but the first one, just, I mean, it just grabs you. You can't not watch it after that. It's really, really well done.

DL: Yeah, they're really fun. Although—oh, go on.

KK: I was going to say the last one, the very last episode, I thought was a bit much.

DL: I don't know that I watched the last season.

KK: Yeah, it was a little…yeah. But you know, still good.

DL: I was reading a couple of the old short stories in preparation for this podcast. Those are also, I highly recommend.

KK: Which ones did you read?

DL: My favorite one that I read recently was, I think it's called the Adventure of the Speckled Band.

KK: Mm hmm.

EL: Oh, yeah.

DL: It's one of the classics.

KK: Right. Yeah. And I think they based one of the episodes on that one, too.

DL: Yeah. that’s right. Yeah.

EL: Yeah, that's a good one. I haven't read all of the Sherlock Holmes it seems like they're practically infinitely many of them. But you know, I had this collection on my Nook and we were moving, so it was like light, and I could read it in the hotel room easily and stuff. And as we were moving to Utah, I think the very first Sherlock Holmes one is set in Utah, or like part of it is set in Utah.

DL: Yeah, maybe the Sign of Four?

EL: Yes, I think it’s the Sign of Four.

DL: Yeah, I think it's one of the first two novellas. So I’ve read every single Sherlock Holmes when I was when I was in high school or something.

EL: Okay. But I was just like, of all things. I didn't know, I hadn't ever read any Sherlock Holmes before. And, like, this British guy writing about this British detective, and it’s set in the state I’m about to move to. It just seemed incredibly improbable to me.

DL: Yeah, I guess he had some kind of fascination with the U.S. because there's that one, which is sort of set in Utah as it was being settled, I guess.

EL: Yeah.

DL: And then there's the case of the five orange pips or something, which actually in a timely way crucially involves the KKK. And so yeah, so there's a lot of sort of interesting interactions with American history.

EL: Yeah, I don't I don't remember if I've read the orange pips.

KK: That figures in the TV series too.

EL: Okay. Yeah, I kind of forgot about those. Those might be a fun thing to go back to, since unlike you, I have not read all of them, and there always seem to be more that I could kind of dive into. I think I kind of tried to read too many at one time, and I just got fed up with what a jerk he is. Self righteous, smug guy.

DL: Yeah, definitely.

EL: Which doesn't make it not entertaining.

DL: If you like this stuff, there's a nother thing I was thinking of pairing. pairing with the theorem. There's a novel by Michael Chabon about a sort of very elderly Sherlock Holmes. Which I don't quite remember the name but part of it is about, you know, what it's like to be Sherlock Holmes when you're 90 and all your friends have left you, and so maybe that might, might appeal to you if you find him sort of an annoying character.

EL: Yeah. Could that be the Yiddish Policeman's Union?

DL: I don't think so. It's a much shorter book.

EL: Okay. That’s the title I could remember.

DL: That one is also excellent. It just doesn't have Sherlock Holmes in it. [Editor’s note: the book is The Final Solution: A Story of Detection.]

EL: Okay. Well, when you were talking earlier about the theorem, you used the word, I think you used the word remake or sequel or something. So I was wondering if you were going to pick movies, or something like that for your pairing. But this kind of works, too, because each one, it’s not a not remakes exactly—I guess with the movies there are remakes, movies and TV shows. But the stories are all, like, some new sequel. Like, here's a slightly different adventure that Sherlock goes on. And slightly different clues that he finds.

DL: Yeah, exactly. That's one thing that I love about math in general is that so much of it is you look at something classic, and then you put a little spin on it. Like I do a little exercise with some of the grad students at UGA in one of our seminars where we take a classic theorem. I think most recently, we did Maschke’s theorem, which is something about representation theory. And then you highlight every word in the theorem that you could change, and then kind of come up with conjectures based on changing some of those words, or questions based on changing some of those words. That's a really fun exercise in, kind of, mathematical remakes.

EL: That does sound fun. And I mean, I think that's one of the things that you learn, especially in grad school, is just how to start looking at statements of theorems and stuff and seeing where might there be some wiggle room here? Or where could I sub out a different space or a different set of assumptions about a function or something and get something new.

DL: Right, exactly. Yeah, definitely. With Dirichlet’s theorem, that happens so many times.

EL: Yeah, well, that's very fun. Thanks for bringing that one up. Thinking about it, I’m a little surprised that we haven't had it already on the podcast.

DL: Yeah, it's classic.

EL: Yeah, it really is.

KK: So we also like to give our guests a chance to plug anything that they're working on. You're very on Twitter.

DL: Yeah, that's right. You can you can follow me @littmath.

KK: Okay.

DL: So what do I want to plug? I think aside from Sherlock Holmes, who maybe needs no plugging, first of all, I would like to plug the Ava DuVernay documentary 13th, which I really liked and I think everyone should should watch.

EL: Yeah, and I saw that's free on YouTube right now. I don't know if that's temporarily, but I’m not a Netflix subscriber.

DL: Yeah, it is on Netflix. And yeah, I don't know if it'll be available on YouTube but for free by the time this comes out, but probably a nominal cost. In terms of things I've done that I think people who listen to this podcast might like, I did a Numberphile video about a year ago on the on it one of Hilbert’s problems about cutting up polyhedra and rearranging them that someone might someone who likes this podcast might enjoy. So if you google “Numberphile the Dehn invariant,” that’ll come up.

EL: Oh, great.

KK: Cool. All right.

EL: We’ll put links to those in the show notes. Yeah.

KK: All right. Well, thanks for joining us.

DL: Thank you guys so much for having me. This was a lot of fun.

KK: I learned something. I learn something every time, but I'm always surprised at what I'm going to learn. So this is this has been great. All right. Thanks, Daniel.

DL: All right. Thank you so much.

On this episode of My Favorite Theorem, we were happy to get to talk to Daniel Litt of the University of Georgia about Dirichlet's theorem on primes in arithmetic progressions. Here are some links you might find useful as you listen:

Litt's website
Litt's Twitter profile
More about the Dirichlet theorem from Wikipedia
Tom Apostol's number theory book
The article Evelyn wrote about surprising biases in the distributions of last digits of prime numbers
Michael Chabon's novella The Final Solution: A Story of Detection
Litt's Numberphile video about the Dehn invariant
Ava DuVernay's documentary 13th

Episode 58 - Susan D'Agostino

10 september 2020 | 25 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about math and so much more. I'm one of your hosts, Kevin Knudson, professor of mathematics at University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. So how are you, Kevin?

KK: I’m fine. It's it's stay at home time. You know, my wife and son are here and we're sheltered against the coronavirus, and we've not really had any fights or anything. It's been okay.

EL: That’s great!

KK: Yeah, we're pretty good at ignoring each other. So that's pretty good. How about you guys?

EL: Yeah, an essential skill. Oh, things are good. I was just texting with a friend today about how to do an Easter egg hunt for a cat. So I think everyone is staying, you know, really mentally alert right now.

KK: Yeah.

EL: She’s thinking about putting bonito flakes in the little eggs and putting them out in the yard.

KK: That’s a brilliant idea. I mean, we were walking the dog earlier, and I was lamenting how I just sort of feel like I'm drifting and not doing anything. But then, you know, I've cooked a lot, and I'm still working. It's just sort of weird. You know, it's just very.

EL: Yeah, time has no meaning.

KK: Yeah, it's it's been March for weeks, at least. I saw something on Twitter, Somebody said, “How is tomorrow finally March 30,000th?”

EL: Yeah.

KK: That’s exactly what it feels like. Anyway, today, we are pleased to welcome Susan D'Agostino to our show. Susan, why don't you introduce yourself?

Susan D’Agostino: Hi. Thanks so much for having me. I really appreciate being here. I’m a great fan of your show. So yeah, I'm Susan D’Agostino. I'm a writer and a mathematician. I have a forthcoming book, How to Free Your Inner Mathematician, which is coming out from Oxford University Press. Actually, it was just released in the UK last week and the US release will be in late May. And otherwise, I write for publications like Quanta, Scientific American, Financial Times, and others. And I'm currently working on an MA in science writing at Johns Hopkins University.

KK: Yeah, that's pretty cool. In fact, I pre-ordered your book. During the Joint Meetings, I think you tweeted out a discount code. So I took advantage of that.

SD: Yes. And actually, that discount code is still in effect, and it's on my website, which I'll mention later.

EL: Great. So you said you're at Hopkins, but you actually live in New Hampshire?

SD: Exactly. Yes. I'm just pursuing the program part-time, and it's a low-residency program. So I’m a full-time writer, and then just one class a semester. It creates community, and it's a great way to meet other mathematicians and scientists who are interested in writing about the subject for the general public.

EL: Nice. I went to Maine for the first time when I was living in Providence last semester and drove through New Hampshire, which I don't think is actually my first time in New Hampshire, but might have been. We did stop at one of the liquor stores there off the highway, which seems like a big thing in New Hampshire because I guess they don't have sales tax.

SD: No sales tax, no income tax, “Live Free or Die.” Yeah, and you probably test right around where I live because I live in New Hampshire has a very short seacoast, about 18 miles, depending on how you measure it. We live right on the seacoast.

EL: Oh yeah, we did pass right there. Wonderful. Yeah, the coast is very beautiful out there.

SD: I love it. Absolutely love it. I'm feeling very lucky because there's lots of room to oo outside these days. So, yeah, just taking walks every day.

EL: Wonderful.

KK: So you used to be a math professor, correct?

SD: Yes.

KK: And you just decided that wasn't for you anymore?

SD: Yeah, well, you know, life is short. There's a lot to do. And I love teaching. I had tenure and everything. And I did it for a decade. And then I thought, “You know, if I don't write the books I have in mind soon, then maybe they won't get done.” I've got my first one out already, only two years into this career pivot to writing, and I’m working on my next one. And I always had in mind, in fact, I have a PhD, but I also have an MFA. So I have a terminal degrees both in math and writing. And I always had one foot in the math world and one foot in the writing world, and I realized I didn't want to only live in one. So this is my effort to live fully in both worlds.

KK: That’s awesome.

EL: Yeah. Nice. So the big question we have now of course, is what is your favorite theorem?

SD: Okay, great. My favorite theorem is the Jordan curve theorem.

KK: Nice.

SD: Yeah. It’s a statement about simple closed curves in a 2-d space. So before I talk about what the Jordan curve theorem is, let's just make sure we're abundantly clear about what a simple closed curve is.

EL: Yes.

SD: So, a curve—you can think about it as just a line you might draw on a piece of paper. It has a start point, it has an end point. It could be straight, it could be bent, it could be wiggly, it could intersect itself or not. The starting point and the end point may be different or not. And because this is audio, I thought maybe we could think about capital letters in a very simple font like Helvetica, or Arial. So for example, the capital letter O is a is a curve. When you draw it, it has a start point and an end point that are the same. The capital letter C is also a curve. That one has a different starting and end point, but that's okay. It satisfies our definition. Capital letter P also. That one intersects itself in the middle, but it's still it's a curve.

Okay, so a simple curve is a curve that doesn't intersect itself along the way. It may or may not have the same starting and end point, but it won't intersect itself along the way. So capital letter O and capital letter C are both simple. But for example, the capital letter B is not simple, because if you were to start at the bottom, go up in a vertical line, draw that first upper loop and then the second upper loop, between the first and second upper bubbles of the B, you will hit that initial vertical line that you drew. So it's not simple because it touches itself along the way.

And a closed curve is a curve that starts and ends at the same point. So the letter O is closed, but the letter C is not because that one starts in one place ends in another.

KK: Right.

SD: Moving forward as we talk about the Jordan curve theorem, let's just keep in mind two great examples of simple closed curves: the letter O, and even the capital letter D. It's fine that that D has some angles, in the bottom left and upper left. So corners are fine, but it needs to start and end in the same place and doesn't intersect itself other than where it starts and ends.

Okay, so the Jordan curves theorem tells us that every simple closed curve in the plane separates the plane into an inside and an outside. So a plane, you might just think of as a piece of paper, you know, an 8 1/2 by 11 piece of paper, let's draw the letter O on it. And when you draw that letter O, you are separating that piece of paper, the surface, into a region that you might call inside the letter O and another region that you might call outside the letter O. And the second part of the Jordan curve theorem tells you that the boundary between this inside and that outside formed by this letter O is actually the curve itself. So if you're standing inside the O, and you want to get to the outside of the O, you've got across that letter O, which is the curve.

Okay, so that doesn’t sound very profound.

KK: It’s obvious. It’s just completely obvious.

EL: Any of us who are big doodlers—like, when I was a kid, at church, I was always doodling inside the letters in the church bulletin. That’s the thing. I know that there's an inside and outside to the letter O.

SD: You do. Yes. And you could ask your kid brother, kid, sister, whoever. Anyone—you probably didn't need a big mathematical theorem to assure you of this somewhat obvious statement when it comes to the letter O. Okay, so, I do want to tell you why I think it's really interesting beyond this fact that it seems obvious. But before I do, I just want to make two quick notes. And one is that you really do need the simple part, and you really do need the closed part of the theorem because, for example, if you think about a non-closed curve, like the letter C, and you're standing on the piece of paper around that letter C, maybe even inside, like where the C is surrounding you, it actually doesn't separate the piece of paper into an inside and an outside. And then you also need the non-simple part because if you think about the letter P, which is not simple because it intersects itself, if you think about the segment of the P that's not the loop, so the vertical bottom part of that P, that is part of the curve, the letter P, and that piece of the curve doesn't separate—so even though that P seems to have a little bit of a bubble up there, in the in the loop of the P, the bottom part of the P is part of the curve, and it's not the boundary between the inside, what you might consider the inside of the P, and the outside of the P. So you really do need the simple part and the closed part.

KK: Right, right.

SD: Okay, so the reason I think it's interesting, in spite of the fact that it seems obvious, is because it actually isn't very obvious. And it's not obvious when you talk about what mathematicians love to call pathological curves.

KK: Yeah. Okay. No, I know, I know, the theorem I just wanted to shrug my shoulders and say, “Oh, look, it's just a special case of Alexander duality.” Right? And so surely it works. But yeah, okay.

SD: And there are other poorly-behaved curves, or misbehaved curves, like another curve you might think about is the Koch snowflake. So one way of thinking about the Koch snowflake is—again, I'm going to wave my hands a little bit here because we're in audio and I can't draw you a picture—but if you think about the outline of a snowflake, and there's a prescribed way to draw the Koch snowflake, but I'm going to simplify it a little bit. Imagine the outline of a snowflake, so not the inside or the outside of the snowflake, just the outline of it. And on a Koch snowflake, that snowflake is going to have jagged edges. It's going to zig and zag as it goes along the outline of the snowflake. The Koch snowflake actually has an infinitely jagged curve, line, to draw it. So it's not that it has 1000 zigs and zags or 1 million or even 1 billion. It has an infinite number of zigs and zags going back and forth. So you know, it's a little bit easier to imagine the— what could loosely be defined as the inside of the Koch snowflake, and the outside of the Koch snowflake when you imagine one being drawn on a piece of paper. You know, right in the heart of the very dead center of that Koch snowflake, you could probably feel pretty confident saying, “Hey, I'm inside the Koch snowflake.” And then far outside, you could be confident saying, “I'm outside of the snowflake.” But if you think about yourself right up against the edge of this Koch snowflake. And put yourself right there. Then as you think about this boundary of the Koch snowflake, the boundary is supposed to be what separates the inside from the outside, but if you're right up close to that boundary, and in the process of drawing an infinite number of constructions to get the ultimate Koch snowflake. You continue zigging and zagging, you add more zigs and zags every time. Then even in the steps that it takes you to get to your drawing of the Koch snowflake, at some point, it might seem like “Hey, I'm inside. Oh wait, now they zigged and zagged and I’m outside. Oh, wait, they zigged and zagged some more. Now I'm inside again.” So it seems like even in the finite steps that you need to take to draw that Koch snowflake, to imagine what the it is in its infinite world, it seems like that boundary is not really clear. So again, another place where it makes you stop and say, “Wait a minute, maybe the Jordan curve theorem is not as obvious as it first looked.”

KK: Right. Why do you love this theorem so much?

SD: Yeah, so I love it. It actually it kind of goes along with your question of what do you pair it well with? So maybe I'll just jump ahead to what's sugar. Yeah. So, um, because even in my book and in the chapter that in which I discuss the Jordan curve theorem, I actually paired it with a poem. And the poem is by a New Hampshire native, Robert Frost, who actually went to Dartmouth, which is where I got my doctorate. And one of my favorite poems by Frost is called “The Road Not Taken.” And in the beginning of the poem, he's standing in front of this fork in the road, essentially, and he's looking at both options, realizing, “Okay, I've got to go left or I've got to go right.” You know, he starts off:

Two roads diverged in a yellow wood, And sorry I could not travel both And be one traveler, long I stood And looked down one as far as I could To where it bent in the undergrowth;

So he's standing here and he's saying, “Well, which path should I take?” And he notices one that he calls you know, “it was grassy and wanted wear” and had no leaves—what was what was the line—“in leaves no step had trodden black.” And he ultimately comes to the conclusion that he's going to take the past path less traveled. You know, at the very end of the poem, he says, “Two roads diverged in a wood and I—/ I took the one less traveled by,/ And that has made all the difference.” And it strikes me that what Frost is telling us, and what the Jordan curve theorem is telling us, is take the paths that are more unusual, that aren't well trodden, that people don't always look at first, that aren't as obvious or as paved for us. Maybe it's a path that's going to make you question whether you're inside or outside. Or maybe it’s going to have what feels like this amorphous boundary that you can't quite put your finger on. I guess it reminds me that sometimes making a non-traditional choice in life, or looking at pathological objects in math, is actually something very engaging to do, and can can make a life a little bit more interesting.

You know, when I first heard about this theorem, I had the same reaction that most everybody else does: Okay, so I can just draw a curve—you know, you say a curve and you think, “Oh, I can just draw a curve.” I'm just going to do a squiggle on a piece of paper. And as long as I make it simple and closed, then it might be the letter O or it might be some blob that doesn't intersect, but at least starts and ends where it ends where it started. You know, I remember thinking, wait, why does this theorem get its own name? Why isn’t it just lemma 113.7?

EL: An observation.

KK: Clearly.

SD: Why did it get its own name? A I remember asking, and a lot of people, at first everybody was happy to recite the theorem and and say what it was and laugh at how obvious it was, but then later, I kept searching and searching, and then finally I ended up discovering that in fact, it wasn't as obvious, but in order to appreciate how it’s not that obvious, you needed to look at the paths not taken, the more unusual lines and curves.

EL: Yeah, so this is a theorem that, of course, I I feel like I've known for a long time, not just in the “it's obvious” sense, but in the sense that it's been stated in classes that I took—and feel entirely unconfident about knowing anything about it's proof, at least in the general case. I feel like the the difference between how much I have used it and relied on it and what I actually understand of how to prove it is very large.

SD: Yeah, honestly I can say the same thing. My background is in coding theory, definitely not topology. And honestly, I never saw topology as my strength. It was always something that I was in awe of, but also found extremely challenging or less intuitive to me. But I had looked at the proof long ago. I haven't looked at them deeply recently. There are a number of different approaches. But yeah, I feel the same, that even—the statement sounds simple and it's not, and to my understanding, the proofs are also non trivial.

KK: Yeah. I mean, I was sort of being glib earlier and saying it's just a special case of Alexander duality, like that's easy to prove.

EL: Yeah. Right.

KK: I mean, I was teaching topology this this semester, and I was proving Poincaré duality, which is a similar sort of thing, and it's highly non-trivial. I mean, you break it into a bunch of steps, and it sort of magically pops out of it. And I think that's kind of the case here. It's like, you break it into enough discrete steps where each thing seems okay. But in the end, it is a lot of heavy machinery. And like even for Poincaré duality, in the end you use Zorn’s lemma I mean, there's some kind of choice going on. I think when when Jordan—actually, did Jordan even state this theorem? Or is this one of those things where where Jordan gets the credit, but it wasn't really him?

SD: Actually, I don’t know, and now I need to know that answer.

EL: I think he did.

KK: Did he?

EL: Yeah, not to toot my own horn but I’m, gonna anyway, the calendar that I published this year, the page-a-day calendar, still available for purchase, I think Camille Jordan’s birthday is pretty early. It's sometime in January, so I've actually even read this not too long ago. And I think he did publish it and did have a proof of it. And there's an interesting article, I believe by Thomas Hales, about his about Jordan’s proof of the Jordan curve theorem, I guess maybe to some extent defending from the claim some people have that that he never had a rigorous proof of it. I did read that for doing the calendar, but it was over a year ago at this point and I don't quite remember. But yeah, you can find a reference to it on my calendar. I will also include that in the show notes.

KK: And also the same Jordan of Jordan canonical form.

SD: Right.

KK: Pretty serious contributions there from one person.

SD: Absolutely.

KK: Yeah. All right. I actually like this pairing a lot.

EL: Yeah.

KK: And and since you live in New Hampshire, it's perfect.

SD: Yes. I have a number of New Hampshire references in my book because I just feel like I wanted to humanize math to the extent that I could, while still tackling pretty substantial ideas. But any time I had an invitation to bring in something from left field that was actually meaningful to me, I just went for it.

EL: Yeah.

SD: I’m sure Evelyn, too, it sounds like you're up on all of the mathematicians’ birthdays at this point because of your calendar.

EL: I know a few of them now. More than I did two years ago.

SD: Right.

KK: So it was like to give our guests a chance to plug anything. You’ve already plugged your book. Any other places we can find you online?

SD: Yeah, well, lately, I've been writing for Quanta magazine, which has been very exciting. And in fact, I have a few math articles already out this year. And I have a very special one—I can't tell you the topic. I'm not supposed to—it should be coming out April 15. And I'm very excited about that article that I believe is going to be on April 15, assuming everything is fine with the publication schedule, given the pandemic. But yeah, listeners can find links to my articles on my website, which is just susandagostino.com. And you can find information about my books and my articles and what I'm up to there. v KK: Cool. Well, thanks so much for joining us, Susan. This was a good one.

EL: Yeah, lovely to chat.

SD: Great. Well, thank you so much. And you know, I love the show, and really, it was my honor to be here. Thank you.

KK: Thanks.

On this episode of My Favorite Theorem, we talked with mathematician and science writer Susan D'Agostino. Here are some links you might find interesting as you listen.

D'Agostino's website
How to Free Your Inner Mathematician, her new book (find a discount code on her website)
Evelyn's article about the Koch snowflake
Thomas Hales' article about Camille Jordan's proof of the Jordan curve theorem (pdf)

Evelyn's page-a-day math calendar

The article D'Agostino was excited about towards the end of the podcast was this interview with Donald Knuth

Episode 57 - Annalisa Crannell

13 augusti 2020 | 33 min

Evelyn Lamb: Welcome to My Favorite Theorem, joining forces today with Talk Math With Your Friends. I'm Evelyn Lamb. I co-host this podcast. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida, where it is boiling hot today, and I’m very happy to be in this—how would they put this on on TV?—crossover event, right?

EL: Yeah.

KK: So like, I think last night on NBC, on Wednesday nights, there are all these shows that take place in Chicago: Chicago Med and Chicago PD and Chicago Fire, Chicago Uber, who knows what. Anyway, sometimes they'll just merge them all into one three-hour super show, right? So here we go. This is the math version of this, right?

EL: Yes. And I realized today that our very first episode of My Favorite Theorem, we published that in late July 2017. So this is our early third birthday! And we're so glad that people came to join us! And we are very happy today to have our guest Annalisa Crannell with us. Hi, Annalisa. Can you introduce yourself and tell us a little bit about yourself?

Annalisa Crannell: So hi, my name is Annalisa Crannell. I profess mathematics at Franklin and Marshall College, which is in south-central, southeastern Pennsylvania. It's a small liberal arts college. I got my PhD working in differential equations, partial differential equations, nonlinear differential equations, switched into discrete dynamical systems, topological dynamical systems, but for the past 10 or 15 years have been really thinking hard about projective geometry applied to perspective art.

KK: That’s quite the Odyssey.

AC: Yeah, I was really influenced by by Paul Halmos saying that one of the marks of a really good mathematician is that they can change fields. And so yeah, I feel like I'm trying to enjoy so many different aspects of what this profession allows us to do.

EL: And a fun story, at least it was fun for me, is that one time you were here in Utah giving a talk at BYU, which is down the street. And we went to an art gallery and you pulled out your chopsticks and showed me how you use your chopsticks to help you know where to stand to best appreciate art, and it was just so amazing to me that that was this thing that you could do. So that was that was a lot of fun. And I think it just, to me, sums up the Annalisa experience.

AC: Thank you. Yeah, summing, I guess, is a good thing for mathematicians. I think everybody should carry chopsticks with them. I mean, it's great. It's frugal. It helps you avoid to trash, but it also helps you do really cool mathematics. So what's what's not to love about them?

EL: Yeah. So what is your favorite theorem?

AC: So if you had asked me about five years ago, I would have said the intermediate value theorem. But today, I am going to say no, Desargues’ theorem. So Desargues’ theorem first came into human knowledge in the 1640s. And it's a theorem that sounds like it's sort of about planar geometry, but I really think of it as being about perspective. So is this when I'm supposed to tell you what the theorem says?

KK: Yes, please.

EL: Yeah. Okay, should we all get out our—so this is one, I feel like I always need like a piece of paper. (I’m trying to hold it up, but I’ve got a Zoom background.) But I got my piece of paper out so I can hopefully follow along at home.

AC: Yeah. If you had a piece of paper or a chalkboard right behind you, you could imagine that you would have a triangle, like, standing up on a glass pane. And then on one side of this glass pane would be maybe a magician or somebody holding a light. Maybe your granddaughter drew the magician. (Okay, for people in the podcast, I'm showing a picture that my granddaughter drew on the chalkboard.) If this light shines on the triangle, then it casts a shadow, and the shadow is also a triangle. And so we say those two triangles are perspective from a point, the point is the light source. And we say that because the individual corners, the corresponding corners, are colinear with the light source. So A and the shadow of A are collinear with a light. B and the shadow of B are colinear with a light. But it turns out that those shadows, the triangle and its shadow, are also perspective from a line. And what that means is that if you think not about the points on the triangles, but the three lines on the triangles, and you really think of them as lines, not line segments, so going on forever, then the corresponding lines will also intersect along a line. And you can think of that second line, which we call the axis, as the intersection between the plane of glass that's sitting up in the air and the ground. So the interesting thing to me about Desargues’ theorem is that it pretends like it's a theorem about planar geometry, because this theorem holds when the two triangles are both in the same plane, in R2 or something, but the best ways of proving it, the most standard ways of proving it, are using essentially perspective, going out into three dimensions and proving it for two completely different planes and then pushing them back down into the regular plane. And so to me, this is a really interesting example of sort of how art informs math rather than the other way around. Or maybe they both inform each other.

EL: So going back a little bit, to me when I've I've looked at Desargues’ theorem before, somehow there's this big conceptual leap to me between perspective from a point and perspective from a line. Perspective from a point seems really easy to think about, and perspective from a line, I just have trouble getting it into my brain.

AC: Yeah, I do think perspective from a point is so much more intuitive. And so, so the minorly intuitive, the somewhat intuitive way of thinking of this axis, is you can sort of pretend like it's a hinge. So if these two triangles will sort of fold on to each other from the hinge—the triangle on the glass and the triangle on the ground can fold along this hinge—then they’re perspective from a line. So if you think about something that's in the real world, a flat thing in the real world, and its mirror image, then those two, it's hard to say whether they're perspective from a point, but the lines in the real world thing and the lines in the mirror will intersect along the line where the mirror hits the ground. And so that's that's another way of thinking of this axis, sort of three dimensionally.

KK: So I want to think about this in projective space, which probably isn't correct. Or maybe it is. I don't know. I mean, so these lines are points in projective spaces. This is this, how one might go at this in some other way? I asked the wrong question. I'm sorry.

AC: So that's not exactly the way that I think of it because I think of the line as a line in projective space.

KK: Okay.

AC: And the point is a point in projective space. So the point comes from, you could say, from a one-dimensional line in R-whatever.

KK: Okay.

AC: And so here's one of the interesting things about this theorem and about me loving this theorem. In 2011, one of my coauthors and I wrote a book on the mathematics of perspective art, and we used Euclidean geometry all the way through. We were giving a MathFest mini-course on this and a young mathematician came up to us and said, “I just love how you use projective geometry in art because I learned projective geometry and felt like it had to have something to do with art. And you guys are the ones that explained to me how it does.” And Mark and I turned to each other. We're like, “What kind of geometry?” So neither of us had ever taken a projective geometry class. Neither of us had ever learned any projective geometry. We did not know that it existed. And so this young mathematician ended up changing our lives. We ended up working with her and really learning a bunch of projective geometry in order to come out with our most recent book, which came out last December. And so when you ask questions that get into really deep projective geometry, I'm like, “Ooh, I have to write that one down because that's something else I have to go learn.” So for those of you young mathematicians out there, I just want to say learning new stuff and not knowing stuff is is really so much fun! Don't be afraid of starting something new, even if you don't know it all.

EL: And how did you first encounter Desargues’ theorem?

AC: Oh, man, so I first encountered Desargues’ theorem when, Fumiko Futamura, this young mathematician, had convinced me I needed to learn it. So I bribed an undergraduate to go through Coxeter’s Projective Geometry with me because it seemed like that was the standard book. And Coxeter is, like, the famous guy in this realm, and he is completely non-intuitive. So I found Desargues’ theorem in there, and I'm like, “I have no idea what this means.” The notation is awful. The diagrams are awful. Everything about this is awful. And so I read through his book trying to say, “What does this have to do with art?” And that was a really fun way to read it. So we just decided Desargues’ theorem is about shadows.

EL: Well, I was wondering. So I remember you have also given a talk about squares that kind of blew my mind, where I guess the the thesis of the talk is that all configurations of four points are a square, if you look at it from the right way. Is Desargues’ theorem related to that theorem? I feel like when you said the word shadow that is what reminded me of that other talk.

AC: Yeah, thank you. So that's really cool. So most of us know what the fundamental theorem of calculus says. Most of us know what the fundamental theorem of algebra says. The fundamental theorem of projective geometry in one sense really ought to be Desargues’ theorem. So you can think about these triangles, these points, these lines as objects. For mathematicians, we care about verbs. So a verb is the function. So you can think of a perspective mapping as mapping one set of points and lines to another set of points and lines with this particular rule that says that corresponding points have to line up with the sun, which you call the center, and corresponding lines have to line up with the axis, this hinge. But there's other functions that take points to points and lines to lines. So we know in linear algebra, you can do this all the time, and in linear algebra sets of parallel lines go to other sets of parallel lines. But there's other kinds of functions that do this. They're called colineations. So the fundamental theorem of projective geometry says that if you have four points and their images, and you know that points go to points and lines go to lines, then the entire rest of the function is pre-determined, we know that.

So Desargues’ theorem says that one kind of colineation is perspective mappings, right? Just, like, a shadow or mapping from the floor, this tiled floor onto your canvas through a window. We know from linear algebra, there are these other affine transformations. And so one really cool theorem that I totally love is if you have something that's not a linear algebra one, that's not an affine transformation, then it's automatically a perspective transformation together with an isometry. So you took a photograph and you moved it. That's this notion that every single thing that you do with four points going to four other points that determines a whole function. So yeah, so anytime you have four points connected by four lines, even if they look like a bow tie, or they look like Captain Kirk’s Star Trek logo, it turns out that's actually a weird perspective image of a square moved around somewhere.

EL: And you just have to figure out where you should stand to see it as a square.

AC: Yes, exactly.

KK: Are you guaranteed to be able to—so if it's on the wall, say, could you have to, like, lift it up into a third dimension to be able to see it correctly?

AC: So one of the weird things that happens is if you have a bow tie, we sort of think of a bow tie is that the inside of the bow tie, you would imagine that has to go to the inside of the square. And that is not actually the way it happens. So let's let's think about something that's much more familiar to us. Can you map a circle through perspective into other weird shapes like an ellipse? Well, Sure you can. So imagine that you've got a lampshade, and you've got a circular lampshade, and the shadow that it projects onto the wall is actually a hyperbola. We know that. And the light from the inside of the shadow goes to the part of the hyperbola that goes off towards infinity. Well, if you have the bow tie, think about the area outside of the “x” as almost a hyperbola. So this is when it would be so wonderful if I could actually draw pictures, but it's a podcast. On the on the bow tie, there's two sides that are parallel to each other, and then there's this weird “x” in the middle. The parallel sides, extend them out towards infinity from the bow tie. Right? That turns out to be where the square goes, so if you had a square lampshade, it would cast a shadow that would look like this outside of the bow tie. So the same way that a circular lampshade casts a shadow that looks like the outside of the hyperbola, the U shape of the hyperbola.

KK: My desk lamp is a rectangle, so I’m trying to see if it’s casting the right shadow here.

EL: Yeah, some experiments you can do. I feel like it's this “expand your mind on what a square is” kind of idea.

KK: Got to get rid of those old ideas, man.

EL: Yeah. I know that we we traveled a little bit from Desargues’ theorem and I want to give you a chance to circle back and for me

KK: Or square back. Sorry.

EL: Or square back, or projectively bow tie back to Desargues’ theorem, and I guess what do you love so much about it?

AC: What do I love so much about Desargues’ theorem? One of the things that I love is that it really tightly connects mathematics informing art and art informing mathematics. So Desargues himself, we don't know if he actually wrote this up and published it. We don't have a copy of his original manuscript. We do have something that came out from one of his, sort of acolytes, one of his followers, a guy named Bosse. And if you look at Bosse, okay, to draw Desargues’ diagram, you need 10 points: the three on the first triangle, the three on the second triangle, the sun, that gives you seven, and then the three along the axis. You also need 10 lines: the three on the triangle, the three on the other triangle, the three light rays, that gives you nine, and then the axis. When Bosse first published his diagram, his diagram was incomprehensible. It had 20 lines and 14 points, and it was just really a mess. And it was hard to even figure out where the heck the triangles were.

KK: Yeah, I don’t see them.

AC: And he ended up proving this not using sort of standard geometry, using using numerical stuff called cross-ratios. But the the proofs that make the most sense, that are convincing our proofs that allow us to think about things in three dimensions and use art. So that's one of the cool things, is that actually drawing, if you go and you shade in Bosse’s diagram in a cool artistic way, all of a sudden it sort of pops into 3-d and you can see it, but his original diagram not so much. The same is true of a lot of different proofs. If you try to imagine them as three-dimensional, if you draw them as three-dimensional, the proof becomes more obvious.

But also Desargues’ theorem is actually useful for artists because if you want to draw the shadow of something, if you want to draw the shadow of a kite, if you want to draw a reflection, shadows and reflections, they are projections, so projective geometry, and how do you know how to draw this? You have to use the fact that the shadow or the reflection, or this this projective image, however, you've made it, is perspective from a point and perspective from a line. So you're constantly using Desargues’ theorem to draw these images of images within your image. It just becomes so incredibly useful.

KK: My wife's an artist, but I can't imagine that she would use this. I mean, if you walked up to a typical artist, are they going to say, “Oh yeah, I use Desargues’ theorem all the time?” Or is it just a sort of an intuitive thing where people who are very good at drawing in perspective, can just kind of naturally draw it that way?

AC: Oh, yeah. So the truth is that Desargues’ theorem has really only pretty much been used by mathematicians, and occasionally misused by mathematicians. There's a description in a book by a guy named Dan Pedoe of Desargues’ theorem to draw the image of a pentagon on the top of a square, and he just completely gets it wrong. And Mark and I think that's hilarious. This book has been reproduced zillions of times. Anyway.

So no, actually artists have this incredible skill. One time, we had asked mathematicians and artists at one of our workshops to try to divide the image of a flag into three equal pieces perspectively. So imagine you're drawing the Italian flag going back into the distance, right? How do you do this? In the real world? This there's the three bars are evenly spaced, but in perspective, they're not. And the artists stood up and said, “Well, you just eyeball it, and you just put them here.” And I was horrified. This is not approved. This is not correct. My colleague Mark said, “Okay, this is good. But for those of us who can't just eyeball it, let's see if we can come up with a construction.” And eventually somebody did. They came up with a really cool geometric construction. And Mark had them put this up over the artist’s solution and it was spot on. As a mathematician, I decided to go take an art class. And one of the things we were supposed to do was to draw cans. And so the top of a can is circular, and so the image was going to be an ellipse. And I could not get the proportions right. My ellipses were so awful. So I would say that disarms is an incredibly useful tool for drawing things that look very accurate for people who do not know art, but who are good at math.

KK: Right.

AC: That’s a really long answer to your question. Yeah, artists don't tend to use it, but it really is a useful thing for drawing things that look correct.

KK: Cool. All right.

EL: And you've incorporated this into a class that you teach to help people, I don't know if the purpose of the class is more math or more perspective drawing, but it seems like an interesting mix.

AC: Yeah, we have a course called Perspective and Projective Geometry. We actually have a book that's come out that has Desargues’ theorem right on the cover up there. And it's aimed at the intro to proofs level. So it really teaches students to make conjectures about what they're seeing in the world and then to try to prove those conjectures, but also to try to draw. Ao it's actually sort of an applied course. So they, this students, when we introduce them to Desargues’ theorem, they're actually drawing the shadow of the letter A, and then discovering Desargues’ theorem, and then proving it using many colors and, yeah, lots of cool lines.

It's so much fun! It's a course that really attracts a very unusual swath of students. They all are students who love math, and who are art-curious. Almost none of them are good at art. But I tend to get more women than men in the class. I have often had my class being highly diverse in terms of races and ethnicities. And so for me, it's a fun class. I didn't do it just for the sake of promoting diversity in the math major, but it's sort of unintentionally has done that. And that's a really good feeling.

KK: Very cool. So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with Desargues’ theorem?

AC: Yeah, so I think I already hinted at this, so anything that you can eat with chopsticks goes really well with Desargues’ theorem because chopsticks allow you to have wonderful food and do math at the same time, and what could be better?

KK: So basically, anything you can eat, then, right, you can eat anything with chopsticks?

AC: Soup is a little bit tricky, but yes.

KK: But you drink the soup, right? They give you the chopsticks, you’ve had ramen, right? There's the chopsticks for the noodles.

AC: Yes. Exactly.

EL: Do you have a favorite food to eat with chopsticks?

AC: Oh my goodness. Pretty much everything. I was just realizing ice cream is not so easy with chopsticks.

EL: Yeah.

AC: I think yesterday was national ice cream day. Yeah, I don't I don't know. I take my chopsticks with me in my in my planner bag, and a spoon. And so when I go to restaurants if they try to give me plastic things, I use my chopsticks. So basically, yes, anything I can eat with chopsticks, I will eat with chopsticks. If I can't, I'll use my spoon.

EL: Nice.

KK: We’re getting Thai takeout tonight. Now I'm really excited.

AC: I’m coming to your house.

KK: Sure, come on down. Although you know with all the COVID, I don't think Florida is really a place you want to be coming these days.

EL: So I guess this would be a good time to open the floor to questions. So Brian, I was thinking that I would be able to keep an eye on it, and I totally couldn't. So I'm glad that you were keeping an eye on it. So do you have any questions that our listeners would like to ask Annalisa?

Brian Katz: I’ve noticed three so far. One is from Joshua Holden, would Desargues’ theorem be useful for computer graphics?

AC: That’s a really good question. If I knew anything about computer graphics, I would be able to answer that better. I do know that my students who have gone on into computer engineering tell me that the course that I offered on projective geometry was one of their most useful courses, that this idea of ray tracing was was super, super helpful. So I don't know if Desargues’ theorem itself is specifically useful, but the idea of projective geometry is certainly how we understand the world through videos.

BK: We got a request from Doug Birbrower asking for you to hold up the line drawing while I asked the next one. I was wondering, so when we're talking about triangles, we have these vertices that are special points. How does this idea translate when you're talking about, say, shadows of more complicated objects that might be smooth? You talked a little about circles, but is there a special that happens when you generalize beyond polygons?

AC: One of the things that makes triangles really awesome is the same reason why triangular stools are so useful, is they're always stable, right? Whereas a four legged stool can wobble. If you try to draw the perspective image of an object With four points like a kite, it's really easy to make it be perspective from the sun without being perspective from a line. And if you do something like that, it'll look like maybe the kite is planar, but the shadow is curved, which might make sense on the ground. So in some ways, it's saying triangles really determine planes. Yeah, the question of drawing other curves is really interesting because of how you do or don't define curves in projective geometry. So one way you could think of a curve is a collection of points. You could also think of it as a collection of tangent lines. And so I think a way to generalize Desargues’ theorem to those would be to be talking about those collections of points and those collections of tangent lines.

BK: And then the third one that got some answers in in the chat was: I have a sense that, like, parallel things that when they're prospective from a point that means the point’s at infinity when we're talking about projective geometry. Is their geometric intuition about what it means for the line, perspective from a line, for that line to be infinity? And TJ suggested that it was that the objects are translations of each other.

AC: So if the line is it that that is at infinity, then either you could think of this as being translations, or you can think of it as being a dilation. And so it's a translation if both the axis where the two triangles meet is infinity and the center, that is what how you shine from one to another, is also off at infinity. And they’re are a dilation if the axis is off at infinity, but the center is what we call an ordinary point.

KK: This is new for us, having a Q and A. It's usually just the three of us, you know, me and Evelyn and whoever we're interviewing, but this is fun. I like this interaction.

EL: Yeah, I like that. And people have good questions. Yeah. Great. Thanks. Are there any more questions from the chat that we want to get to? Okay, looks like I'm seeing no. So I think this will sort of wrap up the…oh, Brian. Yeah.

BK: This one just appeared: Do cylindrical polar coordinates throw any light on this?

AC: Oh, so I was just about to say to everybody, “Thank you so much for asking me questions that I actually know the answers to!” And this one, I have no idea. I don't know. I don't know anything about cylindrical polar coordinates. Sorry. Now I'm going to write that one down and go check it out.

EL: But we can all appreciate the “throwing light” phrase of the question. That was very well done. Thank you.

KK: Clever.

EL: So, to wrap up the podcast portion of this, or the the episode with Annalisa portion of this, we will have show notes that are available. Our podcast listeners probably know where to find that at Kevin's website. And on that will include a link to your website, a link to the books that you have. Do you want to say the titles of the books that you've written that people might be interested in?

AC: So the first one, the one from 2011, is called Viewpoints with a subtitle “mathematical perspective, and fractal geometry in art,” and that's suitable for, like, a first-year seminar in math and art. So students don't need to really know anything at all about mathematics. And then the other one is called Perspective and Projective Geometry, and it came out in 2019.

EL: Yeah, so thank you for joining us, Annalisa. And for your doing it in this different fun format.

AC: I’m really flattered that you invited me to do this. Yeah, it's been so much fun trying to think about how to do this without drying gazillions of pictures. I appreciate that.

EL: Yeah.

KK: Thanks so much.

We were delighted to have a crossover event with Talk Math With Your Friends, an online math seminar that runs on Thursdays at 4 pm Eastern time. You can watch a video of this episode, which includes a collection of "flash favorite" theorems from the audience, here. Our guest for this episode was Annalisa Crannell from Franklin and Marshall College, who talked about Desargues' theorem. Below are some links you might find handy after listening to the episode.
Crannell's academic website
Her collaborator Fumiko Futamura's website
Desargues' theorem on Wikipedia, which includes some helpful diagrams
The Image of a Square, a paper about the theorem that every quadrangle is a square if you look at it the right way. (Also available from Futamura's website.)
Viewpoints: Mathematical Perspective and Fractal Geometry in Art by Crannell and Mark Frantz
Perspective and Projective Geometry by Crannell, Frantz, and Futamura

During the episode, Crannell shared Bosse's original diagram for proving Desargues' theorem. It is here. Below is a version of the diagram colored in, making the triangles a little easier to see.

Episode 56 - Belin Tsinnajinnie

9 juli 2020 | 35 min

Evelyn Lamb: Hello, My Favorite Theorem listeners. This is Evelyn. Before we get to the episode, I wanted to let you know about a very special live virtual My Favorite Theorem taping. If you are listening to this episode before July 16, 2020, you’re in luck because you can join us. We will be recording an episode of the podcast on July 16 at 4 pm Eastern time as part of the Talk Math With Your Friends virtual seminar. Join us and our guest Annalisa Crannell to gush over triangles and Desargues’s theorem. You can find information about how to join us on the My Favorite Theorem twitter timeline, on the show notes for this episode at kpknudson.com, or go straight to the source: sites.google.com/southalabama.edu/tmwyf. That is, of course, for “talk math with your friends.” We hope to see you there!

[intro music]

Hello and welcome to my favorite theorem, the podcasts that will not give you coronavirus…like every podcast because they are podcasts. Just don't listen to it within six feet of anybody, and you'll be safe. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. So if our listeners haven't figured out by now, we are recording this during peak COVID-19…I don’t want to use hysteria, but concern.

EL: Yeah, well, we'll see if it’s peak concern or not. I feel like I could be more concerned.

KK: I’m not personally that concerned, but being chair of a large department where the provost has suddenly said, “Yeah, you should think about getting all of your courses online.” Like all 8000 students taking our courses could be online anytime now… It's been a busy day for me. So I'm happy to be able to talk math a little bit.

EL: Yeah, you know, normally my job where I work by myself in my basement all day would be perfect for this, but I do have some international travel plans. So we'll see what happens with that.

KK: Good luck.

EL: But luckily, it does not impact video conferencing.

KK: That’s right.

EL: So yeah, we are very happy today to be chatting with Belin Tsinnajinnie. Hi, will you introduce yourself?

Belin Tsinnajinnie: Yes, hi. Yá’át’ééh. Shí éí Belin Tsinnajinnie yinishyé. Filipino nishłį́. Táchii’nii báshishchíín. Filipino dashicheii. Tsi'naajínii dashinalí. Hi, everyone. Hi, Evelyn. Hi, Kevin. My name is Belin Tsinnajinnie. I'm a full time faculty professor of mathematics at Santa Fe Community College in Santa Fe, New Mexico. I’m really excited to join you for today's podcast.

EL: Yeah, I'm always excited to talk with someone else in the mountain time zone because it's like, one less time zone conversion I have to do. We're the smallest, I mean, I guess the least populated of the four major US time zones, and so it's a little rare.

BT: Rare for the best timezone.

EL: Yeah, most elevated timezone, probably. Yeah, Santa Fe is just beautiful. I'm sure it's wonderful this time of year. I've only been there in the fall.

BT: Yeah, we're transitioning from our cold weather to weather where we can start using our sweaters and shorts if we want to. We're very excited for the warmer weather we had. We're always monitoring the snowfall that we get, and we had an okay to decent snowfall, and it was cold enough that we're looking forward to warm months now.

EL: Yeah, Salt Lake is kind of the same. We had kind of a warm February, but we had a few big snow dumps earlier. So tell us a little bit about yourself. Like, where are you from? How did you get here?

BT: Yeah. I am Navajo and Filipino. I introduced myself with the traditional greeting. My mother is Filipino, my father is Navajo, and I grew up here in New Mexico, in Na’Neelzhiin, New Mexico, which is over the Jemez mountains here in Santa Fe. I went to high school, elementary school, college here in New Mexico. I went to high school here in Santa Fe. I got my undergraduate degree from the University of New Mexico, and I ventured all the way out over to the next state over, to University of Arizona, to get my graduate degree. While I was over there, I got married and started a family with my wife. We’re both from New Mexico, and one of our biggest goals and dreams was to come back to New Mexico and live here and raise our families where our families are from and where we're from. And when the opportunity presented itself to take a position at the Institute of American Indian Arts here in Santa Fe, it's a tribal college serving indigenous communities from all over the all over the nation and North America, I wanted to take that. I feel very blessed to have been able to work for eight years at a tribal college. And then an opportunity came to serve a broader Santa Fe, New Mexico community, where I also serve communities that are near and dear to my heart, where I've been here for over 30 years. And I'm really excited to have this opportunity to serve my community in a community college setting.

So, going into academia, and going into mathematics, it's not necessarily a typical track that a lot of people have opportunities to take on, but I feel very blessed to be doing math that I love serving communities that I love, and being able to raise my families around the communities that I love to. So I feel like you have a special kind of buy-in by engaging in a career that serves my communities and communities that are going to raise my families as well, too.

KK: That’s great.

EL: Nice. So I see over your shoulder a little bit of a Sierpinski triangle. Is that related to the kind of math you like to think about? Or is it just pretty?

BT: Yeah. One, it’s pretty. When I was at the Institute of American Indian arts, most of the students there, they're there for art. They come from Native communities, and they're not there to do mathematics, necessarily. So part of my excitement was to think about ways to broaden the ideas of mathematics and to build off of their creative strengths. And that piece is a piece that one of my students did. They did their own take on a Sierpinski triangle. I have a few of those items from my office where they integrated visual arts and integrated creative aspects of mathematics from cultural aspects as well, too.

KK: So I always think of Native American artists being kind of geometric in nature. It feels that way to me, I mean, at least the limited bit that I've seen. Is that sort of generally true?

BT: The thing about Native art is that Native cultures are diverse in and of themselves too. So there are over 500 federally-recognized tribes, and in Mexico are over 20 tribes alone, 20 nations alone, and each of them have their own notions of geometry and their own notions of their kinds of mathematics that they engage in with respect to the place that their cultures, their identities, and their languages are rooted in. So, yeah, a lot of it is visual, and geometric, because that's what we see. But there's also many I imagine that we don't see, that's embedded in the languages and the practices. Part of my curiosity is seeing how we can recognize what we do and what our traditions are, how we can recognize that as mathematical. And it might be mathematical in the sense that we, as professional mathematicians, might not be accustomed to seeing or experiencing. And, you know, I'm still trying to understand my own cultures, languages and traditions too. So I know mathematics more than a lot of how I experience my own culture. So on one hand, I'm seeing things from a traditional mathematician brought through academia, but I’m also trying to understand things through the lens of someone who's trying to better understand my cultures and histories.

EL: So what is your favorite theorem?

BT: The theorem I chose today was Arrow’s impossibility theorem.

KK: Nice.

EL: Great. And this will be a timely one, at least for the US, because it will be airing—I mean, I guess the past two years basically have been part of the 2020 presidential season—but really in the thick of it. So yeah, tell us a little bit about what this is.

BT: So I'll say more about why I'm kind of drawn to this theorem. So it's a theorem that basically says that there is no perfect ranked voting system, or no perfect way of choosing a winner and, by extension, for me, it kind of brings up conversations about how democracy itself isn’t perfect and that it's really hard to say that a democratic system can accurately represent the will of the people. And I was drawn to this theorem because as I started thinking about the cultural aspects of mathematics and mathematics education, I'm also interested in the power dynamics and the political dynamics and the sociopolitical aspects of mathematics and math education. And a lot of what's out there and written about math education talks about using quantitative reasoning and quantitative analysis and statistical analysis to really engage in critical dialogues and examining inequities and injustices in the world. And all of that is rich and engaging and needed and necessary ways that we can use mathematics to view the world. But the mathematician part of me still misses the definition-proof-lemma aspect of engaging in mathematics. So this theorem kind of represents a way of engaging in politics through some of the theorem-definition- lemma aspects of it. So the way that I understand Arrow’s theorem, and I mentioned this to you before, that I don't know the ins and outs of this theorem, I just really like the ramifications of it and the discussions that it generates. But it basically starts with the idea that we can describe functions where we're considering a way of choosing a winner of an election from a list of candidates. And we're taking each voter’s ranked preference of those candidates. So one thing that we're assuming is that each voter can rank a list of n candidates, A1 through An, and if everyone can rank their preferences, then a voting system would be a way to take all of those, those ranks, or those ballots, and choosing an overall ranking that is supposed to indicate an overall preference for the group of voters.

And what Arrow’s impossibility theorem talks about is that we want values, and want to describe good ways of what a good voting system is. So we want to describe list of criteria that shows that we have a good voting system. So the list of criteria that involves Arrow’s impossibility theorem talks about 1) and unrestricted domain; 2) social ordering; 3) weak Pareto or unanimity; 4) a non-dictatorship; and 5) independence of irrelevant alternatives. And I'll go through what each one means. So basically, an unrestricted domain means that we want a voting system or a way of choosing a winner to be able to take any set of ballots with any number of candidates and be able to give some overall ordering, that these functions are well-defined. So the unanimity condition talks about if everyone prefers one candidate over another, where every single voter has one candidate ranked over another candidate, then the overall function that turns the ballots into an overall social ordering should indicate that that candidate is preferred over the other candidate. And we also don't want a dictatorship, right? And the idea of that mathematically defined is that we don't want one voter deciding exclusively what the overall social ordering is of the candidates. And so we don't want a dictatorship. And we want an independence of irrelevant alternatives, and what that what a lot of people think about as an example of is a “spoiler” candidate or a third party candidate, where even if everyone prefers one candidate over another, that a change in order of a third or other candidate, without disrupting that other order, shouldn't change the overall outcome of an election. They relate that to how sometimes third party candidates can be a spoiler for an election even though overall, it looks like a plurality of voters might prefer one candidate over another. But certain voting systems can have that characteristic where third or other other set of candidates can disrupt the outcome of that election.

KK: I’ve never heard of that.

EL: Wouldn’t it be terrible if that ever happened? [Note: These statements were delivered somewhat sarcastically, presumably referring to the 2000 Presidential election in the US]

BT: Right, right, right. So what Arrow’s impossibility theorem says is that those all may be desired characteristics of a voting system or a social choice function, but that it's impossible to have all of those criteria in a voting system. So the general outline of the proof is that if we have a system that has the unanimity criterion, and an independence of irrelevant alternatives, that if we have those two criteria in a social choice function, then the voting system must be a dictatorship. So if we add those assumptions, then we can go through and show that there is a voter whose sole ordering determines the overall ordering of the voting group, of the voters.

KK: That’s how I always learned this theorem, is that you set down these minimal criteria, and the only thing that works as a dictatorship, right?

BT: Right.

KK: These criteria are completely reasonable, right?

EL: You can’t have it all.

BT: Right, right. They're not outlandish. They're what we might think of as things that we might value in a democracy. And, of course, these, these things don't perfectly replicate what's going on in the real world, but the outcome is still fascinating to me that mathematically, we can show that we can’t have all these sets of what we think are reasonable criteria in a voting system.

KK: Recently, maybe in the last two years, I’ve been getting interested in gerrymandering questions. And there's there's a similar sort of theorem that got proved in the last year or two, which essentially says that, you know, people don't like these sort of weird-shaped districts, they think that's bad somehow, because it's on unpleasing to the eye. But apparently — and there’s also this idea of the efficiency gap, where you sort of want to minimize wastage. So if you laid out some simple criteria, like you want compact districts, and you want to make the efficiency gap, minimized that, then the theorem is you have to have weird shape districts, right? So it’s sort of an impossibility theorem in that way too. So these these kinds of ideas propagate through all of these these kinds of systems,

EL: The real world is impossible.

BT: Right. And even by extension, you know, in many voting theory classes, there's a districting problem, which relates to a good metric for measuring compactness. But then the apportionment issue as well, that it's very hard, if not impossible, to find a fair way of apportioning a whole number of representatives that's proportionate to the state's population, relative to the overall population of the country.

KK: Yeah.

BT: And so yeah, this is one of my favorite theorems because it kind of opens the door to those conversations and gives me another way of thinking about when representatives, or people who talk about the outcomes of elections, say things like “the people have spoken,” “this is the will of the people,” “we have a mandate now,” that I think these outcomes really complicate those claims and should really give us a critical eye and a critical way of really discussing what the will of the people is, and how those discourses really perpetuate the idea that voting, and voting alone, can accurately indicate the will of the people and that that's to be accepted, and that we move forward with them.

EL: Yeah. So have you gotten to use these Arrow’s paradox or any of these other things in classes?

BT: When I was at the Institute of American Indian Arts, I tried to develop a voting theory class. And we got into that and talked about that. And it interested me too because the voting system on the Navajo Nation, we vote for our own council and our own presidents too, and I use this as a way to think about how we have a certain candidate in Navajo Nation who's always running and is seemingly unpopular. And the voting system for president in Navajo Nation is that we have that two-party runoff system where we vote for our top choices and that the top two vote getters participate in a general runoff election. And for a few consecutive elections, this one candidate that is seemingly unpopular just gets enough votes to get into the top two for the runoff election and then gets overwhelmingly outvoted in the general election. So I think for me it was a fascinating way to engage in these kind of mathematical ideas, or mathematical discourses, while talking about some of the real outcomes that are going on in our nations, in our communities, in our efforts towards our self-determination and sovereignty. So I wanted to tie in something that's mathematical, where we can talk about mathematical discussions, with issues that are contemporary and real to our, our peoples.

EL: It’s something I always wonder about is, you know, we've got a theorem that says voting is impossible — or it says that, you know, it's impossible to actually say, like, this is the will of the people. But do you know if much research has been done about, like, real sets of choices that people have and what voting systems might be — do they really experience this paradox, or in the real world, do they have these strange orders of preferences that that confound ranked choice voting rarely?

BT: I imagine that there is research out there and there are people who have engaged in it much more than I have. But something that makes me curious are some of the underlying assumptions that go into Arrow’s theorem and what has been mathematized as necessary criteria, and the values that those might be representative of for certain groups of people. For example, I guess you could call it an axiom of many these voting theory theorems in mathematics is that one voter is one vote, and you know, there are systems where that might not be true. But one of your criteria is one person, one vote. And that one person votes for their own interests and their own interest only, and there are extensions of these criteria where if we have other non-ranked voting systems, then it can help.

But let me backtrack: one of the outcomes of Arrow’s theorem is that when people know that it's impossible for the outcome to really represent the will of the people, then it could result in people voting for candidates other than their first option because they know that voting for someone other than their true option because we election in favor of something that's not of their desire. So we have people voting against their own actual first choices. And that happens with ranked-choice voting, and some of the extensions of these conversations have been about voting systems that don't require ranked choice. So perhaps giving each candidate a rating, and it helps alleviate some of those issues with ranked-choice voting, and it helps alleviate those issues of third-party candidates, where you can still give your candidate five stars out of five, like an Amazon review, but still really give perhaps a better indication of your true view of the candidates, rather than a linear ranking. So it kind of reveals that there are some issues with just linear ranking of candidates, when the way that we think about in value and understand our preference of candidates might be much more complex than a simple 1 through n ranking. But kind of going back to what I think this could mean for communities and other societal perspectives, is in many democracies, that one vote-one choice is kind of an assumption that that's what we want. But for many communities, perhaps we want to vote for something that does benefit an overall view of the people. What would that look like as a criteria if we allowed for something like that? What would we do if we allow criteria, or embedded in our definitions, some way of evaluating how if when we register a vote, that we're all not only taking into account our own individual interests, but the interests of our land, of our communities, of our nations. So those are cultural values that are not assumed in the current conversations, but for many communities in many Indigenous nations, those are some things that are real and necessary to think about. What would that look like if we expand those and then be critical of those assumptions that are underlying these current conversations on voting theory in mathematics.

EL: So one of the other things we do on this podcast is We ask our guests to pair their theorem with something. What have you chosen to pair with this theorem?

BT: I have a ranking of three pairings.

EL: Great. I’m so glad! Excellent.

BT: So I have 1-2-3. So I'll give my third choice first. The third out of three pairings: green chili cheeseburgers.

EL: Okay.

BT: And in New Mexico, everyone has their favorite place to get a green chili cheeseburger, and we take pride in our green chili, and every year any contest about the green chili cheeseburger and who has the best green chili cheeseburger causes some conversation, and it causes some controversy and rich discussions over who has the best green chili cheeseburger. So, I think about that as a food that has a lot of controversy as to who has the best green chili cheeseburgers in New Mexico. The second pairing is another food item, the Navajo taco.

EL: Oh yeah. Those are good.

KK: What’s in those?

BT: So, well, what we call a Navajo taco is a piece of frybread with toppings often involving meat and cheese, with lettuce and tomato and maybe some chili. And this is another controversial discussion in Native communities because we call it a Navajo taco, but it's not just Navajos who make this kind of dish, because many communities make their own versions of frybread. And so some places call it Indian tacos, and there's a lot of controversy over which community first introduced the Navajo taco and why some people call it the Navajo taco and others call it Indian tacos. And so in Native communities, there's a lot of controversy over what constitutes the best version of this dish. And the other reason I'm pairing that is the frybread itself comes from a time where it was created out of necessity for survival, where the flour that had been rationed out to our communities was rancid, and in order to actually make it edible, it was deep fried. And so on one hand, it represents a point in time where our communities were just fighting for survival, and it also represents their ingenuity, and became a part of our everyday practice. But at the same time, it's a reminder that that was something that was imposed on our communities, much like voting systems nowadays. It's an act of our survival and our sovereignty, the voting systems that we have in place. But I think there's also need to come back and have other conversations about what's good for our communities.

And the first-ranked pairing is mathematics itself with Arrow’s theorem. So we have a lot of conversations about how mathematics is universal, mathematics is for everyone, that everyone can do mathematics, and that everyone can participate in mathematics. But for many people from from equity, justice and diversity perspectives, we want to be critical about who has access to mathematics, whose ideas of mathematics are represented in our mainstream ways of thinking about mathematics. Just like we think about democracy as being the will of the people and being a representation of all the people, that Arrow’s is kind of a critique of that notion of democracy. And I think mathematics, we can take a lesson from this theorem and think about what we mean when we say mathematics is universal or mathematics is for everyone or mathematics is for all, when this term itself is kind of a democratic take on mathematics, that everyone can do mathematics, and everyone can be an equal participant in mathematics. But, you know, we think the same thing about democracy, and this theorem says that there are some issues with that. So I'm interested in seeing how we can take this lesson and how we can think about how we can be more critical about the ways we think about mathematics itself.

EL: Yeah, well, you know, Arrow’s paradox is not about this, but we have issues with people who can't vote for various reasons and should be able to vote, or places that shut down polling places in certain communities to make it so people have to stand in line for six hours. Which is, you know, not easy to do if you've got a job that you need to get to. So yeah, there's so much richness. I love that you paired a ranking of three things with this. And now I feel like we should also vote on these, but I just don't think it's fair for one of them to be math. I mean, you’ve got two mathematicians here, three mathematicians here in total. I think it's going to be a blowout.

KK: No, tacos win every time, don’t they?

EL: I should have known.

KK: This is a really good pairing. I like this a lot.

EL: Yeah.

KK: We also like to give our guests a chance if they want to plug anything. Where can we find you online for example, or can we?

BT: Probably the best way to find me is on Twitter. My Twitter handle is @lobowithacause.

EL: Yeah. You'll see him popping up everywhere. Is that the mascot for the University of New Mexico?

KK: It is, the lobos.

EL: And I believe a talk that you gave at the Joint Math Meetings, is there video of that available somewhere?

BT: I was told that there would be video. I haven't found it yet. There was a video recorded. And I'll follow up with that and see that it gets out. I'll make an announcement on Twitter.

KK: I’ve noticed those have been trickling out kind of slowly. It'll show up, I think.

EL: Yeah, we'll try to dig it up by the time we put the show notes together so people can watch that. Unfortunately, I was still making my way to Denver when that happened, so I didn't get to see it. So selfishly I very much want to see it. I heard really good things about it. So thank you so much for coming on here and giving us a lot to think about.

BT: Oh, it was an honor. And you know, I love your podcasts.

KK: Thanks so much.

BT: I love what you’re doing. I had fun in listening to your other podcasts in preparation for this and loved hearing Henry Fowler and shout out to Moon Duchin too. I heard that you, Kevin, went to that gerrymandering work in Boston a few years ago. I was there too. And I had a great week there.

EL: Oh, nice.

KK: That was a big workshop. There was no way to meet everybody. Yeah,

EL: Thanks for joining us, and have a good rest of your day.

BT: Thank you. Thank you. You too.

In this episode of the podcast, we were happy to talk with Belin Tsinnajinnie, a professor at Santa Fe Community College, about Arrow's impossibility theorem, which basically says that a perfect voting system is impossible. Below are some links you might enjoy as you listen to the episode.
Arrow's impossibility theorem
Cardinal voting, an alternative to voting systems that are based on ranking the options
Our episode with Henry Fowler, who was at the time on the faculty of Diné College and is now at Navajo Technical University
Our episode with Moon Duchin, who studies gerrymandering, among other things
Belin Tsinnajinnie on Twitter

Episode 55 - Rebecca Garcia

11 juni 2020 | 25 min

Evelyn Lamb: Hello and welcome to my favorite theorem. Math podcast. I'm one of your hosts Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right, it is a bright sunny winter day today, so I really like—I mean, I'm from Texas originally, so I'm not big on winter in general, but if winter has to exist, sunny winter is better than cloudy winter.

KK: Sure, sunny winter is great. I mean, it's a sunny winter day in Florida, too, which today means it is currently, according to my watch, 81 degrees.

EL: Oh, great. Yeah.

KK: Sorry to rub it in.

EL: Fantastic. It is a bit cooler than that here.

KK: I’d imagine so.

EL: So yeah. Anything new with you?

KK: No, no. Well, actually so so my I might be going to visit my son in a couple of weeks because he's studying music composition, right? And the the orchestra at his at his university is going to play one of his pieces, and so kind of excited about that.

EL: Very exciting! Yeah, that's awesome.

KK: Yeah, but that's about it. Otherwise, you know, just dealing with downed to trees in the neighborhood. Not in our yard, luckily, but yeah, stuff like that. That's it.

EL: Yeah. Well, we are very happy today to have Rebecca Garcia as a guest. Hi, Rebecca. How are you?

Rebecca Garcia: Hi, Evelyn. Håfa ådai, I should say, håfa ådai, Evelyn, and håfa ådai, Kevin. Thanks for having me on the program.

EL: Okay, and what—håfa ådai, did you say?

RG: Yeah, that's right. That's how we, that's our greeting in Chamorro.

EL: Okay, so you are originally from Guam, and is Chamorro the name of a language or the name of a group of people, or I guess, both?

RG: It’s both actually. Yes. That's right. And so Chamorro is the native language in the island. But people there speak English mostly, and as far as I'm able to tell I think I'm the first Chamorro PhD in pure mathematics.

EL: Well, you’re definitely the first Chamorro guest on our show. I think the first Pacific Island guest also.

KK: I think that's correct. Yeah.

EL: So yeah, how did you—so you currently are not in Guam. You actually live in Texas, right?

RG: I do. I'm a professor at Sam Houston State University, which is in Huntsville, Texas, north of Houston. And I'm also one of five co-directors of the MSRI undergraduate program.

EL: Oh, nice. That seems like it is a great program. So how did you how how did you get from Guam to Huntsville?

RG: Oh my goodness. Wow. That is a that is a long, long journey.

KK: Literally.

RG: I started out as a as a undergraduate at Loyola Marymount University, and I had the thought of becoming a medical doctor. And so I thought we were supposed to do some, you know, life science or you know, chemistry or biology or something along those lines. And so I started out as one of those majors and had to take calculus and fell in love with calculus and the professors in the math department. And I was drawn to mathematics. And that's how I ended up on the mathematics side. And one of the things that I learned in my undergraduate career was these really crazy math facts about the rational numbers. And so that's one of the things that interested me in mathematics, was just the different types of infinities the concept of countable, uncountable, those sorts of things.

EL: Yeah, those those seem to be the kinds of facts that draw a lot of people into this rich world of creativity and math that you might not initially think of as related to math when you're going through school. So I think this brings us to your favorite theorem, or at least the favorite theorem you want to talk about today.

KK: Sounds like it.

EL: Yeah, so what’s that?

RG: Yeah. So it’s more, I would say, more of a fun fact of mathematics that the rationals first of all are countable, meaning they are in one-to-one correspondence with the natural numbers. And so you can kind of, you know, label them, there's a first one and a second one in some way, not necessarily in the obvious way. But then, at the same time, they are dense in the real numbers. So that to me, just blows my mind, that between any two real numbers, there's a rational number.

EL: And yeah, so you can't like take a little chunk of the real line and miss all the rational numbers.

RG: That’s right.

KK: Right.

RG: That to me just blows my mind. Because—and then you just sort of start, you know, your brain just starts messing with you, you know, between zero and one there are infinitely many rational numbers and yet they're still countable. And it just, it just starts to mess with your mind a little bit. Right?

EL: Yeah. Well, and we were we were talking about this a little bit before and it's this weird thing. Like, yeah, there's, like a countable is like a smallness thing. And yet dense is like, they're, you know, they fill up the whole interval this way. I mean, it is really weird. So where did you first encounter this?

RG: This was in a class in real analysis. And, yeah, so that's where I started to…I thought I was going to be a functional analyst. I thought I was that's what I wanted to do this. I love real analysis. That didn't happen either. But it was in that class where we were talking about just these strange facts, like the Cantor set: that set is a subset of the reals that is uncountable and yet it’s sparse.

KK: Totally disconnected, as the topologists say.

RG: Totally disconnected. There you go. Yeah. Right. And so then all these weird things are happening. And you're just in this world where you thought you understood the real line, and then they throw these things at you like, the reals are dense. I mean, the rationals are dense in the reals, you have these weird uncountable sets that are totally disconnected. What's going on? Yeah, so that's where I started to hear about all these weird things happening.

KK: Right. So one of two things happens when people learn these things, right? It either blows their minds so much they can't keep going. Or it intrigues them so much that you want to learn more. But not be an analyst. Right?

RG: [laughing] That’s right. At some point I fell in love with computational algebraic geometry and these Gröbner bases, and how you can really get your hands on some of these things and their applications to combinatorics. So I ended up, I had an algebraist’s heart, but I was exposed to some really good analysts early in my career. And so I was very confused. But I've always, I stay true to my algebraic heart and follow that mostly.

EL: And so is this a fact that you get to teach to your students now ever?

RG: So no, this is not, but I do like to talk about the the different infinities and things along those lines. And I like to, before class I come in early, and I'll have a little chat with them about just the fact that—you know, they they don't understand that math is not “done.” So, there's still so much to do. And they have no idea that, you know, there's what, what is research like? What does that mean? And so I talk about open questions. And I bring some of that in the beginning of class. And these concepts that had also drawn me in, about the different kinds of infinities and these weird concepts about the rationals being dense and, you know, just things like that. I do get to talk about it, but it's not in a class that I would teach the material on.

EL: Yeah, just going back to this idea that you've got the rationals that are dense, so it's this, like, measure zero small set, but it's like everywhere. And then you've got the Cantor set, which is uncountable and sparse. It's like, we've got these various ways of measuring these sets. And you think that they line up in some natural way. And yet they don't. It's just like, you know, the density is measuring a different type of property of the numbers than the measure is.

RG: That’s exactly.

EL: And actually, I guess countability is a different thing. Also, I mean, it's, yeah, it's so weird. And it's hard to keep all these things straight. My husband does a lot of analysis and like has, yeah, all of these, like, what kinds of sets are what.

RG: And what properties they have. And yeah, I don’t have that completely straight.

KK: This is why I’m a topologist.

EL: But I mean, topology is like,

KK: Oh, it's weird too.

EL: It’s secretly analysis.

KK: Well…

EL: Analysis wishes it was topology, maybe.

KK: So my old undergraduate advisor—who passed away last summer, and I was really sad about that—but he always he always referred to topology as analysis done right.

EL: Shots fired.

KK: Which is cheap, of course, right? Because you prove all this stuff in topology Oh, the image of a connected set is connected. Yeah, that's easy now go off to the real line and prove that the connected sets are the intervals. That's the hard part. Right? So yeah, he's being disingenuous, but it was. It's a good line. Right.

RG: Right.

EL: So you said that you ran into this, was this an undergraduate class where you first saw these notions of countability and everything?

RG: Right, it was an undergraduate class where I ran into those notions and I was a junior, well, I guess it was in my second semester as a junior, where we were talking about these strange sets. And that's when I had also thought about going on to graduate school and wanting to do mathematics for the rest of my life. I mean, I was a major by then, of course, but I just didn't know what I was going to do. But it wasn't until then, when I learned about, well, this is this could be a career for you. This may be something you like to do. And of course, this was many, many years ago. And nowadays, you can do so much more with mathematics, obviously. I mean, we know that we can do so much more, I should say. We've always been able to do so much more. We just haven't been able to share that with our students so much. We never really spent the time to let them know there's so many careers and mathematics that one can do. But anyway, at that, at that time I was I was drawn into really thinking about becoming a mathematician, and that was one of the experiences that that made me think that there's so much more to this than than I originally thought.

EL: Yeah, well, I talk to a lot of people, you know, in my job writing and doing podcasts and stuff about math, and there's so many people who don't realize that, like, math research is a career you can do.

RG: Right.

EL: And the more we can share these kinds of “aha” moments and insights, the better and, you know, just show like, well, you can use, you know, kind of the logic and the rules of the game to like, find out these really surprising aspects of numbers.

RG: Right. And I think also, one of the experiences that I've had as an undergrad that really just sort of sealed the deal—I’m going to go into mathematics—was doing an undergraduate research program as a student. Well wasn't really at the time an undergraduate research program, it was just another summer program. This is many years ago, almost before all of that. And I had the chance to spend a summer just thinking about mathematics at a higher level with a cohort of other students who were like-minded as well, you know. And it was really—it was it was like, “Oh, I can do this for the rest of my life? Like how amazing is that?” And so, I was part of a summer program as an undergrad. And then when I was a graduate student, my lifelong mentor, Herbert Medina, was running a program in Puerto Rico and asked me to be a TA while I was a grad student. And so these were some of the things that led me to do what I do now, working with undergraduates, doing research and mathematics.

EL: And so that ties in to the MSRI program that you are part of, right?

RG: Right.

EL: I guess it I've seen it written like MSRI-UP. So I guess that's undergraduate program?

RG: Yes. Undergraduate Program. That's right. Yeah. Well, that that's sort of like, a different stage that I'm at now. But yeah, before that, I started my own undergraduate research program together with colleagues in Hawaii, at the University of Hawaii at Hilo. And we ran an undergraduate research program called PURE Math, and that was Pacific Undergraduate Research Experience in Mathematics. And we ran that for five years. And then, and then I ended up moving into the co-director role at MSRI-UP.

EL: Nice.

KK: That’s a great program.

EL: Yeah. So the other thing we like to ask our guests to do, is to pair their theorem with something. You know, just like the right wine can enhance that meal, you know, what would you recommend enjoying the density of the rationals with?

RG: Well, I did think about this a bit. And one of the things that I think, you know, you think the rationals are dense but they really shouldn't be? So, I think of foods that are dense, but they really shouldn't be, and one of those foods that comes to mind, especially being here in Texas, but also being married to a mathematician who is from Mexico, is tamales. So tamales really should not be dense. They should be fluffy and sumptuous, but here in Texas, you find really dense the most, unfortunately. But it It was strange to also discover that growing up in Guam, we also have our own version of tamales, and a lot of the foods are related in some way to foods from Mexico. So I feel like there's this huge rich connection between myself being from Guam, my husband being Mexican and there's just this strange richness that we share this culture, that I don't know, it just blows my mind too. So the same way that the rational is being dense in the reals blows my mind.

EL: All right, well, I have to ask more about this tamale like creation from in traditional Guam cuisine. What, is that wrapped in, like, banana leaves or something like that?

RG: It ought to be, and maybe traditionally it was. I think that nowadays it's not that way. They usually serve it in aluminum foil, and it's made—it's a mixture like tamales. So tamales in Mexico are made with corn, right?

KK: I was about to ask this. What are they made of in Guam?

RG: Yeah, yeah. And so in Guam we actually use, like, a rice product.

EL: Okay.

RG: It's ground up just like corn. And so instead of corn, we're using rice, and it's flavored in different ways.

KK: Interesting.

EL: All right. I have kind of in my mind because I'm more familiar with this like almost, is it kind of like a mochi texture? Because, I mean, that's a rice product, but maybe it's not maybe that's like more gelatinous than this would be.

RG: Yeah, I guess mochi is really pounded and yeah, so yeah, that's more chewy. I think that the tamal, well, you wouldn't say it like that, but the tamales in Guam are very soft and, gosh, I don't know how to describe it. But it's a very soft textured food.

KK: I would imagine the rice could be softer, and I mean, corn can get very dense, especially when you start to put lard in it and things like that.

RG: Yes.

KK: I mean, it’s delicious.

RG: It is delicious. And oh my, I can’t get enough tamales. Oh, well.

KK: Yeah, maybe you can.

RG: Yeah, I should learn.

EL: Yeah, well, nice. I unfortunately, we do have a couple restaurants in Salt Lake that are Pacific Island restaurants, but we have more people from Samoa and Tonga here. I don't know if we have a lot of people from Guam here. Yeah, there's actually like a surprising number of like, Samoans who live in Salt Lake. Who knew?

RG: Right.

EL: But yeah, it's it's because of like the history of Mormon missionaries.

KK: That’s what I was gonna say.

EL: Yeah, the world is very interesting, but yeah I don't know if I've seen this kind of food there. I will just have to, you know, if I'm ever in Huntsville I’ve got to get you to make me some of this. I’m just inviting myself over for dinner now. Hope you don't mind.

RG: That would be great. It would be wonderful to have you here.

EL: Is there anything else you'd like to share? We'd like to give our guests a chance to like, share, you know if they've got a website or blog or book or anything, but also if you want to share information about MSRI-UP, application information, anything like that for students? Anything you'd like to share?

RG: Oh, wow. That's a lot of stuff.

EL: Yeah, I know. I just rattled off a ton of things.

RG: Well, yes, I do have, I guess I would like to say for the undergraduate listeners in the audience, please consider applying to our MSRI-UP program, and just in general apply to a research program in the summer. These are paid opportunities for you to expand your mind and do some mathematics in a great environment, and so I highly recommend considering applying for that. And so this is the time right now of course by the time the listeners hear this, I’m sure it will be over, but consider doing some undergraduate research or using your summer wisely.

KK: I parked cars in the summer in college. I did.

EL: Well, you never know the connections that might happen though because I was talking to someone one time who basically his big break to get to go to grad school came because, like, somehow he was involved in like parking enforcement somewhere, and some math professor called in to complain about, like, getting a ticket, and one thing led to another and then he ended up in grad school. So really, you never know. Maybe that's not the ideal route to take. There are more direct routes, but yeah, there are many paths.

RG: Yes, there are. And there's also another, I guess another thing to flag would be, well, contributed to a book that Dr. Pamela Harris and others have put together on undergraduate research. So that just I guess that was just released. I'm not entirely sure now. I think it was accepted, and I don't know if if one is able to purchase it, but if you if you consider working with your students on undergraduate research, this is a great resource to use to get you going, I guess.

KK: Great.

EL: Oh, awesome. So this is like a resource for like faculty who want to work with undergraduates? Oh, that's great.

RG: Yes.

EL: We will find a link to that and put that in the show notes for people.

RG: That sounds good.

EL: Okay, great. Thanks so much for joining us.

KK: It’s been great.

RG: Thank you so much.

On this episode of My Favorite Theorem, we were happy to talk with Rebecca Garcia, a mathematician at Sam Houston State University, about the density of the rational numbers in the reals. Here are some links you might find helpful.

Her website
A biography of Garcia for SACNAS
MSRI-UP
A Project-Based Guide to Undergraduate Research in Mathematics, the book she mentioned contributing to

Episode 54 - Steve Strogatz

14 maj 2020 | 33 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida, and here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer based in Salt Lake City, where it is snowy, but I understand not as snowy as it is for our guest.

KK: I know, and we've been trying to make this one happen for a long time. So I'm super excited that this is finally going to happen. So today we are pleased to welcome Professor Steve Strogatz. Steve, why don't you introduce yourself?

Steve Strogatz: Well, wow, thank you. Hi, Kevin. Hey, Evelyn. Thanks for having me on. Yeah, I've wanted to be on the show for a very long time. And I think it's true what Evelyn just said, we have a very big snowstorm here today in not-so-sunny Ithaca, New York, upstate. I just took my dog out for a walk, and the snow was over my boots and going into them and making my feet wet.

KK: See, I have a Florida dog. She wouldn't know what to do. Actually, we were in North Carolina a few years ago at Christmas, and it snowed, and she was just alarmed. She had no idea what to do. And she's small, too, she just couldn't take it.

SS: Yeah, well, it would be more like tunneling than running.

KK: Right.

EL: Yeah. So we actually met quite a few years ago at this point — actually, I know the exact date because it was, like, two days before my brother's wedding the first time we met because you were on the thesis committee for my sister in law, who is a physicist, many years ago, and so we have this weird, it was when I had just moved to New York to work at Scientific American for the first time. So it was at the very beginning of my life as a math writer. And I remember just being floored by how generous you were with being willing to meet with a nobody like me.

SS: Well that’s nice.

EL: At this time when I was first starting.

SS: But actually, I had a crystal ball, and I knew you were going to become the voice of mathematics for the country, practically. I mean, so I let me brag on Evelyn’s behalf a little bit. If you go on Twitter, you—I wonder if you know this, Kevin, do you know this little factoid I'm going to unreel?

KK: I bet I do.

SS: You know where I'm going. On Twitter, if you ask “What mathematician do other mathematicians follow?” I think Evelyn is the number one person the last time I checked.

KK: She is indeed number one. That's right.

SS: Yeah.

EL: I like to say I'm the queen of math, Twitter, although I don't actually like to say this because it feels really weird.

SS: Well that’s okay. You didn't say it. But yeah, I do remember our meeting that day in my office. And right, it was on this happy occasion of a family, of a wedding. Okay, sorry, I interrupted you, Kevin.

KK: Oh, I don't know. I was going to say with the Twitter thing. I think you're not far behind, right? Like, aren't you number two, probably?

SS: I think the last time I looked I was number two.

KK: Yeah.

SS: So look at that. Okay, so look at that, the two tweet monsters here.

KK: And now the funny thing is I'm not even on that list. So here we go.

SS: Okay. Yeah, well you could catch up. I'm sure you'll be coming right on our heels.

KK: Maybe. I have over 1000 followers now, but apparently not that many mathematicians. So this is how this goes. Anyway, what weird times we live in, right?

SS: It's very weird. I mean, I don't know what this can get us, a cup of coffee or what.

KK: Maybe, maybe. Okay. Let's talk theorems. So Steve, you must have a favorite theorem. What is it?

SS: Yeah, I have a very sentimental attachment to a theorem and complex analysis called Cauchy’s theorem, or sometimes called Cauchy’s integral theorem.

KK: Oh, I love that theorem.

SS: It’s a fantastic theorem. And so I don't know. I mean, I feel like I want to say what I like about it mathematically and what I like about it personally. Does that work?

EL: Yeah, that’s exactly what we want.

SS: Well, okay. So then, the scene is, it's my sophomore year of college. Maybe I'll start with the emotional.

KK: Okay.

SS: It’s my sophomore year of college. I've just gotten very demoralized in my freshman year, taking the the honors linear algebra course that a lot of universities offer as a kind of first introduction to what college math is really going to be like. You know, a lot of kids in high school have done perfectly well in their precalculus and calculus courses, and then they get to college and suddenly it's all about proofs and abstraction. And it can be—I mean, we sometimes call it a transition course, right? It's a transition into the rigorous world of pure math. And so it was a shock for me. I had a lot of trouble with that course. I couldn't read the book very well, it didn't have pictures. And I'm kind of visual. And so I was always at a loss to figure out what was going on. And being a freshman I didn't have any sense about, why don't I look at a different book, you know, or maybe, maybe I should switch sections. Or I could ask my teaching assistant, or I could go to office hours. I didn't know to do any of that stuff.

So anyway, this is not my favorite theorem. I was very demoralized after this experience in linear algebra. And then when I took a second semester, also an honors course, that was a rigorous calculus course with the Heine-Borel theorem, and, you know, like, all kinds of—again, no formulas, it was all about, I remember hearing this stuff about “every open cover has a finite subcover,” and I thought, “I want to take a derivative! I can't do anything here. I don't know what to do!” So anyway, after that first year, I thought, “I don't have the right stuff to be a mathematician. And so maybe I'll try physics,” which I also always loved. I say all that as preamble to this complex analysis course that I was taking in sophomore year, which, you know, I still wanted to take math, I heard complex variables might be useful for physics, I thought it would be an interesting course. I don't know. Turned out it was a really great course for me because it really looked a lot like calculus, except it was f(z) instead of f(x).

KK: Right.

SS: You know, but everything else was kind of what I wanted. And so I was really happy. I had a great teacher, a famous person actually named Elias Stein.

KK: Oh.

SS: So Stein is a well-known mathematician, but I didn't know that. To me, he was a guy who wore Hush Puppies and, you know, had always kind of a rumpled appearance, came in with his notes. And he seemed nice, and I really liked his lectures. But so one day, he starts proving this thing, Cauchy’s theorem, and he draws a big triangle on the board. And he's going to prove that the integral of an analytic function f around this triangle is zero no matter what f is. All he needs is that it's analytic, meaning that it has a derivative in the sense of a function of a complex variable. It's a little more stringent condition—actually a lot more stringent than to say a function of a real variable is differentiable, but I didn't appreciate that at the time. I mean, that's sort of the big reveal of the whole subject.

KK: Right.

SS: That this is an unbelievably stringent condition. You can’t imagine how much stuff follows from this innocuous-looking assumption that you could take a derivative, but okay, so I'm kind of naive. Anyway, he says he's going to prove this thing, only assuming that f is analytic on this triangle and inside it. And that's enough. And then, you know, I feel like you don't have enough information, there's nothing to do! So then he starts drawing a little triangle inside the big triangle, and then little triangles inside the little triangle. And it starts making a pattern that today I would call a fractal, though I didn't know it at the time, and he didn't say the word fractal. And actually, nobody ever says that when they're doing this proof. But it’s—right, they don’t—but it's triangles inside of triangles in a self-similar way that doesn't actually play any particular role in the proof, other than it's just this bizarre move, like, What is going on? Why is he drawing these triangles inside of triangles? And by the end, I mean, I won't go into the details of the proof, but he got the whole thing to work out, and it was so magnificent that I started clapping.

And at that point, every kid in the room whipped their head around to look at me, and the professor looked at me, like what is wrong with you? You know, and yet, I thought, “Wow, why are you guys looking at me?” This was the most amazing theorem and the most amazing proof.” You know, so anyway, to me, it was a very significant moment emotionally because it made me feel that math was, first of all, something I could do again, something I could appreciate and love, after having really been turned off for a year and having a kind of crisis of confidence. But also, you know, aside from any of that, it's just, I think people who know would regard this proof —this is actually by a mathematician named Goursat, a French mathematician who improved on Cauchy’s original proof. Goursat’s proof of Cauchy’s theorem is just one of the great— you know, it's from “The Book” in the words of Paul Erdős, right? If God had a proof of this theorem, it would be this proof. Do you guys have any thoughts about that? I mean, I'm assuming you know what I'm talking about with this theorem and this proof.

KK: Well, this is one of my favorite classes to teach because everything works out so well. Right? Every answer is zero because of Cauchy’s theorem, or it's 2πi because you have a pole in the middle, right?

SS: Yeah.

KK: And so I sort of joke with my students that this is true. But then the things you can do with this one theorem, which does—you’re right, it's very innocuous-looking, you know, you integrate an analytic function on a closed curve, and you get zero. And then you can do all these wonderful calculations and these contour integrals and, like, the real indefinite integrals and all this stuff. I just love blowing students’ minds with that, and just how clean everything is.

EL: Yeah, I kind of—I feel like I go back and forth a little bit. I mean, like, in my Twitter bio, it does have “complex analysis fangirl.” And I think that's accurate. But sometimes, like you said, it's so many of these, you know, you're you're like teaching it or reading it and you're like, “Oh, this is complex analysis is so powerful,” but in another way, it's like our definition of derivative in the complex plane is so restrictive that like, we're just plucking the very nicest, most well-behaved things to look at and then saying, “Oh, look what we can do when we only look at the very most well-behaved things!” So yeah, I kind of go back and forth, like is it really powerful or are we just, like, limiting ourselves so much in what we think about?

KK: And I guess the real dirty secret is that when you try to go to two complex variables, all hell breaks loose.

SS: Ah, see, I've never done that subject, so I don't appreciate that. Is that right?

KK: I don't, either. Yeah. But I mean, apparently, once you get into two variables, like none of this works.

SS: Ohhh. But that's a very interesting comment you make there, Evelyn, that—you know, in retrospect, it's true. We've assumed, when we make this assumption that a function is analytic, that we are living in the best of all possible worlds, we just didn't realize we were assuming that. It seems like we're not assuming much. And yet, it turns out, it's enormously restrictive, as you say. And so then it's a question of taste in math. Do you like your math really surprising and really beautiful and everything works out the way it should? Or do you like it thorny and full of rich counterexamples and struggles and paradoxes? And I feel like that's sort of the essential difference between real analysis and complex analysis.

EL: Yeah.

SS: In complex analysis, everything you had dreamed to be true is true, and the proofs are relatively easy. Whereas in real analysis, sort of the opposite. Everything you thought was true is actually false. There are some nasty counterexamples, and the proofs of the theorems are really hard.

EL: Yeah, you kind of have to MacGyver things together. “Yeah, I got this terrible epsilon and like, you know, it's got coefficients and exponents and stuff, but okay, here you go. I stuck it together.

KK: But but that's interesting, Steve, that this is your favorite theorem because, you know, you're very famous for studying kind of difficult, thorny mathematics, right? I mean, dynamics is not easy.

SS: Huh, I wouldn't have thought that, that's interesting that you think that. I don't think of myself as doing anything thorny.

KK: Okay.

SS: So that's interesting. I mean, yes, dynamical systems in the hands of some practitioners can be very subtle. I mean, those are people who have a taste for those those kinds of issues. I've never been very sophisticated and haven't really understood a lot of the subtleties. So I like my math very intuitive. I’m on the very applied end of the applied-pure spectrum, so that sometimes people will think I'm not really a mathematician at all. I look more like a physicist to them, or maybe even, God forbid, a biologist or something. So yeah, I don't really have much taste for the difficult and the subtle. I like my math very cooperative and surprising. I like—well, not surprising for mathematical reasons, but more surprising for its power to mirror things in the real world. I like math that is somehow tapping into the order in the world around us.

EL: Yeah, so this it's interesting to me, also that you picked this because, yeah, as you say, you are a very applied mathematician. And I think of complex analysis as a very pure—I actually, I'm trying to not say “pure” math, because I think it's this weird, like, purity test or something. But you know, that like a very theoretical thing. So does it play into your field of research at all?

SS: Well, uh, not particularly. Yeah. So that's a good question. I mean, I have to say I was a little intimidated by the title of the podcast. If you ask me what's my favorite theorem, the truth is for me, theorems are not my favorite things.

KK: Okay.

SS: My favorite things are examples or mathematical models. Like there’s a model in my field called the Kuramoto model after a Japanese physicist Yoshiki Kuramoto. And if you asked me what's my favorite mathematical object, I would say the Kuramoto model, which is a set of differential equations that mirrors how fireflies can get their flashes in sync, or how crickets can chirp in sync, or how other things in nature can self-organize into cooperative, collective oscillation. So that's my favorite object. I've been studying that thing for 30 years. And I suppose there are theorems attached to it, but it's the set of equations themselves and what they do that is my favorite of all. So I don't know, maybe that's my real answer.

KK: Well, that’s fine. So yeah, it's true. We've had people who've done that in the past, they didn't have a favorite theorem, but they had a favorite thing.

SS: But still, I mean, I am still a mathematician, part of me is, and I do have theorems that I love, and one of the things I love about Cauchy’s theorem is that in the proof, with this drawing of all the nested triangles inside the big triangle, you end up using a kind of internal cancellation. The triangles touch other triangles except on their common edge, sometimes you're going one way, and sometimes you're going in the opposite direction on that same edge. And so those contributions end up cancelling. And you end up, the only thing that doesn't cancel is what's going on around the boundary. And then that can be sort of pulled all the way into a tiny triangle in the interior, which is where you end up using the local property that is the derivative condition to get everything that you need to prove the result about the big triangle on the outside.

But the reason I'm going into all that is that this is a principle, this internal cancellation, that is at the heart of another theorem that's been featured on your show, the fundamental theorem of calculus, which uses a telescoping sum to convert what's happening on the boundary to what's happening when you integrate over the interior. This idea of telescoping I think, is really deep. I mean, it's what we use to prove Stokes’ theorem. It's what we would use to prove all the theorems about line integrals. It comes up in topology when you're doing chains and cochains. So this is a principle that goes beyond any one part of math, this idea of telescoping. And I've been thinking I want to write an article, someday (I haven't written it yet) called “Calculus Through the Telescope” or “A Telescopic View of Calculus” or something like that, that brings out this one principle and shows its ramifications for many parts of math and analysis and topology. I think some people get it, people who really understand differential forms and topology know what I'm talking about. But no one ever really told me this, and I feel like maybe it should be mentioned, even though it is well-known to the people who know it.

KK: Right, it's the air we breathe, right? So we don't we don't think about it.

SS: I guess, but like, I think there are probably high school teachers, or others who are teaching calculus—like for instance, when I learned about telescoping series in my first calculus course, that's just seems like a trick to find an exact sum of a certain infinite series of numbers. You know, they show you, “Okay, you could do this one because it's a telescoping series.” And it seems like it's an isolated trick, but it's not isolated. This one idea—you can see the two- dimensional version of it in Cauchy’s theorem, and you can see the three-dimensional version of it in the divergence theorem, and so on. Anyway, so I like that. I feel like this idea has tentacles spreading in all directions.

EL: Yeah. Well, this makes me want to go back and think about that idea more because, yeah, I wouldn't say that I would necessarily have thought to connect it to this many other things. I mean, you did preface your statement with “those who really understand differential forms,” and my dark secret is that the word “form” really scares me. It's a tough one. It's somehow, that was one of those really hard things, when I started doing more, like, hard real analysis. It's like, I feel like I always had to just kind of hold on to it and pray. And you get to the end of it. You're like, “Well, I guess I did it.” But I feel like I never really got that full deep understanding of forms.

SS: Huh. I don't I don't claim that I have either. I'm reminded of a time I was a teaching assistant for a freshman course for the the whiz kids that—you know, every university has this where you throw outrageous stuff at these freshmen, and then they rise to the occasion because they don't know what you're asking them to do is impossible. But so I remember being in a course, like I say, as a teaching assistant, where it was called A Course in Mathematics for Students of Physics, based on a book by Shlomo Sternberg, at Harvard, and Paul Bamberg, who's a physicist there too, and a very good teacher. And that book tried to teach Maxwell's equations and other parts of physics with the machinery of differential forms and homology and cohomology theory to freshmen. But what was amazing is it sort of worked, and the students could do it. And in the course of teaching it, I came to this appreciation of integrating forms, and how it really does amount to this telescoping sum trick. And, anyway, yeah, it's true, that maybe it's not super widely appreciated. I don't know. I don't know if it is, I don't want to insult people who already know what I'm talking about. But I I do feel like there's a story to tell here.

KK: Okay. Well, we'll be looking for that.

EL: Yeah.

SS: Someday.

KK: In the New York Times, right?

SS: Well.

KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. And we might have sprung this on you, but you seem to have thought of a solution here. So what have you chosen to paired with Cauchy’s theorem?

SS: Cubist painting.

KK: Oh, excellent. Okay. Explain.

EL: Yeah, tell us why.

SS: Well, I'm thinking of Cubism. I don’t—look, I don't know much about art. So it might be a dumb pairing. But what I'm thinking is there's a there's a painting. I think it's by Georges Braque of a guy, or maybe it's Picasso. Someone walking down stairs. And maybe it's called a Nude Descending a Staircase, or something like that. You're nodding, do you know what I mean?

EL: I'm a little nervous about saying, I think it is Picasso, but I'm looking it up on my phone surreptitiously.

SS: I could try too. For some reason, I'm thinking it's George Braque, but that may be wrong. But so I'll describe the painting I have in my head and it may be totally not—

EL: No, it’s Marcel Duchamp!

SS: Oh, it's Marcel Duchamp?

EL: Yeah.

SS: And what's the name of it?

EL: Nude descending a staircase, number two. I think.

SS: Yeah, that's the one. Would that be considered Cubism?

EL: Yeah.

SS: It says according to Wikipedia, it’s widely considered a modernist classic. Okay, I don't know if it's the best example of what I'm thinking. But it's, let me just blow it up and look at it here. So, what hits me about it is it's a lot of straight lines. It's very rectilinear. And you don't see anything that really looks curved like a human form. You know, people are made of curved surfaces, our faces, our cheeks are, you know. What I like is this idea that you can build up curved objects out of lots of things made of straight lines. You know what you can do? mesh refinement on it. For instance, there's an old proof of the area of a circle where you chop it up into lots of pizza-shaped slices, right, and then you add up the areas of all those. And they can be approximated by triangles, and if you make the triangles thin enough, then those slivers can fill out more and more of the area, the method of exhaustion proof for the area of a circle. So this idea that you can approximate curved things with triangles, reminds me of this idea in Cauchy’s theorem that you first you prove it for the triangle, and then later Professor Stein proved the result for any smooth curve by approximating it with triangles, you know, a polygonal approximation to the curve, and then he could chop up the interior into lots of triangles. So I sort of think it pairs with this vision of the human form and it's sinuous descent down. You know, this person is smooth and yet they're being built out of these strange Cubist facets, or other shapes. I mean, think of other Cubist paintings you you represent smooth things with gem-like faceted structures, it sort of reminds me of Cauchy’s theorem.

KK: Okay, good pairing. Yup.

EL: Yeah, glad we got to the bottom of that before we made false statements about art on this math podcast.

SS: Yeah, it may not be the best Cubist example. But what are you gonna do? You invited a mathematician.

KK: So we also like to let our guests make pitches for things that they're doing. So you have a lot going on. You have a new podcast.

EL: Yeah, tell us about it.

SS: Okay. Yeah, thank you for mentioning it. I have a podcast with the confusing name Joy of X. Confusing because I also wrote a book by that name. And before that I had written an article by that name.

KK: Yes.

SS: So I did not choose that name for the podcast. But my producer felt like it sort of works for this podcast because it's a show where I interview scientists and mathematicians—in spirit, very similar to what we're doing here. And I talk to them about their lives and their work. And it's sort of the inner life of a scientist, but it could be a neuroscientist, it could be a person who studies astrophysics, or a mathematician. It's anything that is covered by Quanta Magazine. So Quanta Magazine, some of your listeners will know, is an online magazine that covers fundamental parts of math and science and computer science. Really, it's quite terrific. If people haven't read it, they might want to look at it online. It's free. And anyway, so Quanta wanted to start a podcast. And they asked me to host it, which was really fun because I get to explore all these parts of science. I've always liked all of the different parts of science, as well as math. And so yeah, that's the show. It's called the Joy of X where here, X takes on this generalized meaning of the unknown, not just the unknown in algebra, but anything that's unknown, and the joy of doing science and the scientific question. We'll be sure to link to that.

EL: Yeah.

KK: Also, I think Infinite Powers came out last year, right? 2019?

SS: That’s true. Yes, I had a book, Infinite Powers, about calculus. And that was an attempt to try to explain to the general public what's so special about calculus, why is it such a famous part of math. I try to make the case that it really did change the world and that it underpins a lot of modern science and technology as well as being a gateway to modern math. I really do think of it as one of the greatest ideas that human beings have ever come up with. Of course, that raises the question, did we discover it or invent it? But that’s a good one.

EL: Put that on a philosophy podcast somewhere. We don’t need that on this math podcast.

SS: Yeah, I don't really know what to say about that. That's a good timeless question. But anyway, yes, Infinite Powers was a real challenge to write because I'm trying to tell some of the history, but I'm not a historian of math. I wanted to really teach some of the big ideas for people who either have math phobia or who took calculus but didn't see the point of it, or just thought it was a lot of, you know, doing one integral after another without really understanding why they're doing it. So it's my love song to calculus. It really is one of my favorite parts of math, and I wanted other people to see what's so lovable and important about it.

KK: Yeah.

SS: The book, as I say, was hard because I tried to combine history and applications and big ideas without really showing the math.

KK: Yeah, that's hard.

SS: And make it fun to read.

KK: Right. It is. It's a very good book, though. I did read it.

SS: Oh, thanks.

KK: And I enjoyed it quite a bit.

EL: Well, it is on my table here under a giant pile of books to read, because people need to just stop publishing.

SS: That’s right.

EL: There’s too much. We just need to have a year to catch up, and then we could start going again but what's what's

KK: What’s that Japanese word, sort of the joy of having unread books? [Editor’s note: Perhaps tsundoku, “aquiring reading materials but letting them pile up in one’s home without reading them.”] There's a Japanese concept of like these books that you’ll, well, maybe even never read. But that you should have stacks and stacks of books. Because, you know, maybe you'll read them. Maybe you won't. But the potential is there.

SS: Nice.

KK: So I have a nightstand, on the shelf of my nightstand there's probably 20 books there right now, and I haven't read them all. I've read half of them, maybe, but I'm going to read them. Maybe.

SS: Yeah, yeah.

KK: Actually, you know, when you were talking about your sort of emotional feelings about Cauchy’s theorem, it reminded me of your—I don't know if it was your first book, but The Calculus of Friendship, about your relationship with your high school teacher.

SS: Well, how nice of you to mention it.

KK: Yeah. That was interesting to it, because it reminded me a lot of me, in the sense of, I thought I knew everything too when I was 18. Like, I thought, “Calculus is easy.” And then I get to university and math wasn't necessarily so easy. You know. And so these same sort of challenges, you know?

SS: Well, I appreciate that, especially because that book is pretty obscure. As far as I know, not many people read it. And it's very meaningful to me because I love my old teacher, Mr. Joffrey, who is now, let’s see, he's 90 years old. And I stayed in touch with him for about 35 years after college, and we wrote math problems to each other, and solutions. And it was really a friendship based on calculus. But over the course of those 35 years, a lot happened to both of us in our lives. And yet, we didn't tend to talk about that. It was like math was a sanctuary for us, a refuge to get away from some of the ups and downs of real life. But of course, real life has a way of making itself, you know, insinuating itself whether you like it or not. And so it's it's that story. The subtitle of the book is “what a teacher and a student learned about life while corresponding about math.” And I sometimes think of it as, like, there's a Venn diagram where there's one circle is people who want to read math books with all the formulas, because I include all the formulas from our letters.

KK: Yeah.

SS: And then there's people who want to read books about emotional friendships between men. And if you intersect those two circles, there's a tiny sliver that apparently you're one of the people in it.

KK: And your book might be the unique book in that in that Venn diagram too.

SS: Maybe. I don't know. But yeah, so it was it was clear it would not be a big hit in any way. But I felt like I couldn't do any other work until I wrote that book. I really wanted to write it. It was the easiest book to write. It poured out of me, and I would sometimes cry while I was writing it. It was almost like a kind of psychoanalysis for myself, I think, because I did have a lot of guilty feelings about that relationship, which, you know, if you do read the book, anyone listening, you'll see what I felt guilty about, and I deserved to feel guilty. I needed to grow up, and you see some of that evolution in the course of the book.

KK: Yeah. All right. Anything else you want to pitch? I mean?

SS: Well, how about I pitch this show? I mean, I'm very delighted to be on here. Really, I think you guys are doing a great thing helping to get the word out about math, our wonderful subject. And so God bless you for doing that.

KK: Well, this has been a lot of fun, Steve, we really appreciate you taking time out of your snow day. And so now do you have to shovel your driveway?

SS: Oh, yeah, that may be the last act I ever commit.

KK: Don’t you still have a teenager at home? Isn't that what they're for?

SS: My kids, I do have—you know what, that's a good point. I have one daughter who is still in high school and has not left for college yet, so maybe I could deploy her. She's currently making oatmeal cookies with one of her friends.

KK: Well, that's a useful, I mean that that's helping out the family too, right? I mean,

SS: They’re both able bodied, strong young women. So I should get them out there and with me, and we could all shovel ourself out. Yeah.

KK: Good luck with that. Thank you. Thanks for joining us.

SS: My pleasure. Thanks for having me.

On this episode of My Favorite Theorem, we were happy to talk with Steve Strogatz, an applied mathematician at Cornell University, about the Cauchy integral theorem. Here are some links you might find helpful.

Strogatz’s website, which includes links to information about his books and article
The Joy of X, the podcast he hosts for Quanta Magazine
The Cauchy integral theorem on Wikipedia
The Kuramoto model
Nude Descending a Staircase no. 2 by Marcel Duchamp

Episode 53 - Ruthi Hortsch

9 april 2020 | 29 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the podcast that was already quarantined. I’m one of your hosts, Evelyn Lamb. I am holed up in my house in Salt Lake City, Utah, where I'm a freelance writer. So, honestly, I have worked in my basement, you know, every day for the past five years, and that hasn't changed. This is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida, which is open for business…But you can't go to campus.

EL: Okay.

KK: Yeah, we moved all of our classes online two weeks ago, I'm just teaching a graduate course this term, so that's sort of easier for me. I feel bad for the people who have to actually lecture and figure out how to do this all at once. My faculty have actually been great. They really stepped up. And, remarkably, I've had very few complaints from students, and I'm the chai,r so you know, they would come to me. And it's just really not—I mean, everybody has really taken the whole thing in stride. A lot of anxiety out there, though, among our students. Really, this is a really challenging time for everybody. And I just encourage my faculty to, you know, be kind to their students and to themselves. So let’s shelter in place and get through this thing, right?

EL: Yup. Yeah, we had an earthquake a week and a half ago to just, like, shake things up, literally. So it's just like, oh, as if I pandemic sweeping through town was not enough. We'll just literally shake your house for a while.

KK: Yeah, well, you know, we can go outside. We have a Shelter in Place Order, but it's been 90 degrees every day for the last week. And so you know, I like to go bird watching, but my favorite bird watching spot is a city park, and it's closed. So I have to just kind of sit on my back porch and see what's up. Yeah. Oh, well,

EL: Well, yes, we're making it through it. And I hope—I mean by the time this is—we have a bit of a backlog in our past episodes, and so who even knows what's going to be happening when this is airing. [Editor’s note: We decided to publish this one out of order, so we actually recorded it pretty recently.] But whatever is happening, I know our guests will be very thrilled to be listening to Ruthi Hortsch! Hi, Ruthi. How are you today?

Ruthi Hortsch: Hey, I'm managing.

EL: Yeah.

RH: It’s a weird time.

EL: Definitely. So what do you do, and where are you?

RH: Yeah, so I'm in New York City right now, which is kind of right now the hotbed of lots of new infections. But I've been in my apartment for the last two and a half weeks and haven't really directly been experiencing that.

I work for an organization called Bridge to Enter Advanced Mathematics. So we're a education nonprofit. We work with low-income and historically marginalized youth. And we're trying to create a realistic pathway for them to become mathematicians, scientists, engineers, programmers.

We start working with students when they're in middle school and we try to figure out, like, what are the things you need to get you to a place where you'll have a successful STEM career? And so we do a lot of different things, but they all are to that purpose.

EL: Yeah, and I'm so glad that we have you on the show to talk about this. Because, yeah, I've been thinking like, we really need to get someone from BEAM on here because I think BEAM is just such a great program. My spouse, and I donate to it every year. I mean, obviously not every year, I don't even know how old it is. But you know, we've made that part of our yearly giving, and yeah, I just think it does great work. So, does that have programs in both New York and LA now?

RH: Yes. So we started in New York City in 2011. And a few years ago, we expanded to LA. So the LA programs are still pretty new. They're building up, kind of starting with students in the first year of contact, and then adding in programming for the older students as that first class gets older. So they now have eighth graders, and that's their oldest class, and they'll continue to add in the ninth grade and the 10th grade program, et cetera, as it goes on. The other kind of exciting thing is, last year, we got a grant from the Gates Foundation. And that grant was to partner with other local programs and other cities to help them build up programs that could do some of the same things we do. So it's not the same comprehensive, really intensive support that we give our students in New York City and LA. But assuming summer camps don't get canceled this summer because of corona, there are going to be day camps in Albuquerque and Memphis that are advised by us.

EL: Oh, that's so great. Yeah, because that's the one thing about it is that it is so localized and, of course, important places for it to be localized. But, you know, the more the, the wider, the better. So that's awesome. And what's your role there? What do you do?

RH: Yeah, I have a hard time answering this question. So I work in programs, which is like, I work on things that are directly affecting students. I run one of our summer camps in the summer. So I run a sleepaway camp at Union College, in which students learn proof-based mathematics for the first time. The students at the sleepaway camp are all rising eighth graders, and so they get to learn number theory and combinatorics and group theory. They also do some modeling and programming and stuff.

During the year I do some managing our other programs team, so supporting other staff. I also do all of our faculty hiring. So certainly we hire a lot of people just for the summer, and most of them are—so we hire college, university students, we hire grad students, we hire professors in various different roles. And I handle all of the, like, hiring people to teach math courses.

EL: Wow.

KK: That’s a lot. Are your programs sort of face to face, or are they online? Is it sort of a combination of stuff?

RH: Yeah, so our summer we run six in-person summer camps each summer. So there's two in upstate New York that are sleepaway, one in Southern California that's sleepaway, and then one day camp in LA and two day camps in New York City. And those are all in-person, face to face. And then during the school year, we also have Saturday classes, which is a mix of life skills and enrichment. And we also do in-person advising. So we have office hours where students can come ask us anything, and then also kind of more intensive. Like, how do you apply to college? How do you get into other summer programs or other STEM opportunities? So most of our programs are face to face. Right now, we've had to cancel a bunch of our year-round stuff. So we don't have Saturday classes right now. We are doing one class for the eighth graders virtually, because we really thought it was critical. And at the moment, we're hoping the summer programs will still run, but it's really hard to say what's going to be going on in two weeks.

KK: Yeah, well, fingers crossed.

EL: But as wonderful as it is to talk about BEAM, what we're dying to know is what is your favorite theorem?

RH: Yeah, so this was actually really fast for me to think of. My favorite theorem is Falting’s theorem. So Falting’s theorem is also actually known as the Mordell conjecture, because Mordell originally conjectured it in the same paper in which he proved Mordell’s theorem, I believe, or at least during the same process of research for him.

EL: Yeah, and so for longtime listeners, was it Mathilde Lalín who, that was her favorite theorem?

RH: Mm-hmm.

EL: Okay, that's right. So we're kind of dovetailing right in.

RH: Yeah. So Mordell’s theorem is about—so when you look at elliptic curves, they have a finitely-generated abelian group. And Mordell’s theorem is the theorem that proves that it actually is finitely-generated.

KK: Right.

RH: So when I say the finitely-generated part, it's actually only looking at the rational points on the curve. So we care about algebraic curves, kind of in general. And then we want to think about, like, how do different algebraic curves behave differently? And because I'm trained as a number theorist, I also specifically care about how many rational points are on that curve and how they behave. So this intersects also with algebraic geometry. And in some sense, this is a statement about how the arithmetic part of the curves—the rational points—interacts with the geometry of it.

So one thing that people care about a lot in geometry is the notion of a genus. This is one of the ways to classify things. And of course, when you're looking at visual shapes, one way of thinking about the genus is how many holes does it have? So if you're just looking at a shape that’s, like, a big sphere, there's no way of poking a hole through it without actually breaking it apart. And so that has genus zero because there are zero holes. But if you're looking at a doughnut, a torus, that has one hole because there's like one place where you can poke something through. And then you can generalize from there that having more holes is higher genus. And so that's kind of a wishy-washy way of looking at things, and a very visual way. There are ways to define that formally in the algebraic sense, but in the places where both definitions make sense, the definition is the same.

And so when you look at algebraic curves, we can ask ourselves, how do genus zero curves act differently than genus one curves, act differently than genus two curves, and does that tell us anything about the number of rational points? And so it turns out that with genus zero curves, genus zero curves are actually really just conic sections. So basically the nice lines that you study in like algebra in high school. And those have infinitely many rational points, right? So when I say rational point, you can kind of think of it as being like the points where the components have rational values.

And genus one curves are actually exactly elliptic curves. So in that case, that's when Mordell’s theorem kicks in and the rational points are this finally generated abelian group. And sometimes they have infinitely many rational points, and sometimes they don't, and it kind of depends on what this algebraic structure, this algebraic group structure, looks like. So that's the most complicated weird point. And for genus two or higher curves, it turns out to be true that there are only finitely many rational points on a genus two or higher curve. And that's the statement of Falting’s theorem.

EL: Okay, and so I, there's something that I, you know, you hear like genus two or higher. And I always wonder, is there a limit to how high the genus can be of these curves? Or, like, is there a maximum complexity that these curves can have?

RH: So no. And actually, there's a statement in algebraic geometry that makes it really easy-ish— you know, “ish”— to calculate the genus, which is called Riemann-Roch. And it gives you a relationship between the degree of the equation defining it and the genus. And essentially, the genus grows quadratically with the degree. There's an asterisk on everything I'm saying. It’s mostly true.

KK: It’s mostly true.

EL: So if I'm remembering correctly, Mordell’s—let’s see, Mordell’s conjecture, Falting’s theorem—was really important for proving Fermat’s last theorem. Is that correct?

RH: I don't think so, no. But all of these things are related to each other.

EL: Okay.

RH: A lot of the common definitions and theorems that play into all these things, they share a lot, but it's not directly, like, one thing implied the other.

EL: Okay, yeah.

RH: In particular, Fermat’s Last Theorem was reduced to a statement about elliptic curves, which is about genus one curves, while Falting’s theorem is really a statement about genus two or higher curves.

EL: Okay.

KK: So was this a love at first sight kind of theorem?

RH: I think no. I think part of the reason that I really started appreciating it was because I had a mentor in undergrad who was really excited about it. And I didn't really understand the full implications and the context, but I was like, “Okay, this mentor I have is really about it, so I'm going to be really about it.”

And we actually used Falting’s theorem as a black box for the REU project I was working on. So we assumed it was true and then used that to show other things. And then later on in grad school, I had a number of things that I was really interested in that Falting’s theorem was related to. One of the things that I think is really cool that's being researched right now is there’s a bunch of like, tropical geometry that is being studied. And this is, like, relating algebraic verbs to kind of more combinatorial objects. So you can actually translate these lcurves that have a more—I don't want to say analytic, but a smooth structure, and then turning them into a question about, like, counting more straight-edged structures instead.

One of the things about Falting’s proof of Falting’s theorem is that it's not, it doesn't actually give you a bound. So it tells you that there are only finitely many points, but it doesn't give you a constructive way of saying, like, what does it actually bounded by, the number of finite points? And using tropical geometry, people have been able to make statements about bounds in certain situations, which is really cool.

KK: Okay, I always like these tropical pictures, you know, because suddenly everything just looks almost like Voronoi diagrams in the plane, these piecewise linear things. So I guess the idea of genus probably still makes sense there in some way, once you define it properly. Right?

RH: Yeah. And there's a correspondence between, there’s a notion of a tropical curve, which still looks like one of those Voronoi diagrams. There’s an actual correspondence, this curve in classical algebraic geometry gives you this particular diagram.

EL: Nice. And so you say it was very easy to choose this theorem. So what's your, like, elevator sales pitch for this theorem? Keeping in mind that no one is going to be in an elevator with anyone else anytime soon. We're staying far apart, but you know.

RH: Yeah. So, I think it’s kind of amazing that geometry can tell you something about the arithmetic of a curve. I think this is what drew me to arithmetic algebraic geometry, that there is this kind of relationship. When you think, okay, arithmetic, geometry, those are totally different fields, people study them in totally different ways, but in fact, it turns out that the geometry of a curve can tell you information about the arithmetic. And that's just bizarre, and also very powerful in that you can make a statement about how many rational solutions there are to an equation using correspondence in geometry.

The REU project that I worked on actually is a statement that I think is really easy to understand. If you have a rational polynomial, that gives you a function from the rationals to the rationals, right?

And so you can ask yourself: how many-to-one is that function? How many points gets sent to the same point? And if you look at only rational points, our REU project showed that it can't be more than four-to-one off a finite number points.

So if you are willing to ignore some finite number of points, then no rational polynomial is ever more than four-to-one.

KK: Interesting.

RH: And that feels like a very powerful statement. And it's because we had this hammer of Falting’s theorem to just smash it in the middle.

KK: That’s really fascinating. So no matter how high the degree it's no more than four-to-one? I wouldn’t have guessed that.

RH: Off a finite number of points.

KK: Yeah, sure. Generically. Yeah. Right. Interesting.

RH: I think the real powerful thing there is that Falting’s theorem comes in.

KK: Yes.

RH: Oh, actually, higher degree means high complexity means high genus.

KK: Okay, cool. So another thing we like to do is ask our guests to pair their theorem with something. So what pairs well with Falting’s theorem?

RH: Yeah, so this is a maybe a little bit of a stretch, but I've been living in New York City for four years, and I love bagels. They’re definitely one of the best parts of living in New York City. I'm always two blocks away from a really good bagel. Traditionally, bagels are genus one, so it's actually not quite appropriate. You have to, I don't know, do the fancy cut to increase the genus—there’s a way to cut a bagel to get higher genus. But I still think since we're thinking about genuses, we're thinking about complexity of things.

EL: Yeah. Well, like, you cut the bagel in in half, you know, to get like the cream cheese surface, and then just stick them together and you've got a genus two. Put a little cream cheese on the side. You know?

RH: Yeah. I mean, if we're cutting holes we can cut as we want.

EL: That’s true. So, are you more—what do you put on the bagel? What kind of bagel, also, do you prefer?

RH: Ao I mostly like everything bagels.

EL: Of course. Yeah. Great bagel.

RH: There is a weird thing that goes on where some bagel shops put salt on their everything bagel and some don't. And I feel like the salt is important.

KK: Yeah. Agree.

EL: As long as it's not too much. Like just the right amount of salt is—

RH: Yeah. It’s definitely important.

KK: Well a salt bagel is a pretzel.

EL: Yes.

RH: And I don't actually eat cream cheese. So I do eat fish sometimes, but I generally don't eat dairy. And I so I usually get, like, tofu scallion spread. And the tofu spread that gets sold in the bagel shops here is actually really good.

KK: Well yeah, I'm not surprised. I can't get a decent bagel in Gainesville. I mean, there's a couple of bagel shops, but they're no good.

RH: Yeah. This is what you get for leaving New York City.

KK: Right, right.

EL: Yeah, it's funny, actually one of our quarantine projects we're thinking about is making bagels. I've made bagels one other time. But, yeah.

KK: It's kind of a nuisance. You know that. That boiling step is really—I mean, it's crucial, but it just takes so much time and space.

EL: Yeah, I mean, they were not nearly as good as a real bagel shop bagel, but fun to play with.

KK: Yeah. So what's everyone doing to keep themselves occupied? So far I've got a batch of sauerkraut fermenting. I just started a batch of limoncello that'll be ready in a month. I made scones. Maybe that’s it. Yeah. How about you guys?

RH: Well, I'm still trying to work 40 hours a week.

KK: Yeah, I'm doing that too.

RH: We're still trying to help our students respond to the crisis and helping support them both academically, but holistically also.

KK: Yeah, it's very stressful.

RH: And at the moment, we're still doing all of our prep work for the summer, which is a huge undertaking? But when I have free time, I've been cooking more. And I'm actually also working on writing a puzzle hunt.

EL Ooh, cool. Well if that happens, we'll include a link to that in the show notes—if it's the kind of thing that you can do out of a particular geographical place.

RH: Yeah, so the puzzle hunt I'm helping write is actually for Math Camp.

EL: Okay.

RH: So before I worked for BEAM I worked for Canada-USA Math Camp, and in theory, they're running a camp this summer, and one of the traditional events there is [the puzzle hunt]. I think the puzzle hunt often gets put up after the summer, but I’m not sure.

EL: Oh, cool. The last thing that I, or library book that I got out from the library—it was actually supposed to be due, like, the day after the library shut down here—was 660 Curries, which is an Indian cookbook that—we don’t really cook meat at home, but it's got, I don't know, maybe a hundred-page section of legume curries and a bunch of vegetable curries, so we've been kind of working through that. We made one last night that was great. It was a mixture of moong dal and masoor dal. Yeah, we’ve been eating a lot of curry, and it just makes my early-this-year plan of, like, “Oh, I want to make more dal, so I've got to go stock up on lentils and rice,” brilliant plan, really has made it a lot easier. So yeah.

RH: I love dal, and I don't feel like anybody around me ever likes dal as much as I do.

KK: This is a dal-lover convention right here. It's one of my favorite things to eat. Yeah.

EL: Oh, yeah. Well, I can recommend, if you get a chance to get 660 Curries, I don't remember if it's called mixed red and lentil dal with garlic and curry leaves, or something like that.

KK: Yeah, I'm actually making curry tonight, but chicken curry so we'll we'll see.

EL: Yeah, so other than that, just panicking most of the time. It’s been a big pastime for me.

RH: I’ve had to, like, ban myself from reading the news in the evening.

KK: Good call.

EL: That is very smart.

RH: I haven’t done a good job keeping to it.

EL: Yeah, I have not done a good job with my self-control with that. So, I’m really trying to do that. I'm hoping to do some sewing projects too, maybe making some masks that I can leave out for people in the neighborhood to take. Obviously not medical grade, but maybe make people feel a little better.

KK: So yeah, Ellen, my wife, started doing that yesterday. She made, you know, probably 15 of them yesterday real quick.

EL: Nice.

KK: I went to the store yesterday and you know—

EL: Hopefully it gives people a little peace of mind and maybe decreases droplet transmission.

KK: Let’s hope.

EL: I’ve refrained from armchair epidemiology, which I encourage everyone to do. So yeah, I hope everyone stays safe and tries to keep keep a good spirit and help the people in your lives. I hope our listeners can do that too. And I hope they find some enjoyment in thinking about math for a little while with us.

KK: So yeah, thanks for joining us, Ruthi. We really appreciate it.

EL: Yeah, everyone go find BEAM online if you want to learn more about that.

RH: Yeah. Follow us on social media.

EL: Yeah. So what are the handles for that?

RH: Yeah, I should have this memorized. You can find it on our website. They're all linked to on our website, beammath.org. If you're in New York or LA, we have trivia night, which is a puzzle-y, mathy trivia, usually in the fall, that you can buy tickets to. So I definitely recommend that. And otherwise, sign up for our newsletter, which you can also do on our website.

EL: And you're on Twitter also, right?

RH: Yes, I am. You do have to know how to spell my last name, though.

EL: Okay.

RH: Yeah, I'm @ruthihortsch.

EL: All right. And that's H-O-R-T-S-C-H?

RH: Good job!

EL: Yeah, it’s funny, I was actually in a Zoom spelling bee last night. So yeah, I got second place.

KK: Good for you.

EL: Got knocked out on diaphoresis.

KK: Diaphoresis. Wow. Yeah, that's pretty—okay, anyway. All right. Well, thanks for joining us and take care everyone.

RH: Right. Yeah, it was nice to meet you.

EL: Bye.

[outro]

On today's episode of My Favorite Theorem, we had the privilege to talk with Ruthi Hortsch, a program coordinator at Bridge to Enter Advanced Mathematics (BEAM), a math program for low-income and historically marginalized middle- and high-school students. Dr. Hortsch lives in New York City, which is currently being hit hard by covid-19. We love all our listeners and guests, and right now we are especially thinking about those in New York and other virus hot spots. You may be sick, you may be worried about loved ones, you may be suddenly parenting or caregiving in ways you hadn't expected. We wish you the best, and we hope you enjoy thinking about math for a little bit instead of the news cycle. Stay strong and healthy, friends!

As you listen to this episode, you may find these links helpful.
The Bridge to Enter Advanced Mathematics website, Twitter, Facebook, and Instagram pages.
Ruthi Hortsch on Twitter
Faltings’s theorem, Dr. Hortsch's favorite theorem
Our episode with Matilde Lalín, whose favorite theorem was the closely-related Mordell's theorem.
660 Curries

Canada/USA Mathcamp

Tropical Geometry wikipedia page

Episode 52 - Ben Orlin

12 mars 2020 | 27 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer, usually based in Salt Lake City, but currently still in Providence. I'll be leaving from this semester at ICERM in about a week. So trying to eat the last oysters that remain in the state before I leave and then head back.

KK: Okay, so you actually like oysters.

EL: Oh, I love them. Yeah, they're fantastic.

KK: That is one of those, it’s a very binary food, right? You either love them—and I do not like them at all.

EL: Oh, I get that, I totally get it.

KK: Sure.

EL: They’re like, in some sense objectively gross, but I actually love them.

KK: Well, I'm glad you've gotten your fill in. Probably—I imagine they're a little more difficult to get in Salt Lake City.

EL: Yeah, you can but it’s not like you can get over here.

KK: Might be slightly iffy. You don't know how long they've been out of the water, right?

EL: Yeah. So there's one place that we eat oysters sometimes there, yeah, that's the only place.

KK: Yeah, right. Okay. Well, today we are pleased to welcome Ben Orlin. Ben, why don't you introduce yourself?

Ben Orlin: Yeah, well, thanks so much for having me, Kevin and Evelyn. Yes, I'm Ben Orlin. I’m a math teacher, and I write books about math. So my first book was called Math with Bad Drawings, and my second one is called Change Is the Only Constant.

EL: Yeah, and you have a great blog of the same name as your first book, Math with Bad Drawings.

BO: Yeah, thank you. And I think our blogs are, I think almost birthday, not exactly but we started them within months of each other, right? Roots of Unity and Math with Bad Drawings.

EL: Oh, yeah.

BO: Began in, like, spring of 2013 which was a fertile time for blogs to begin.

EL: Yeah. Well, in a few years ago, you had some poll of readers of like, what other things they read and, and stuff and my blog was like, considered the most similar to yours, by some metric.

BO: Yeah, I did a reader survey and asked people, right, what what other sources they read, and mostly I was looking for reading recommendations. So what else do they consider similar? Overwhelmingly it was XKCD. Not so much—just because XKCD, it’s like if you have a little light that you're holding, a little candle you're holding up, and you're like, what does this remind you of? And like a lot of people are going to say the sun because they look up, and that’s where they see visible light.

KK: Sure.

BO: But I think in terms of actually similar writing, I think Toots of Unity is not so different, I think.

EL: Yeah. So I thought that was interesting because I have very few drawings on on mine. Although the ones that I do personally create are definitely bad. So I guess there’s that similarity.

BO: That’s the key thing, committing to the low quality.

KK: Yeah, but that's just it. I would argue they're actually not bad. So if I tried to draw like you draw, it would be worse. So I guess my book should just be Math with Worse Drawings.

BO: Right.

KK: You actually get a lot of emotion out of your characters, even though they're they're simple stick figures, right? There’s some skill there.

BO: Yeah, yeah. So I tried. I tried to draw them with a very expressive faces. Yeah, they're definitely still bad drawings is my feeling. Sometimes people say like, “Oh, but they've gotten so much better since you started the blog,” which is true, but it's one of these things where they could they could get a lot better every five-year interval for the next 50 years and still, I think not look like professional drawings by the end of it.

EL: Right. You're not approaching Rembrandt or anything.

KK: All right, so we asked you on here, because you do have bad drawings, but you also have thoughts about mathematics and you communicate them very well through your drawings. So you must have a favorite theorem. What is it?

BO: Yeah. So this one is drawn from my second book, actually, the second book is about calculus. And I have to confess I already kind of strayed from the assignment because it's not so much a favorite theorem as a favorite construction.

KK: Oh, that’s cool.

EL: You know, we get rule breakers on here. So yeah, it happens.

BO: Yeah, I guess that's the nature of mathematicians, they like to bend the rules and imagine new premises. So pretending that this were titled My Favorite cCnstruction, I would pick Weierstrass’s function. So that you know, first introduced in 1872. And the idea is it's this function which is continuous everywhere and differentiable nowhere.

EL: Yeah. Do you want to describe maybe what this looks like for anyone who might not have seen it yet?

BO: Yeah, sure. So when you're picturing a graph, right, you're probably picturing—it varies. I teach secondary school. So students are usually picturing a fairly small set of possibilities, right? Like you're picturing a line, maybe you're thinking of a parabola, maybe something with a few more squiggles, maybe as many squiggles as a sine wave going up and down. But they all have a few things in common one is that almost anything that students are going to picture is continuous everywhere. So basically, it's made of one unbroken line. You can imagine drawing it with your pencil without picking the pencil up. And then the other feature that they have is that they—this one's a little subtler, but there will be almost no points that are jagged, or sort of crooked, or, you know, if I picture an absolute value graph, right, it sort of is a straight line going down to the origin from the left, and then there's a sharp corner at the origin, and then it rises away from that sharp corner. And so those kind of sharp corners, you may have one or two in a graph a student would draw, but that's sort of it. You know, like sharp corners are weird. You don't can't draw all sharp corners. It feels like between any two sharp corners on your graph, there's going to have to be some some kind of non-sharp stuff connecting it, some kind of smooth bits going between them.

KK: Right.

BO: And so what sort of wild about about Weierstrass’s function is that you look at it, and it just looks very jagged. It’s got a lot of sharp corners. And you start zooming in, and you see that even between the sharp corners, there are more sharp corners. And you keep zooming in and there's just sharp corners all the way down. It's what we today call it fractal. Although back then that word wasn't around. And it's just it's the entire thing. Every single point along this curve is in some sense, a sharp corner.

EL: Yeah, it kind of looks like an absolute value everywhere.

BO: Yeah, exactly. It has that cusp at every single point you could look at.

KK: Right? So very pathological in nature. And, you know, I'm sure I've seen the construction of this. Is it easy to say what the construction is? Or is this going to be too technical for an audio format?

BO: It’s actually not hard to construct. There are there whole families of functions that have the same property. But Weierstrass’s is pretty simple. He starts with basically just a cosine curve. So you sort of have cosine of πx. So picture, you know, a cosine wave that has a period of two. And then you do another one that has a much shorter period. So you can sort of pick different numbers. But let's say the next one that you add on has a period that's 21 times faster. So it's sort of going up and down much quicker. And it's shorter, though, we've shrunk the amplitude also. So it's only about a third, let's say, as tall. And so you add that onto your first function. So now we've got—we started with just a nice, gentle wave. And now we've got a wave that has lots of little waves kind of coming off of it. And then you keep repeating that process. So the next, the second one in the iteration has a period of 21 cycles for two units. The next one has 212 cycles. And it's 1/9 the height of the original.

KK: Okay.

BO: And then after that, you're going to do you know, 213 cycles in the same span, 214 cycles. And so it goes—I don't know if you can hear my daughter is crying in the background, because I think she she finds it sort of upsetting to imagine the function that's has this kind of weird property.

EL: Fair.

BO: Especially because it's such a simple construction. Right? It's just, like, little building blocks for her that we're putting together. And one of the things I like about the construction, is it at no step, do you have any non-differentiable points, actually. It's a wave with a little wave on top of it and lots of little waves on top of that, and then tons and tons of little waves on top of that, but these are all smooth, nice, curving waves. And then it's only in the limit, sort of at the at the end of that infinite bridge, that suddenly it goes from all these little waves to its differentiable nowhere.

KK: I mean, I could see why that would be true, right?

BO: Yeah, right. Right. It feels like it's getting worse. And you can do—Weierstrass’s function is really a whole family of functions. He came up with some conditions that you need, basically that’s the basic idea. You need to pick an odd number for the number of cycles and then a geometric series for for the amplitude.

KK: So what's so appealing about this to you? It's just you can't draw it well, like you have to draw it badly?

KK: Yeah, that's one thing, right. Exactly. I try to push people into my corner, force them to have to drop badly. I do like that this is something—right, graphs of functions are so concrete. And yet this one you really can't draw. I've got it in my book, I have a picture of the first few iterations. And already, you can't tell the difference between the third step and the fourth step. So I had to, I had to, you know, do a little box and an inset picture and say, actually, in this fourth step, what looks like one little wave is really made up of 21 smaller waves. So I do sort of like that, how quickly we get into something kind of unimaginable and strange. And also, you know, I'm not a historian of mathematics. And so I always wind up feeling like I'm peddling sort of fairy tales about about mathematical history more than the complicated truth that is history. But the role that this function played in going from a world where it felt like functions were kind of nice and were something we had a handle on, into opening up this world where, like, oh no, there are all these pathological things going on out there. And there are just these monsters that lurk in the world of possibility.

KK: Yeah.

EL: Right. And was this it—Do you know, was this maybe one of the first, or the first step towards realizing that in some measure sense, like, all functions are completely pathological? Do you know kind of where it fell there, or, like, what the purpose was of creating it in the first place?

BO: Yeah, I think that's exactly right. I don't know the ins and outs of that story. I do know that, right, if you look in spaces of functions, that they sort of all have this property, right, among continuous functions, I think it's only a set of measure zero that doesn't have this property. So the sort of basic narrative as I understand it, leading from kind of the start of the 19th century to the end of the 19th century, is basically thinking that we can mostly assume things are good, to realizing that sometimes things are bad (like this function), culminating in the realization that actually basically everything is bad. And the good stuff is just these rare diamonds.

EL: Yeah, I guess maybe this slight, I don't know, silver lining, is that often we can approximate with good things instead. I don't know if that's like the next step on the evolution or something.

BO: Right. Yeah, I guess that's right. Certainly, that's a nice way to salvage some a silver lining, salvage a happy message. Because it's true, right? Even though, a simpler example, the rationals are only a set of measure zero and the reals, you know, they're everywhere, they're dense. So at least, you know, if you have some weird number, you can at least approximate it with a rational.

EL: Yeah, I was just thinking when you were saying this, how it has a really nice analogy to the rationals. And, and even algebraic numbers and stuff like, “Okay, start naming numbers,” you'll probably name whole numbers, which are, you know, this sparse set of measure zero. It’s like, o”h, be more creative,” like, “Okay, well, I'll name some fractions and some square roots and stuff.” But you're still just naming sets of measure zero, you’re never naming some weird transcendental function that I can't figure out a way to compute it.

BO: Yeah, it is funny, right? Because in some sense, right? We've imagined these things called numbers and these things called functions. And then you ask us to pick examples. And we pick the most unlikely, nicest hand-picked, cherry-picked examples. And so the actual stuff—we’ve imagined this category called functions, and most of what's in that category that we developed, we came up with that definition, most of what's in there is stuff that's much too weird for us to begin to picture.

EL: Yeah.

BO: Which says something about, I guess, our reach exceeding our grasp or something. I don't really know, but they are our definitions can really outrun our intuition.

EL: Yeah. So where did you first encounter this function?

BO: That’s a good question. I feel like probably as a kind of folklore bit in maybe 12th grade math. I feel like when I was probably first learning calculus, it was sort of whispered about. You know, my teacher sort of mentioned it offhand. And that was very enticing, and in some sense, that's actually where my whole second book comes from, is all these little bits of folklore, not exactly the thing you teach in class, but the little, I don't know, the thing that gets mentioned offhand. And you go “Wait, what, what was that?” “Oh, well, don't worry. You'll learn about that in your real analysis class in four years.” I don't want to learn about that in four years. Tell me about that now. I want to know about that weird function. And then I think the first proper reading I did was probably in a William Dunham’s book The Calculus Gallery, which is a nice book going through different bits of historical mathematics, beginning with the beginnings of calculus through through like the late 19th century. And he has the here's a nice discussion of the function and its construction.

KK: So when we were preparing for this, you also mentioned there are connections to Brownian motion here. Do you want to mention those for our audience?

BO: Yeah, I love that this turns out—so I have some quotes here from right when this function was sort of debuted, right when it was introduced to the world. You have Émile Picard, his line was, “If Newton and Leibniz had thought that continuous functions do not necessarily have a derivative, the differential calculus would never have been invented.” Which I like. If Newton and Leibniz knew what you were going to do to their legacy, they would never have done this! They would have rejected the whole premise. And then Charles Hermite? [Pronounced “her might, wonders if the pronunciation is correct]

KK: Hermite. [Pronounced “her meet”]

BO: That sounds better. Sounds good. Sure. Right. His line was, and I don't know what the context was, but, “I turn away with fright and horror from this lamentable evil of functions that do not have derivatives.” Which is really layering on I like the way people spoke in the 19th century. There was more, a lot more flavor to their their language.

EL: Yeah.

BO: And Poincaré also, he was saying 100 years ago prior to Weierstrass developing it, such a function would have been regarded as an outrage to common sense. Anyway, so I mention all those. You mentioned Brownian motion, right? The instinct when you see this function is that this is utterly pathological. This is math just completely losing touch with physical reality and giving us these weird intellectual puzzles and strange constructions that can't possibly mean anything to real human beings. And then it turns out that that's not true at all, that Brownian motion—so you look at pollen dancing around on the surface of some water, and it's jumping around in these really crazy aggressive ways. And it turns out our best models of that process, you know, of any kind of Brownian motions—you know, coal dust in the air or pollen on water—our best model to a pretty good approximation has the same property. The path is so jagged and surprising and full of jumps from moment to moment that it's nowhere differentiable, even though the particle obviously sort of has to be continuous. It can’t be discontinuous, I mean, it's jumping, like literally transporting from one place to another. So that's not really the right model. But it is non-differentiable everywhere, which means, weirdly, that it doesn't have a speed, right? Like, a derivative is a is a velocity.

EL: So that means maybe an average speed but not a speed at any time.

BO: Yeah, well, actually, even—I think it depends how you measure. I’d have to looked back at this, because what it means sort of between any two moments according to the model, between any two points in time, is traversing an infinite distance. So I guess it could have an average velocity, but the average speed I think winds up being infinite rates. Over a given time interval, you can just take how far it travels that time interval and divide by time, but I think the speed, if you take the absolute value of the magnitude? I think you sort of wind up with infinite speed, maybe? But really, it's just that you can’t—speed is no longer a meaningful notion. It's moving in such an erratic way. that n you can't even talk about speed.

KK: Well, because that tends to imply a direction. I mean, you know, it’s really velocity. That always struck me as that's the real problem, is that you can't figure out what direction it's going, because it's effectively moving randomly, right?

BO: Yeah, I think that's fair. Yeah. The only way I can build any intuition about it is to picture a single—imagine a baseball having a single non-differentiable moment. So like, you toss it up in the air. And usually what would happen is that it goes up in the air, it kind of slows down and slows down and slows down. There's that one moment when it's kind of not moving at all. And then it begins to fall. And so the non-differentiable version would be, like, you throw it up in the air, it's traveling up at 10 meters per second, and then a trillionth of a second later, it's traveling down at 10 meters per second. And what's happening at that moment? Well, it's just unimaginable. And now for Brownian motion, you've got to picture that that moment is every moment.

KK: Right. Yeah. Weird, weird world.

BO: Yeah.

KK: So another thing we like to do on this podcast is ask our guests to pair their, well in your case construction, with something. What does the Weierstrass function pair with?

BO: Yeah. So I think, I have two things in mind, both of them constructions of new things that kind of opened up new new possibilities that people could not have imagined before. So the first one, maybe I should have picked a specific dish, but I'm picturing basically just molecular gastronomy, this movement in in cooking where you take—one example I just saw recently in a book was, I think it was WD-50, a sort of famous molecular gastronomy restaurant in New York, where they had taken, the comes to you and it looks like a small, poppyseed bagel with lox. And then as it gets closer, you realize it's not a poppyseed bagel with lox, it's ice cream that looks almost identical to a poppyseed bagel with lox. So that's sort of weird enough already. And then you take a taste and you realize that actually, it tastes exactly like a poppyseed bagel with lox, because they've somehow worked in all the flavors into the ice cream.

KK: Hmm.

BO: Anyway, so molecular gastronomy basically is about imagining very, very weird possibilities of food that are outside our usual traditions, much in the way that Weierstrass’s function kind of steps outside the traditional structures of math.

EL: Yeah, I like this a lot. It's a good one. Partly because I'm a little bit of a foodie. And like, when I lived in Chicago, we went to this restaurant that had this amazing, like, molecular gastronomy thing. I’m trying to remember one of the things we had was this frozen sphere of blue cheese. And it was so weird and good. Yeah, you’d get you get like puffs of air that are something, and there’s, like, a ham sandwich, but it was like the bread was only the crust somehow there's like nothing inside. Yeah, it was all these weird things. Liquefied olive that was like in inside some little gelatin thing, and so it was just like concentrated olive taste that bursts in your mouth. So good.

BO: That sounds awesome to me the the molecular gastronomy food. I have very little experience of it firsthand.

KK: So you mentioned a second possible pairing. What would that be?

BO: Yeah, so the other one I had in mind is music. It's a Beatles album, Revolver.

KK: Great album.

BO: One of my favorite albums, and much like molecular gastronomy shows that the foods that we're eating are actually just a tiny subset of the possible foods that are out there, similarly what revolver did for for pop music and in ’65 whenever it came out.

KK: ’66.

BO: Okay. 66 Alright, thank you for that.

EL: I am not well-versed in albums of The Beatles. You know, I am familiar with the music of the Beatles, don’t worry. But I don't know what's on what album. So what is this album?

BO: So Kevin and I can probably go to track by track for you.

KK: I’d have to think about it, but it's got Norwegian Wood on it, for example.

BO: Oh, that's rubber sole, actually.

KK: Oh, that’s Rubber Soul. You're right. Yeah, I lost my Beatles cred. That's right. My bad. I mean, some would argue that—so Revolver was, some people argue, was the first album. Before that, albums had just been collections of singles, even in the case of the Beatles, but Revolver holds together as a piece.

BO: Yeah, that’s one thing. Which again, there's probably some an analogy to Weierstrass’s function there. Also, it begins with this kind of weird countdown where, I don’t remember if it's John or George, but they’re saying 1234 in the intro into Taxman.

KK: Yeah. Into Taxman, which is probably, it's not my favorite Beatles song, but it's certainly among the top four. Right.

BO: Yeah. So that one, already right there it’s a pop song about taxes, which is already, so lyrically, we're exploring different parts of the possibility space than musicians were before. Track two is Eleanor Rigby, which is, the only instrumentation is strings. Which again is something that you didn't really hear in pop. You know, Yesterday had brought in some strings, that was sort of innovative. Other bands have done similar things but, but the idea of a song that’s all strings, and then I’m Only Sleeping as the third track, which has this backwards guitar. They recorded the guitar and just played it backwards. And then Yellow Submarine, which is, like, this weird Raffi song that somehow snuck onto a Beatles album. Yeah, and then For No One has this beautiful French horn solo. Yes, every track is drawn from sort of a distant corner of this space of possible popular music, these kind of corners that had not been explored previously. Anyway, so my recommendation is, is think about the Weierstrass function while eating, you know, a giant sphere of blue cheese and listening to Taxman.

EL: Great. Yeah. I strongly urge all of our listeners to go do that right now.

BO: Yeah, if anyone does it, it'll probably be the first time that that set of activities has been done in conjunction.

EL: Yeah. But hopefully not the last.

BO: Hopefully not the last. That's right. Yeah. And most experiences are like that, in fact.

KK: So we also like to let our guests plug things. You clearly have things to plug.

BO: I do. Yeah. I'm a peddler of wares. Yes, so the prominent thing is my blog is Math with Bad Drawings, and you're welcome to come read that. I try to post funny, silly things there. And then my two books are Math with Bad Drawings, which kind of explores how math pops up in lots of different walks of life, like, you know, in thinking about lottery tickets or thinking about the Death Star is another chapter, and then Change Is the Only Constant is my second book, and it's all about calculus, and it’s sort of calculus through stories. Yeah, that one just came out earlier this year, and I'm quite proud of that one. So you should check it out.

KK: Yeah, so I own both of them. I've only read Math with Bad Drawings. I've been too busy so far to get to Change Is the Only Constant.

EL: And there were there been a slew of good pop—or I assume good because I haven't read most of them yet—pop math books that have come out recently, so yeah I feel like my stack is growing. It’s a fall of calculus or something.

BO: It’s been a banner year. And exactly, calculus has been really at the forefront. Steve Strogatz’s Infinite Powers was a New York Times bestseller, and then David Bressoud [Calculus Reordered] and others who I'm blanking on right now have had one. There was another graphic, like, cartoon calculus that came out earlier this year. So yeah, apparently calculus is kind of having a moment.

EL: Well, and I just saw one about curves.

KK: Curves for the Mathematically Curious. It's sitting on my desk. Many of these books that you've mentioned are sitting on my desk.

EL: So yeah, great year for reading about calculus, but I think Ben would prefer that you start that reading with Change Is the Only Constant.

BO: It's very frothy, it's very quick and light-hearted and should be—you can use it as your appetizer to get into the the, the cheesier balls of the later books.

KK: But it's highly non-trivial. I mean, you talk about really interesting stuff in these books. It's not some frothy thing. I mean it's lighthearted, but it's not simple.

BO: I appreciate that. Yeah, the early draft of the book I was doing pretty much a pretty faithful march through the AP Calculus curriculum. And then that draft wasn't really working. And I realized that part of what I wasn't doing that should be doing was since I'm not teaching, you know, you had to execute calculus maneuvers. I'm not teaching how to take derivatives. I can talk about anything as long as I can explain the ideas. So we've got Weierstrass’s function in there. And there's a little bit even on Lebesgue integration, and other sort of, some stuff on differential equations crops up. So since I'm not actually teaching a calculus course and I don't need to give tests on it, I just got to tell stories.

EL: Well, yeah, I hope people will check that out. And thanks for joining us today.

BO: Yeah, thanks so much for having me.

KK: Yeah. Thanks, Ben.

[outro]

Our guest on this episode, Ben Orlin, is a high school math teacher best-known for his blog and popular math books. He told us about Weierstrass’s construction of a function that is continuous everywhere but differentiable nowhere. Here is a short collection of links that might be interesting.

Ben’s Blog, Math with Bad Drawings

Math with Bad Drawings, the book

Change is the Only Constant

Episode 51 - Carina Curto

13 februari 2020 | 29 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math theorem with no test at the end. I think I decided I liked that tagline. [Editor’s note: Nope, she really didn’t notice that slip of the tongue!]

Kevin Knudson: Okay.

EL: So we’re going to go with that. Yeah. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is the other host.

KK: I’m Kevin Knudson, a professor of mathematics at the University of Florida. How are you doing?

EL: I’m doing well. Yeah, not not anything too exciting going on here. My mother-in-law is coming to visit later today. So the fact that I have to record this podcast means my husband has to do the cleaning up to get ready.

KK: Wouldn’t he do that anyway? Since it’s his mom?

EL: Yeah, probably most of it. But now I've got a really good excuse.

KK: Yeah, sure. Well, Ellen and I had our 27th anniversary yesterday.

EL: Oh, congratulations.

KK: Yeah, we had a nice night out on the town. Got a hotel room just to sit around and watch hockey, as it turns out.

EL: Okay.

KK: But there's a pool at the hotel. And you know, it's hot in Florida, and we don't have a pool. And this is absurd—which Ellen reminds me of every day, that we need a pool—and I just keep telling her that we can either send the kid to college or have a pool. Okay.

EL: Yeah.

KK: I mean, I don't know. Anyway, we're not here talking about that, we're talking about math..

EL: Yes. And we're very excited today to have Carina Curto on the show. Hi, Carina, can you tell us a little bit about yourself?

Carina Curto: Hi, I'm Carina, and I'm a professor of mathematics at Penn State.

EL: Yeah, and I think I first—I don't think we've actually met. But I think the first time I saw you was at the Joint Meetings a few years ago. You gave a really interesting talk about, like, the topology of neural networks, and how your brain has these, like, basically kind of mental maps of spaces that you interact with. It was really cool. So is that the kind of research you do?

CC: Yeah, so that was—I remember that talk, actually, at the Joint Meetings in Seattle. So that was a talk about the uses of typology for understanding neural codes. And a lot of my research has been about that. And basically, everything I do is motivated in some way by questions in neuroscience. And so that was an example of work that's been motivated by neuroscience questions about how your brain encodes geometry and topology of space.

KK: Now, there's been a lot of a lot of TDA [topological data analysis] moving in that direction these last few years. People have been finding interesting uses of topology in neuroscience and studying the brain and imaging stuff like that, very cool stuff.

CC: Yeah.

EL: And did you come from more of a neuroscience background? Or have you been kind of picking that up as you go, coming from a math background?

CC: So I originally came from a mathematical physics background.

EL: Okay.

CC: I was actually a physics major as an undergrad. But I did a lot of math, so I was effectively a double major. And then I wanted to be a string theorist.

KK: Sure, yeah.

CC: I started grad school in 2000. So this is, like, right after Brian Greene’s The Elegant Universe came out.

EL: Right. Yeah.

CC: You know, I was young and impressionable. And so I kind of went that route because I loved physics, and I loved math. And it was kind of an area of physics that was using a lot of deep math. And so I went to grad school to do mathematical string theory in the math department at Duke. And I worked on Calabi-Yaus and, you know, extra dimensions and this kind of stuff. And it was, the math was mainly algebraic geometry, is what right HD thesis was in So this had nothing to do with neuroscience.

EL: Right.

CC: Nothing. And so basically about halfway through grad school—I don't know how better to put it, then I got a little disillusioned with string theory. People laugh now when I say that because everybody is.

KK: Sure.

CC: But I started kind of looking for other—I always wanted to do applied things, interdisciplinary things. And so neuroscience just seemed really exciting. I kind of discovered it randomly and started learning a lot about it and became fascinated. And so then when I finished my PhD, I actually took a postdoc in a neuroscience lab that had rats and, you know, was reporting from the cortex and all this stuff, because I just wanted to to learn as much neuroscience as possible. So I spent three years working in a lab. I didn't actually do experiments. I did mostly computational work and data analysis. But it was kind of a total cultural immersion sort of experience, coming from more of a pure math and physics background.

EL: Right. Yeah, I bet that was a really different experience

CC: It was really different. So I kind of left math in a sense for my first postdoc, and then I came back. So I did a second postdoc at Courant at NYU, and then started getting ideas of how I could tackle some questions in neuroscience using mathematics. And so ever since then, I've basically become a mathematical neuroscientist. I guess I would call myself.

KK: So 2/3 of this podcast is Duke alums. That's good.

CC: Oh yeah? Are you a Duke alum?

KK: I did my degree there too. I finished in ’96.

CC: Oh, yeah.

KK: Okay. Yeah.

CC: Cool.

EL: Nice. Well, so what is your favorite theorem?

CC: So I have many, but the one I chose for today is the Perron-Frobenius theorem.

KK: Nice.

EL: All right.

CC: And so you want to know about it, I guess?

KK: We do. So do our listeners.

CC: So it's actually really old. I mean, there are older theorems, but Perron proved it, I think in 1907 and Frobenius in 1912, so it carries both of their names. So it's over 100 years old. And it's a theorem and linear algebra. So it has to do with eigenvectors and eigenvalues of matrices.

KK: Okay.

CC: And so I'll just tell you quickly what it is. So, if you have a square matrix, so like an n×n square matrix with all positive—so there are many variations of that theorem. I'm going to tell you the simplest one—So if all the entries of your matrix are positive, then you are guaranteed that your largest eigenvalue is unique and real and is positive, so a positive real part. So eigenvalues can be complex. They can come in complex conjugate pairs, for example, but when we talk about the largest one, we mean the one that has the largest real part.

EL: Okay.

KK: All right.

CC: And so one part of the theorem is that that eigenvalue is unique and real and positive. And the other part is that you can pick the corresponding eigenvector for it to be all positive as well.

EL: Okay. And we were talking before we started taping that I'm not actually remembering for sure whether we've used the words eigenvector and eigenvalue yet on the podcast, which, I feel like we must have because we've done so many episodes, but yeah, can we maybe just say what those are for anyone who isn't familiar?

CC: Yeah. So when you have a matrix, like a square matrix, you have these special vectors. So the matrix operates on vectors. And so a lot of people have learned how to multiply a matrix by a vector. And so when you have a vector, so say your matrix is A and your vector is x, if A times x gives you a multiple of x back—so you basically keep the same vector, but maybe scale it—then x is called an eigenvector of A. And the scaling factor, which is often denoted λ, is called the eigenvalue associated to that eigenvector.

KK: Right. And you want x to be a nonzero vector in this situation.

CC: Yes, you want x to be nonzero, yes, otherwise it's trivial. And so I like to think about eigenvectors geometrically because if you think of your matrix operating on vectors in some Euclidean space, for example, then what it does, what the matrix will do, is it will pick up a vector and then move it to some other vector, right? So there's an operation that takes vectors to vectors, called linear transformations, that are manifested by the matrix multiplication. And so when you have an eigenvector, the matrix keeps the eigenvector on its own line and just scales, or it can flip the sign. If the eigenvalue is negative, it can flip it to point the other direction, but it basically preserves that line, which is called the eigenspace associated. So it has a nice geometric interpretation.

EL: Yeah. So the Perron-Frobeius theorem, then, says that if your matrix only has positive entries, then there's some eigenvector that's stretched by a positive amount.

CC: So yeah, so it says there's some eigenvector where the entries of the vector itself are all positive, right, so it lies in the positive orthant of your space, and also that the the corresponding eigenvalue is actually the largest in terms of absolute value. And the reason this is relevant is because there are many kind of dynamic processes that you can model by iterating a matrix multiplication. So, you know, one simple example is things like Markov chains. So if you have, say, different populations of something, whether it be, say, animals in an ecosystem or something, then you can have these transition matrices that will update the population. And so, if you have a situation where if your matrix that's updating your population has—whatever the leading eigenvalue is of that matrix is going to control somehow the long-term behavior of the population. So that top eigenvalue, that one with the largest absolute value, is really controlling the long-term behavior of your dynamic process.

EL: Right, it kind of dominates.

CC: It is dominating, right. And you can even see that just by hand when you sort of multiply, if you take a matrix times a vector, and then do it again, and then do it again. So instead of having A times x, you have A squared times x or A cubed times x. So it's like doing multiple iterations of this dynamic process. And you can see how, then, what’s going to happen to the to the vector if it's the eigenvector. Well, if it's an eigenvector, well, what's going to happen is when you apply the matrix once, A times x, you're going to get λ times x. Now apply A again. So now you're applying A to the quantity λx, but the λ comes out, by the linearity of the of the matrix multiplication, and then you have Ax again, so you get another factor of λ, so you get λ^2 times x. And so if you keep doing this, you see that if I do A^k times x, I get λ^k times x. And so if that λ is something, you know, bigger than 1, right, my process is going to blow up on me. And if it's less than 1, it's going to converge to zero as I keep taking powers. And so anyway, the point is that that top eigenvector is really going to dominate the dynamics and the behavior. And so it's really important if it's positive, and also if it's bigger or less than 1, and the Perron-Frobenius theorem basically tells you that you have, it gives you control over what that top eigenvalue looks like and moreover, associates it to an all-positive eigenvector, which is then a reflection of maybe the distribution of population. So it's important that that be positive too because lots of things we want to model our positive, like populations of things.

KK: Negative populations aren't good. Yeah,

CC: Yes, exactly. And so this is one of the reasons it's so, useful is because a lot of the things we want to model are—that vector that we apply the matrix to is reflecting something like populations, right?

KK: So already this is a very non-obvious statement, right? Because if I hand you an arbitrary matrix, I mean, even like a 2×2 rotation matrix, it doesn't have any eigenvalues, any real eigenvalues. But the entries aren't all positive, so you’re okay.

CC: Right. Exactly.

KK: But yeah, so a priori, it's not obvious that if I just hand you an n×n matrix with all real entries that it even has a real eigenvalue, period.

CC: Yeah. It's not obvious at all, and let alone that it's positive, and let alone that it has an eigenvector that's all positive. That's right. And the positivity of that eigenvector is really important, too.

EL: Yeah. So it seems like if you're doing some population model, just make sure your matrix has all positive entries. It’ll make your life a lot easier.

CC: So there's an interesting, so do you do you know what the most famous application of the Perron-Frobenius theorem is?

EL: I don't think I do.

KK: I might, but go ahead.

CC: You might, but I’ll go ahead?

KK: Can I guess?

CC: Sure.

KK: Is it Google?

CC: Yes. Good. Did you Google it ahead of time?

KK: No, this is sort of in the dark recesses of my memory that essentially they computed this eigenvector of the web graph.

CC: Right. Exactly. So back in the day, in the late ‘90s, when Larry Page and Sergey Brin came up with their original strategy for ranking web pages, they used this theorem. This is like, the original PageRank algorithm is based on this theorem, because they're, they have again the Markov process where they imagine some web—some animal or some person—crawling across the web. And so you have this graph of websites and edges between them. And you can model the random walk across the web as one of these Markov processes where there's some matrix that that reflects the connections between web pages that you apply over and over again to update the position of the of the web crawler. And and so now if you imagine a distribution of web crawlers, and you want to find out in the long run what pages do they end up on, or what fraction of web crawlers end up on which pages, it turns out that the Perron-Frobenius theorem gives you precisely the existence of this all-positive eigenvector, which is a positive probability that you have on every website for ending up there. And so if you look at the eigenvector itself, that you get from your web matrix, that will give you a ranking of web pages. So the biggest value will correspond to the most, you know, trafficked website. And smaller values will correspond to less popular websites, as predicted by this random walk model.

EL: Huh.

CC: And so it really is the basis of the original PageRank. I mean, they do fancier things now, and I'm sure they don't reveal it. But the original PageRank algorithm was really based on this. And this is the key theorem. So I think it's a it's kind of a fun thing. When I teach linear algebra, I always tell students about this.

KK: Linear Algebra can make you billions of dollars.

CC: Yes.

KK: That’ll catch students’ attention.

CC: Yes, it gets students’ attention.

EL: Yes. So where did you first encounter the Perron-Frobenius theorem?

CC: Probably in an undergrad linear algebra class, to be honest. But I also encountered it many more times. So I remember seeing it in more advanced math classes as a linear algebra fact that becomes useful a lot. And now that I'm a math biologist, I see it all the time because it's used in so many biological applications. And so I told you about a population biology application before, but it also comes up a lot in neural network theory that I do. So in my own research, I study these competitive neural networks. And here I have matrices of interactions that are actually all negative. But I can still apply the theorem. I can just flip the sign.

EL: Oh, right.

CC: And apply the theorem, and I still get this, you know, dominant eigenvalue and eigenvector. But in that case, the eigenvalue is actually negative, and I still have this all-positive eigenvector that I can choose. And that's actually important for proving certain results about the behavior of the neural networks that I study. So it's a theorem I actually use in my research.

EL: Yeah. So would you say that your appreciation of it has grown since you first saw it?

CC: Oh for sure. Because now I see it everywhere.

EL: Right.

CC: It was one of those fun facts, and now it’s in, you know, so many math things that I encounter. It's like, oh, they're using the Perron-Frobenius theorem. And it makes me happy.

EL: Yeah, well, when I first read the statement of the theorem, it's not like it bowled me over, like, “Oh, this is clearly going to be so useful everywhere.” So probably, as you see how many places it shows up, your appreciation grows.

CC: Yeah, I mean, that's one of the things that I think is really interesting about the theorem, because, I mean, many things in math are like this. But you know, surely when Perron and Frobenius proved it over 100 years ago, they never imagined what kinds of applications it would have. You know, they didn't imagine Google ranking web pages, or the neural network theory, or anything like this. And so it's one of these things where it's like, it's so basic. Maybe it could look initially like a boring fact of linear algebra, right? If you're just a student in a class and you're like, “Okay, there's going to be some eigenvector, eigenvalue, and it's positive, whatever.” And you can imagine just sort of brushing it off as another boring fact about matrices that you have to memorize for the test, right? And yet, it's surprisingly useful. I mean, it has applications in so many fields of applied math and in pure math, and so it's just one of those things that gives you respect for even seemingly simple and not obviously, it doesn't bowl you over, right, you can see the statement and you're not like, “Wow, that's so powerful!” But it ends up that it's actually the key thing you need in so many applications. And so, you know, it's earned its place over time. It's aged nicely.

EL: And do you have a favorite proof of this theorem?

CC: I mean, I like the elementary proofs. I mean, there are lots of proofs. So I think there's an interesting proof by Birkhoff. There are some proofs that involve the Brouwer fixed point theorem, which is something maybe somebody has chosen already.

EL: Yes, actually. Two people have chosen it!

CC: Two people have chosen the Brouwer fixed point theorem. Yeah, I would imagine that's a popular choice. So, yeah, there are some proofs that rely on that, which I think is kind of cool. So those are more modern proofs of it. That's the other thing I like about it, is that it has kind of old-school elementary proofs that an undergrad in a linear algebra class could understand. And then it also has these more modern proofs. And so it's kind of an interesting theorem in terms of the variety of proofs that it admits.

KK: So one of the things we like to do on this podcast is we like to invite our guests to pair their theorem with something. So I'm curious, I have to know what pairs well with the Perron-Frobenius theorem?

CC: I was so stressed out about this pairing thing!

KK: This is not unusual. Everybody says this. Yeah.

CC: What is this?

KK: It’s the fun part of the show!

CC: I know, I know. And so don't know if this is a good pairing, but I came up with this. So I went to play tennis yesterday. And I was playing doubles with some friends of mine. And I told them, I was like, I have to come up with a pairing for my favorite theorem. So we chatted about it for a while. And as I was playing, I decided that I will pair it with my favorite tennis shot.

EL: Okay.

CC: So, my favorite shot in tennis is a backhand down the line.

KK: Yes.

CC: Yeah?

KK: I never could master that!

CC: Yeah. The backhand down the line is one of the basic ground strokes. But it's maybe the hardest one for amateur players to master. I mean, the pros all do it well. But, you know, for amateurs, it's kind of hard. So usually people hit their backhand cross court. But if you can hit that backhand down the line, especially when someone's at the net, like in doubles, and you pass them, it's just very satisfying, kind of like, win the point. And for my tennis game, when my backhand down the line is on, that's when I'm playing really well.

EL: Nice.

CC: And I like the linearity of it.

EL: Right, it does seem like, you know, you're pushing it down.

CC: Like I'm pushing that eigenvector.

KK: It’s very positive, everything's positive about it.

CC: Everything’s positive. The vector with the tennis ball, just exploding down the line. It's sort of maybe it's a stretch, but that's kind of what I decided.

EL: A…stretch? Like with an eigenvalue and eigenvector?

CC: Right, exactly. I needed to find a pairing that was a stretch.

EL: I think this is a really great pairing. And you know, something I love about the pairing thing that we do—other than the fact that I came up with it, so of course, I'm absurdly proud of it—is that I think, for me at least it's built all these bizarre connections with math and other things. It's like, now when I see the mean value theorem, I'm like, “Oh, I could eat a mango.” Or like, all these weird things. So now when I see people playing tennis, I'll be like, “Oh, the Perron-Frobenius theorem.”

CC: Of course.

EL: So are you a pretty serious tennis player?

CC: I mean, not anymore. I played in college for a little bit. So when I was a junior, I was pretty serious.

EL: Nice. Yeah, I’m not really a tennis person I've never played or really followed it. But I guess there's like some tennis going on right now that's important?

CC: The French Open?

EL: That’s the one!

KK: Nadal really stuck it to Federer this morning. I played obsessively in high school, and I was never really any good, and then I kind of gave it up for a long time, and I picked up again in my 30s and did league tennis when I lived in Mississippi. And my team at our level—we were just sort of very intermediate players, you know—we won the state championship two years in a row.

CC: Wow.

KK: And then and then I gave it up again when I moved to Florida. My shoulder can't take it anymore. I was one of these guys with a big booming serve and a pretty good forehand and then nothing else, right?

CC: Yeah.

KK: So you know, if you work my backhand enough you're going to destroy me.

EL: Nice. Oh, yeah, that's a that's a lot of fun. And I hope other our tennis appreciator listeners will now have have an extra reason to enjoy this theorem too. So yeah, we also like to give our guests a chance, like if they have a website or book or anything they want to mention—you know, if people want to find them online and chat about tennis or linear algebra— is there anything you want to mention?

CC: I mean, I don't have a book or anything that I can plug, but I guess I wanted to just plug linear algebra as a subject.

KK: Sure.

CC: I feel like linear algebra is one of the grand achievements of humanity in some ways. And it should really shine in the public consciousness at the same level as calculus, I think.

EL: Yeah.

KK: Maybe even more.

CC: Yeah, maybe even more. And now, everybody knows about calculus. Every little kid knows about calculus. Everyone is like, “Oh, when when are you going to get to calculus?” You know, calculus, calculus. And linear algebra—it also has kind of a weird name, right, so it sounds very elementary somehow, linear and algebra—but it's such a powerful subject. And it's very basic, like calculus, and it's used widely and so I just want to plug linear algebra.

EL: Right. I sometimes feel like there are basically—so math can boil down to like, doing integration by parts really well or doing linear algebra really. Like, I joked with somebody, like, I didn't end up doing a PhD in a field that used a lot of linear algebra, but I sort of got my PhD in applied integration by parts, it's just like, “Oh, yeah. Figure out an estimate based on doing this.” And I think linear algebra, especially now with how important social media and the internet are, it is really an important field that, I agree, more people should know about. It is one of the classes that when I took it in college, it's one of the reasons I—at that time, I was trying to get enough credits to finish my math minor. And I was like, “Oh, yeah, actually, this is pretty cool. Maybe I should learn a little more of this math stuff.” So, yeah, great class.

CC: And you know, it's everywhere. And you know, there are all these people, almost more people have heard of algebraic topology than linear algebra, outside, you know, because it's this fancy topology or whatever. But when it comes down to it, it's all linear algebra tricks. With some vision of how to package them together, of course, I’m not trying to diminish the field, but somehow linear algebra doesn't get it’s—it’s the workhorse behind so much cool math and yeah, doesn't get its due.

EL: Yes, definitely agree.

KK: Yeah. All right. Well, here's to linear algebra.

EL: Thanks a lot for joining us.

CC: Thank you.

KK: It was fun.

[outro]

Our guest on this episode, Carina Curto, is a mathematician at Penn State University who specializes in applications in biology and neuroscience. She talked about the Perron-Frobenius theorem. Here are some links you may find useful as you listen to this episode.

Curto’s website
A short video of Curto talking about how her background in math and physics is useful in neuroscience and a longer interview in Quanta Magazine
An article version of Curto’s talk a few years ago about topology and the neural code
Curto ended the episode with a plug for linear algebra as a whole. If you’re looking for an engaging video introduction to the subject, check out this playlist from 3blue1brown.

Episode 50 - aBa

9 januari 2020 | 35 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer, usually in Salt Lake City, Utah, currently in Providence, Rhode Island. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics, almost always at the University of Florida these days. How's it going?

EL: All right. We had hours of torrential rain last night, which is something that just doesn't happen a whole lot in Utah but happens a little more often in Providence. So I got to go to sleep listening to that, which always feels so cozy, to be inside when it's pouring outside.

KK: Yeah, well, it's actually finally pleasant in Florida. Really very nice today and the sun's out, although it's gotten chilly—people can't see me doing the air quotes—it’s gotten “chilly.” So the bugs are trying to come into the house. So the other night we were sitting there watching something on Netflix and my wife feels this little tickle on her leg and it was one of those big flying, you know, Florida roaches that we have here.

EL: Ooh

KK: And our dog just stood there wagging at her like, “This is fun.” You know?

EL: A new friend!

KK: “Why did you scream?”

EL: Yeah, well, we’re happy today to invite aBa to the show. ABa, would you like to introduce yourself?

aBa Mbirika: Oh, hello. I’m aBa. I'm here in Wisconsin at the University of Wisconsin Eau Claire. And I have been here teaching now for six years. I tell them where I'm from?

EL: Yeah.

KK: Sure.

aM: Okay. I am from, I was born and raised in New York City. I prefer never to go back there. And then I moved to San Francisco, lived there for a while. Prefer never to go back there. And then I went up to Sonoma County to do some college and then moved to Iowa, and Iowa is really what I call home. I'm not a city guy anymore. Like Iowa is definitely my home.

EL: Okay.

KK: So Southwestern Wisconsin is also okay?

aM: Yeah, it's very relaxing. I feel like I'm in a very small town. I just ride my bicycle. I still don't know how to drive, like all my friends from New York and San Francisco. But I don't need a car here. There's nowhere to go.

EL: Yeah.

aM: But can I address why you just called me aBa, as I asked you to?

EL: Yeah.

aM: Yeah, because maybe I'll just put this on the record. I mean, I don't use my last name. I think the last time I actually said some version of my last name was grad school, maybe? The year 2008 or something, like 10 years ago was the last time anyone's ever heard it said. And part of the issue is that it's It's pronounced different depending on who's saying it in my family. And actually it's spelled different depending on who’s in the family. Sometimes they have different letters. Sometimes there's no R. Sometimes it’s—so in any case, if I start to say one pronunciation, I know Americans are going to go to town and say this is the pronunciation. And that's not the case. I can't ask my dad. He's passed now, but he didn’t have a favorite. He said it five different ways my whole life, depending on context. So he doesn't have a preference, and I'm not going to impose one. So I'm just aBa, and I'm okay with that.

EL: Yeah, well, and as far as I know, you're currently the only mathematician named aBa. Or at least spelled the way yours is spelled.

aM: Oh yeah, in the arXiv. Yeah, on Mathscinet that it’s. Yeah, I'm the only one there. Recently someone invited me to a wedding and they were like, what's your address? And I said, “aBa and my address is definitely enough.”

EL: Yeah, so what theorem would you like to tell us about?

aM: Oh, okay, well I was listening actually to a couple of you shows recently, and Holly didn’t have a favorite theorem, Holly Krieger. I'm exactly the same way. I don't even have a theorem of, like, the week. She was lucky to have that. I have a theorem of the moment. I would like to talk about something I discovered when I was in college, that’s kind of the reason. but can I briefly say some of my like, top hits just because?

EL: Oh yeah.

KK: We love top 10 lists. Yeah, please.

aM: Okay. So I'm in combinatorics, loosely defined, but I have no reason—I don't know why people throw me in that bubble. But that's the bubble that that I've been thrown in. But my thesis—actually, I don’t ever remember the title, so I have to read it off a piece of paper—Analysis of symmetric function ideals towards a combinatorial description of the cohomology ring of Hessenberg varieties.

KK: Okay.

aM: Okay, all those words are necessary there. But my advisor said, “You're in combinatorics.” Essentially, my problem was, we were studying an object and algebraic geometry, this thing called a Hessenberg variety. To study this thing we used topology. We looked at the cohomology ring of this, but that was very difficult. So we looked at this graded ring from the lens of commutative algebra. And I studied the algebra the string by looking at symmetric functions, ideals of symmetric functions, and hence that's where my advisor said, “You're in combinatorics.” So it was the main tool used to study a problem an algebraic geometry that we looked at topology. Whatever, so I don't know what I am. But any case for top 10 hits, not top 10, but diagram chasing. Love it. Love it.

EL: Wow, I really don't share that love, but I’m glad somebody does love it.

aM: Oh, it's just so fun for students.

KK: So the snake lemma, right?

aM: The snake lemma, yes. It's a little bit maybe above the level of our algebra two class that I teach here for undergrads, but of course I snuck it in anyways. And the short five lemma. Those are like, would be my favorites if the moment was, like, months ago. In number theory I have too many faves, but I’m going to limit it to Euler-Fermat’s theorem that if a and n are coprime, then a to the power of the Euler totient function of n is congruent to 1 mod n. But that leads to Gauss’s epically cool awesome theorem on the existence of primitive roots. Now, this is my current craze.

EL: Okay.

aM: And this is just looking at the group of units in Z mod nZ, or more simply the multiplicative group of units of integers modulo n. When is this group cyclic? And Gauss said it's only cyclic when n is the numbers 2, or 4, or an odd prime to a k power, or twice an odd prime to some k-th power. And basically, those are very few. I mean, those are very little numbers in the broad spectrum of the infinity of the natural numbers. So this is very cool. In fact, I'm doing a non-class right now with a professor who retired maybe 10 years ago from our university, and I emailed him and said, “Want to have fun on my like my research day off?” And we’re studying primitive roots because I don't know anything about it. Like, my favorite things are things I know nothing about and I want to learn a lot about.

EL: Yeah, I don't think I've heard that theorem before. So yeah, I'll have to look that up later.

aM: Yes. And then the last one is from analysis, and I did hear Adrianna Salerno talked about it and in fact, I think also someone before her on your podcast, but Cantor’s theorem on uncountability of the real numbers.

EL: Yeah, that's that's a real classic.

aM: I just taught that two days ago in analysis, and like, it's like waiting for their heads to explode. And I think, I don't know, my students’ heads weren't all exploding. But I was like, “This is so exciting! Why are you not feeling the excitement?” So yeah, yeah, it was only my second time teaching analysis. So maybe I have to work on my sell.

EL: Yeah, you'll get them next time.

aM: Yeah. It's so cool! I even mentioned it to my class that’s non-math majors, just looking at sets, basic set theory. And this is my non-math class. These students hate math. They're scared of math. And I say, “You know, the infinity you know, it's kind of small. I mean, you're not going to be tested on this ever. But can I please take five minutes to like, share something wonderful?” So I gave them the baby version of Cantor’s theorem. Yeah, but that's it. I just want to throw those out there before I was forced to give you my favorite theorem.

EL: Yes. So now…

KK: We are going to force you, aBa. What is your favorite theorem?

EL: We had the prelude, so now this is the main event.

aM: Okay, main event time. Okay, you were all young once, and you remember—oh, we’re all young, all the time, sorry—but divisibility by 9. I guess when we're in high school—maybe even before that—we know that the number 108 is divisible by 9 because 1+0+8 is equal to 9. And that's divisible by 9. And 81 is divisible by 9 because 8+1 is 9, and 9 is divisible by 9. But not just that, the number 1818 is divisible by 9 because 1+8+1+8 is 18. And that's divisible by 9. So when we add up the digits of a number, and if that sum is divisible by 9, then the number itself is divisible by 9. And students know this. I mean, everyone kind of knows that this is true. I guess I was a sophomore in college. That was maybe a good 4 to 6 years after I started college because, well, that was hard. It's a different podcast altogether, but I made some choices to meet friends who made it really hard for me to go to school consistently in San Francisco—part of the reason why I'm kind of okay not going back there much anymore. Friends got into trouble too much.

But I took a number theory course and learned a proof for that. And the proof just blew my mind because it was very simple. And I wasn't a full-blown math major yet. I think I was in physics— I had eight majors, different majors through the time—I wasn't a math person yet. And I was on a bus going from—Oh, this is in Sonoma County. I went to Sonoma State University as my fourth or fifth college that I was trying to have a stable environment in. And this one worked. I graduated from there in 2004. It definitely worked. So I was on a bus to visit some of my bad friends in San Francisco—who I love, by the way, I'm just saying of the bad habits—and I was thinking about this theorem of divisibility by 9 and saying, what about divisibility by 7? No one talks about that. Like, we had learned divisibility by 11. Like the alternating sum of the digits, if that's divisible by 11, then the number is divisible by 11. But what about 7? You know, is that doable? Or why is it not talked about?

EL: Yeah.

aM: So it was an hour and a half bus ride. And I figured it out. And it was extremely, like, the same exact proof as the divisibility by 9, but boiled down to one tiny little change. But it's not so much that I love this theorem. I actually haven't even told it to you yet. But that I did the proof, that it changed my life. I really—that’s the only thing I can go back to and say why am I an associate professor at a university in Wisconsin right now. It was the life-changing event. So let me tell you the theorem.

EL: Yeah.

aM: It’s hardly a theorem, and this is why I don't know if it even belongs on this show.

EL: Oh, it totally does!

aM: Okay, so I don't even think I had calc 2 yet when I discovered this little theorem. All right, so here we go. So look at the decimal representation of some natural number. Call it n.

EL: I’ve got my pencil out. I'm writing this down.

aM: Oh, okay. Oh, great. Okay, I'm reading off a piece of paper that I wrote down.

EL: Yeah, you said something about it to us earlier. And I was like, “I'm going to need to have this written down.” It’s funny that I do a podcast because I really like looking at things that are written down. That helps me a lot. But let's podcast this thing.

aM: Okay, so say we have a number with k+1 digits. And so I'm saying k+1 because I want to enumerate the digits as follows: the units digit I'm going to call a0, the tens digit I’ll call a1 the hundreds place digit a2 etc, etc, down to the k+1st digit, which we’ll call ak. So read right to left, like in Hebrew, a0, a1 a2 … (or \cdots, you LaTeX people) ak-1 then the last far left digit ak.

EL: Yeah.

aM: So that is a decimal representation of a number. I mean, we're just, you know, like number 1008. That would be a0 is the number 8, a1 is the number 0, a2 is number 0, a3 is the number 1. So we just read right to left. So we can represent this number, and everybody knows this when you're in junior math, I guess in elementary school, that we can write the number—now I'm using a pen—123 as 3 times 1 plus—how many tens do we have? Well, we have two tens. So 2 times 10. How many hundreds do we have? Well, we have one of those. So 1 times 100. So just talking about, yeah, this is mathematics of the place value system in base 10. No surprise here. But a nicer way to write it as a fat sum, where i, the index goes from 0 to k of ai times 10i.

EL: Yeah.

aM: That’s how we in our little family of math nerds, how we compactly write that. So when we think about when does this number divisible by 7? It suffices to think about when what is the remainder when each of these summands is—when we divide each of these summands by 7, and then add up all those remainders and then take that modulo 7. So the key and crux of this argument is that what is 10 congruent to mod 7? Well, 10 leaves the remainder of 3 when you divide by 7. In the great language of concurrences—Thank you, Gauss—10≡3 mod 7. So now we can look at this, all of these tens we have. We have a0 ×100+ a1 ×101 + a2 ×102, etc, etc. When we divide this by 7, this number really is now a0 ×30 — because I can replace my 100 with 30 —plus a1 ×31 instead of—because 101 is the same as 31 in modulo 7 land—plus a2 ×32, etc. etc…. to the last one, ak ×3k. Okay, here I am on the bus thinking, “This is only cool if I know all my powers of 3.”

EL: Yeah. Which are not really that much easier than figuring it out in the first place.

aM: Okay, but I'm young mathematically and I'm just really super excited. So one little example, I guess this is not, I can't remember what I did on the bus, but 1008 is is a number that's divisible by seven. And let's just perform this check, using this check on this number. So is 1008 really divisible by 7? What we can do is according to this, I take the far right digit, the units digit, and that's 8 ×30, so that's just the number 8, 8×1, plus 0×31. Well, that's just 0, thankfully. Then the next, the hundreds place, that’s 0×32. So that's just another 0. And then lastly, the thousands place, 1×33 and that's 27. Add up now my numbers 8+0+0+27. And that's 35. And that's easy to know that the divisibility of. 7 divides 35 and thus 7 divides 1008. And, yeah, I don't know, I’m traveling back in time, and this is not a marvelous thing. But everybody, unfortunately, who I saw in San Francisco that day, and the next day, learned this. I just had to teach all my friends because I was like, “Well, this is not what I'm doing for college. This is something I figured out on the bus. This math stuff is great.”

EL: Yeah, just the fact that you got to own that.

aM: Yeah. And that also it wasn't in the book, and actually it wasn't in subsequently any book I've ever looked in ever since. But it's still just cute. I mean, it's available. And what it did, I guess it just touched me in a way, where I guess I didn't know about research, I didn't know about a PhD program. My end goal was to get a job, continue at the photocopy place that was near the college, where I worked. I really told my boss that, and I really believed that I was going to do that. And our school never really sent people to graduate programs. I was one of the first. And I don't know, it just changed me. And there were a lot of troubles in my life before then. And this is something that I owned. And that's my favorite theorem on that bus that day.

KK: It’s kind of an origin story, right?

aM: Yes, because people ask me, how did you get interested in math? And I always say the classic thing. Forget this story, but I'm also not speaking to math people. My usual thing is the rave scene. I mean, that was what I was involved in in San Francisco, and then, I don't know if you know what that is, but electronic dance music parties that happen in beaches and fields and farms and houses.

EL: What, you don’t think we go to a lot of raves?

aM: I don’t know if raves still happen!

EL: You have accurately stereotyped me.

aM: Okay. Now, I have to admit my parents were worried about that. And they said, “Ecstasy! Clubs!” and I was like, “No, Mom. That's a different rave. My people are not indoors. We’re outdoors, and we're not paying for stuff, and there's no bar, and there's no drinking. We're just dancing and it's daytime. It was a different thing. But that's really why I got involved in this math thing. In some sense, I wanted to know how all of that music worked, and that music was very mathematical.

EL: Oh.

aM: But then I kind of lost interest in studying the math of that because I just got involved in combinatorics and all the beautiful, theoretical math that fills my spirit and soul. But the origin story is a little bit rave, but mostly that bus.

EL: Yeah. A lot of good things happen on buses.

aM: You guys know about the art gallery theorem? Guarding a museum.

EL: Yeah. Yeah.

aM: What’s the minimum number of guards? Okay, I took the seat of someone—my postdoc was at Bowdoin college, and sadly the person who passed away shortly before I got the job was a combinatorialist named Steve Fisk (I hope I’ve got the name right). In any case, he's in the Proofs from the Book, for coming up with a proof for that art gallery theorem. You know, the famous Proofs from the Book, the idea that all the beautiful proofs are in some book? But yeah, guess where he came up with that, he told the chair of the math department when I started there: on a bus! And he was somewhere in Eastern Europe on a bus, and that's where he came up with it. And it's just like, yeah, things can happen on a bus, you know?

EL: Yeah. Now I want our listeners to, like, write in with the best math they've ever done on a bus or something. A list of bus math.

aM: You also have to include trains, I think, too.

EL: Yeah. Really long buses.

aM: All public transportation.

EL: Yeah. So something that we like to do on this podcast is ask our guests to pair their theorem with something. So what have you chosen to pair with your favorite theorem?

aM: Oh my gosh, I was supposed to think about that. Yes. Okay. Oh, 7.

EL: I feel like you have so many interests in life. You must you must have something you can think of.

aM: Oh, no, it's not a problem. I do currently a lot of mathematics. I'm in my office, sadly, a lot of hours of the day, but sometimes I leave my office and go to the pub down the road. And I call it a pub because it's really empty and brightly lit and not populated by students. It's kind of like a grown up bar. But I do a lot of recreational math there, especially on primitive roots recently. So I think I would pair my 7 theorem with seven sips of Michelob golden draft light. It's just a boring domestic beer. And then I would go across the street to the pizza place that's across from my tavern, and I would eat seven bites of a pizza with pepperoni, sausage, green pepper, and onion.

EL: Nice.

aM: I have a small appetite. So seven people would say yes, he can probably do seven bites before he’s full and needs to take a break.

EL: Or you could you could share it with seven friends.

aM: Yes. Oh, I'm often taking students down there and buying pizza for small sections of research students or groups of seven. Yes.

EL: Nice. So I know you wanted to share some other things with us on this podcast. So do you want to talk about those? Or that? I don't know exactly what form you would like to do this in.

aM: Oh, I wrote a poem. Yeah, I just want to share a poem that I wrote that maybe your listeners might find cute.

EL: Yeah. And I'd like to say I think the first time—I don't think we actually met in person that time, but the first time I saw you—was at the poetry reading at a Joint Math Meeting many years ago.

aM: Oh my gosh! I did this poem, probably.

EL: You might have. I’ll see I remember you. Many people might have seen you because you do stand out in a crowd. You know, you dress in a lot of bright colors and you have very distinctive glasses and hair and everything. So you were very memorable at the time. Yes, right now it's pink, red, and yeah, maybe just different shades of pink.

aM: Yes.

EL: But yeah, I remember seeing you do a poem at this this joint math poetry thing and then kept seeing you at various things and then we met, you know, a few years ago when I was at Eau Claire, I guess, we actually met in person then. But yeah, go ahead, please share your poem with us.

aM: Okay, this is part of the origin story again. This was just shortly after this seven thing from the bus. I was introduced to a proofs class, and they were teaching bijective functions. And I really didn't get the book. It was written by one of my teachers, and I was like, you know, I wrote a poem about it. And I think I understand my poem a little bit more than what you wrote in your book. And like, they actually sing this song now. So they recite it, so say the teachers at Sonoma State, each year to students who are taking this same course. But here it is, I think it's sometimes called a rap because I kind of dance around the room when I sing it. So it's called the Bijection Function Poem. And here you go. Are you ready?

EL: Yes.

KK: Let’s hear it.

aM: All right.

And it clearly follows that the function is bijective
Let’s take a closer look and make this more objective
It bears a certain quality – that which we call injective
A lovin’ love affair, Indeed, a one-to-one perspective.
Injection is the stuff that bonds one range to one domain
For Mr. X in the domain, only Miss Y can take his name
But if some other domain fool should try to get Miss Y’s affection,
The Horizontal Line Police are here to check for 1 to 1 Injection.

(Okay, that’s a little racy.)

Observe though, that injection does not alone grant one bijection
A function of this kind must bear Injection AND Surjection
Surjection!? What is that? Another math word gone surreal
It’s just a simple concept we call “Onto”. Here‟s the deal:
If for EVERY lady ‘y’ who walks the codomain of f
There exists at least one ‘x’ in the Domain who fancies her as his sweet best.
So hear the song that Onto sings – a simple mathful melody:
“There ain’t a Y in Codomain not imaged by some X, you see!”
So there you have it 2 conditions that define a quality.
If it’s injective and surjective, then it’s bijective, by golly!

(So this is the last verse. And there's some homework problems in my last verse, actually.)

Now if you’re paying close attention to my math-poetic verse
I reckon that you’ve noticed implications of Inverse
Inverse functions blow the same tune – They biject oh so happily
By sheer existence, inverse functions mimic Onto qualities (homework problem 1)
And per uniqueness of solution, another inverse golden rule (homework problem 2)
By gosh, that’s one-to-one & Onto straight up out the Biject School!
Word!

aM: Yeah, I never tire that one. I love teaching a proofs class.

EL: Yeah. And you said you use it in your class every time you teach it?

aM: Every time I have to say bijection. I mean, the song works, though. My only drawback in recent times is my wording long ago for “Mr. X in the domain” and “Miss Y can take his name” and the whole binary that this thing is doing. So I do have versions, I have a homosexual version, I have a this version—this is the hetero version—then I have the yet-to-be-written binary-free version, which I don't know how to make that because I was thinking for “Person X in the domain, only Person Y can take his name,” but you know person doesn't work. It's too long syllabically so I'm working on that one.

EL: Yeah.

aM: I’m working on that one.  EL: Well, yeah, modernize it for for the times we live in now.

aM: Yes. I kind of dread reading and reciting this is purely hetero version, you know? And also there's not necessarily only one Miss Y that can take Mr. X’s name. I mean, you know, there's whole different relation groups these days.

EL: Yeah.

aM: But I'm talking about the injection and surjection.

EL: Yeah, the polyamorous functions are a whole different thing.

KK: Those are just relations, they’re not functions. It’s a whole thing.

aM: Oh, yes, relations aren't necessarily functions, but certain ones that be called that right?

EL: Yeah. Well, thank you so much for joining us. Is there anything else you would like to share? I mean, we often give our guests ways to find—give our listeners ways to find our guests online. So if there's anything, you know, a website, or anything you’d like to share.

aM: Can you just link my web page or should I tell you it? [Webpage link here] Actually googling “aBa UWEC math.” That's all it takes. UWEC aBa math. Whenever students can’t find our course notes, I just say like, “I don't know, Google it. There's no way you cannot find our course notes if you remember the name of your school, what you're studying and my name.” Yeah.

EL: We’ll put a link to that also in the show notes for people.

aM: Yeah, one B, aBa, for the listeners.

EL: Yes, that's right. We didn't actually—I said it was the only one spelled that way but we didn't spell it. It's aBa, and you capitalize the middle, the middle and not the first letter, right?

aM: No, yes, that's fine. It looks more symmetric that way.

EL: Yeah. You could even reverse one of them.

aM: I usually write the B backwards. Like the band, but I can't do that usually, though. I don't want to be overkill to the people that I work around. But yes, at the bottom of my webpage, I have the links to videos of me singing various songs to students, complex analysis raps, PhD level down to undergraduate level, just different raps that I wrote for funs.

And I wanted to plug one thing at JMM. I mean, not that it's hard to find it in the program, but I'm an MAA invited speaker this time, and I'm actually scared pooless a little bit to be speaking in one of those large rooms. I don't know how I got invited. But I said yes.

KK: Of course you said yes!

aM: Well, I'm excited to share two research projects that I've been doing with students. Because I like doing research just for the sheer joy of it. And I think the topic of my talk is “A research project birthed out of curiosity and joy” or something like that, because one of the projects I'm sharing wasn't even a paid research project. I just had a student that got really excited to study something I noticed in Pascal's triangle, and these tridiagonal real symmetric matrices. I mean, it was finals week, and I was like, “You want to have fun?” And we spent the next year and a half having fun, and now she's pursuing graduate school, and it's great. It's great, research for fun. But one thing I'm talking about that I'm really excited about is the Fibonacci sequence. And I know that's kind of overplayed at times, but I find it beautiful. And we're looking at the sequence modulo 10. So we're just looking at the last, the units digits.

EL: Yeah, last digits.

aM: And whenever you take the sequence mod anything, it's going to repeat. And that's an easy proof to do. And actually Lagrange knew that long, long ago. But recently, in 1960, a paper came out studying these Fibonacci sequences modulo some natural number, and proved the periodicity bit and proved—there’s tons of papers in the Fibonacci Quarterly related to this thing. But what I'm looking at in particular is a connection to astrology—which actually might clear the room, but I'm hoping not—but the sequence has a length of periods 60. So if you lay that in a circle, it repeats and every 15th value in the Fibonacci number ends in 0. That's something you can see with the sequence, but it’s a lot easier to see when you're just looking at it mod 10. and that's something probably people didn't know now. Every 15th Fibonacci number ends in 0.

KK: No, I didn't know that.

aM: And if it ends in 0, it's a 15th Fibonacci number. And so, it’s an if and only if. And every 5th Fibonacci number is a multiple of five. So in astrology, we have the cardinal signs: Aries, Cancer, Libra and Capricorn. And you and you lay those on the zeros. Those are the zeros. And then the fixed and mutable signs, like Taurus, Gemini, etc, etc. As you move after the birth of the astrological seasons, those ones lay on the fives, and then you can look at aspects between them. Actually, I'm not going to say much astrology, by the way, in this talk. So people who are listening, please still come. It's only math! But I'm going to be looking at sub-sequences, but it got inspired by some videos online that I saw by a certain astrologer. And I—there was no mathematics in the videos and I was like, “Whoa, I can fill these gaps.” And it's just beautiful. Certain sub-sequences in the Fibonacci sequence mod 10 give the Lucas sequences mod 10. The Lucas sequence, and I don't know if your listeners or you guys know what the Lucas sequence is, but it's the Fibonacci sequence, but the starting values are 2 and then 1.

KK: Right.

aM: Instead of zero and one.

EL: Yeah.

aM: And Edward Lucas is the person, actually, who named the Fibonacci sequence the Fibonacci sequence! So this is a big player. And I am really excited to introduce people to these beautiful sub-sequences that exist in this Fibonacci sequence mod 10. It's like, just so sublime, so wonderful.

EL: I guess I never thought about last digits of Fibonacci numbers before, but yeah, I hope to see that, and we'll put some information about that in the show notes too. Yeah, have a good rest of your day.

aM: All right, you too, both of you. Thank you so much for this invitation. I’m happy to be invited.

EL: Yeah, we really enjoyed it. v KK: Thanks, aBa.

aM: All right. Bye-bye.

On this episode of My Favorite Theorem, we talked with aBa Mbirika, a mathematician at the University of Wisconsin Eau Claire. He told us about several favorite theorems of the moment before zeroing in on one of his first mathematical discoveries: a way to determine whether a number is divisible by 7.

Here are some links you may find interesting after listening to the episode.

aBa’s website at UWEC
Snake lemma
Short five lemma
Euler-Fermat’s theorem
Gauss’s primitive roots
Adriana Salerno’s episode of the podcast
Steve Fisk’s “book proof” of the art gallery theorem
Information on aBa’s MAA invited address at the upcoming Joint Mathematics Meetings

Episode 49 - Edmund Harriss

12 december 2019 | 31 min

Kevin Knudson: Welcome to My Favorite Theorem, math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined today by your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer, usually based in Salt Lake City, but today coming from the Institute for Computational and Experimental Research in Mathematics at Brown University in Providence, Rhode Island, where I am in the studio with our guest, Edmund Harriss.

KK: Yeah. this is great. I’m excited for this, this new format where we're, there's only two feeds to keep up with instead of three.

EL: Yeah, he even had a headphone splitter available at a moment's notice.

KK: Oh, wow.

EL: So yeah, this is—we’re really professional today.

KK: That’s right.

EL: So yeah, Edmund, will you tell us a little bit about yourself?

Edmund Harriss: I was going to say I'm the consummate unprofessional. But I'm a mathematician at the University of Arkansas. And as Evelyn was saying, I'm currently at ICERM for the semester working on illustrating mathematics, which is an amazing program that's sort of—both a delightful group of people and a lot of very interesting work trying to get these ideas from mathematics out of our heads, and into things that people can put their hands on, people can see, whether they be research mathematicians or other audiences.

EL: Yeah. I figured before we actually got to your theorem, maybe you could say a little bit about what the exact—or some of the mathematical illustration that you yourself do.

EH: So, yeah, well, one of the big pieces of illustration I've done will come up with a theorem,

EL: Great.

EH: But I consider myself a mathematician and artist. And a part of the artistic aspect, the medium—well, both the medium but more than that, the content, is mathematics. And so thinking about mathematical ideas as something that can be communicated within artwork. And one of the main tools I've used for that is CNC machines. So these are basically robots that control a router, and they can move around, and you can tell it the path to move on and carve anything you like. So even controlling the machine is an incredibly geometric operation with lots of exciting mathematics to it. When I first came across—so one of the sorts of machine you can have is called a five-axis machine. That's where you control both the position, but also the direction that you're cutting in. So you could change the angle as its as its cutting. And so that really brings in a huge amount of mathematics. And so when I first saw one of these machines, I did the typical mathematician thing, and sort of said, “Well, I understand some aspects of how this works really well. How hard can the stuff I don't understand be?” It took me several years to work out just how hard some of the other problems were. So I've written software that can control these machines and turn—in fact, even turn a hand-drawn path into a something the machine can cut. And so to bring it back to the question, which was about illustrating mathematics: One of the nice things about that idea is it takes a sort of hand-drawn path—which is something that's familiar to everyone, especially people in architecture or art, who are often wanting to use these machines, but not sure how—and the mathematics comes from the notion that we take that hand-drawn path, and we make a representation of that on the computer. And so you've got a really interesting function, they're going from the hand drawn path through to the the computer representation, you can then potentially manipulate it on the computer before then passing it again back to the machine. And so now the output of the machine is something in the real world. The initial hand-drawn path was in the real world, and we sort of saw this process of mathematics in the middle.

Amongst other things, I think this is a really sort of interesting view on a mathematical model. you have something in the real world, you pull it into an abstract realm, and then you take that back into the world and see what it can tell you. In this case, it's particularly nice because you get a sense of really what's happening. You can control things, both in the abstract and in the world. And I think, you know, to me that really speaks to the power of thinking and abstraction of mathematics. Of course, also controlling these machines allows you to make mathematical models and objects. And so a lot of my my work is sort of creating mathematical models through that, but I think the process is a more interesting, in many ways, mathematical idea, illustration of mathematics, that the objects that come out

KK: Okay, pop quiz. What's the configuration space of this machine? Do you know what it is?

EH: Well, it depends on which machine.

KK: The one you were describing, where you can where you can have the angles changing. That must affect the topology of the configuration space.

EH: So it’s R3 crossed with a torus.

KK: Okay.

EH: And so even though you're changing the angle of the bit, you really need to think about a torus. It's really also a subset of a torus because you can't reach all angles.

KK: Sure, right.

EH: But it is a torus and not a sphere.

KK: Yeah. Okay.

EH: So if you think about how to get from one position of the machine to another, you really want to—if you think about moving on a sphere, it's going to give you a very odd movement for the machine, whereas moving along a torus gives the natural movement.

KK: Sure, right. All right. So, what's your favorite theorem?

EH: So my favorite theorem is the Gauss-Bonnet.

KK: All the way with Gauss-Bonnet!

EL: Yes. Great theorem. Yeah.

EH: And I think in many ways, because it speaks to what I was saying earlier about the question: as we move to abstraction, that starts to tell us things about the real world. And so the Gauss-Bonnet theorem comes at this sort of period where mathematics is becoming a lot more abstract. And it's thinking about how space works, how we can work with things. You're not just thinking about mathematics as abstracted from the world, but as sort of abstraction in its own right. On the artist side, a bit later you have discussion of concrete art, which is the idea that abstract art starts with reality and then strips things away until you get some sort of form, whereas concrete art starts from nothing and tries to build form up. And I think there's a huge, nice intersection with mathematics. And in the 19th century, you've got that distinction where people were starting to think about objects in their own right. And as that happens, suddenly this great insight, which is something that can really be used practically—you can think about the gospel a theorem, and it's something that tells you about the world. So I guess I should now say what it is.

EL: Yeah, that would be great. Actually, I guess it must have been almost two years ago at this point, we had another guest who did choose the Gauss-Bonnet theorem, but in case someone has not religiously listened to every single episode—

KK: Right, this was some time ago.

EL: Yeah, we should definitely say it again.

EH: So the gospel out there links the sort of behavior of a surface to what happens when you walk around paths on that surface. So the simplest example is this: I start off, I’m on a sphere, and I start at the North Pole and I walk to the equator. At the equator, I turn 90 degrees, I walk a quarter of the way around the Earth, I turn 90 degrees again, and I walk back to the North Pole. And if I turn a final 90 degrees, I’m now back where I started facing in the same direction that I started. But if I look at how much I turned, I didn't go through 360 degrees. So normally if we go around a loop on a nice flat sheet, if you come back to a started pointing in the same direction, you've turned through 360 degrees. So in this path that I took on sphere, I turned through 270 degrees, I turned through too little. And that tells me something about the surface that I'm walking on. So even if I knew nothing about the surface other than this particular loop, I would then know that the surface inside must be mostly positively curved, like a sphere.

And similarly if I did the same trick, but instead of doing it on the sphere, I took a piece of lettuce and started walking around the edge of a piece of lettuce, in fact, I’d find that when I got back to where I started, I’d turned a couple of hundred times round, instead of just once, or less than once, as in the case the sphere. And so in that case, you've got too much turning. And that tells you that the surface inside is made up of a lot of saddles. It's a very negatively curved surface. And one of the motivations of creating this theorem for Gauss, I believe—I always find it dangerous to talk about history of mathematics in public because you never know what the apocryphal stories are—one of the questions Gauss was interested in was not whether or not the earth was a sphere. Well, actually, whether or not the earth was a sphere. So not whether or not it was round, or topologically a ball, but whether it was geometrically really a perfect sphere. And now we can go up into space and have a look back at the earth, and so we can sort of do a three-dimensional version of that, regard the earth as a three dimensional sphere, but Gauss was stuck on the surface of the earth. So he really had this sort of two dimensional picture. And what you can do is create different triangles and ask, for those triangles, what’s the average amount of curvature? So I look at that turning, I look at the total area, the size of the triangle, and ask does that average amount of curvature change as I draw triangles in different places around the earth? And at least to Gauss’s measurements—again, in the potentially apocryphal story I heard—the earth appeared to be a perfect sphere up to the level of measurement, they were able to do then. I think now, we know that the earth is an oblate spheroid, in other words, going between the poles is a slightly shorter distance than across the equator.

KK: Right.

EH: I believe that it was only a couple of years ago that we managed to make spheres that were more perfect than the Earth. So it was sort of, yeah, the Earth is one of the most perfect spheres that anyone has experience of, but it's not quite a perfect sphere when your measurements are fine enough.

KK: So what's the actual statement of Gauss-Bonnet?

EH: So, the statement is that the holonomy, which is a fancy word for the amount of turning you do as you go around a path on the surface, is equal to—now I’m forgetting the precise details—so that turning is closely related to the integral of the Gaussian curvature as you go over the whole surface.

KK: Right.

EH: So it's relating going around that boundary—which is a single integral because you're just moving around a path—to the double integral, which is the going over every point in the surface. And the Gaussian curvature is the notion of whether you're like a sphere, whether you're flat, or whether you're like a saddle at each individual point.

KK: And the Euler characteristic pops up in here somewhere if I remember right.

EH: Yeah. So the version I was giving was assuming that you’re bounding a disk in the surface, and you can do a more powerful version that allows you to do a loop around something that contains a donut.

EL: Yeah, and it relates the topology of a surface, which seems like this very abstract thing, to geometry, which always seems more tangible.

EH: Yeah. Yeah, the notion that the total amount of curvature doesn't change as you shift things topologically.

EL: Right.

EH: Even though you can push it about locally.

KK: Yeah. So if you're if you're pushing it in somewhere, it has to be pooching out somewhere else. Right? That's essentially what's going on, I guess. Right?

EH: Yeah. You know, another thing that's really nice about the the Gauss-Bonnet theorem, it links back to the Euler characteristic and that early topological work, and sort of pulls the topology in this lovely way back into geometric questions, as Evelyn said. And then the Euler characteristic has echoes back to Descartes. So you're seeing this sort of long development of the mathematics that's coming out. It’s not something that came from nowhere. It was slowly developed by insight after insight, of lots of different thinking on the nature of surfaces and polyhedra and objects like that.

EL: Yeah. And so where did you first encounter this theorem?

EH: So this is rather a confession, because—when I was a undergraduate, I absolutely hated my differential equations course. And I swore that I would never do any mathematics involved in differential equations. And I had a very wise PhD advisor who said, “Okay, I'm not going to argue with you on this, but I predict that at some point, you will give me a phone call and say you were wrong. And I don't know when that will be. But that's my prediction.”

KK: Okay.

EH: It did take several years. And so yes, many years later, I'd learned a lot of geometry, and I wanted to get better control over the geometry. So I sort of got into doing differential geometry not through the normal route—which is you sort of push on through calculus—but through first understanding the geometry and then wanting to really control—specifically thinking about surfaces that were neither the geometry of the sphere, the plane, or the hyperbolic plane. Those are three geometries that you can look at without these tools. But when you want to have surfaces that have saddles somewhere and positive curvature—I mean, this relates back to the CNC because you're needing to understand paths on surfaces there in order to take our tool and produce surfaces.

And so I realized that the answers to all my questions lay within differential equations, and actually differential equations were geometric, so I was foolish to dislike them. And I did call up my advisor and say, “Your prediction has come true. I'm calling you to say I was wrong.”

EL: Yeah.

EH: So basically, I came to it from looking at geometry and trying to understand paths on surfaces and realizing from from there that there was this lovely toolkit that I had neglected. And one of the real gems of this toolkit was this theorem. And I think it's a real shame that it's not something that's talked about more. I’ve said this is a bit like the Sistine Chapel of mathematics. You know, most people have heard of the Sistine chapel.

KK: Sure.

EH: Quite a lot of people can tell you something that's actually in it.

EL: Right.

EH: And slowly, only a few people have really seen it. And certainly a very few people have studied it and really looked and can tell you all the details. But in mathematics, we tend to keep everything hidden until people are ready to hear the details. And so I think this is a theorem that you can really play with and see in the world. I mean, it's not a—there are some models and things you can build that are not great for podcasts, but it's something you can really see in the world. You can put it put items related to this theorem into the hands of people who are, you know, eight or nine years old, and they can understand it and do something with it and and see how what happens because all you have to do is give people strips of paper and ask them to start connecting them together, just controlling how the angles work at the corners.

And depending on whether those angles add up to less than 360 degrees—well not the angles at the corner—depending on whether the turning gives you less than 360, exactly 360, or more than 360, you're going to get different shapes. And then you can start putting those shapes together, and you build out different surfaces. And so you can then explore and discover a lot of stuff in a sort of naive way You certainly don't need to understand what an integral is in order to have some experience of what the Gauss-Bonnet theorem is telling you. And so this is sort of it's that aspect, that this is something that was always there in the world. The sort of experiments, the sort of geometry you can look at, through differential geometry and things like the Gauss-Bonnet, that was available to the whole history of mathematics, but we needed to make a break from just geometry as a representation of the world to then sort of step back and look at this result that is a very practical, hands-on one.

You know, if you really want to control things, then you do need to have solid multivariate calculus. So generally, the three-semester course of calculus is often meant to finish with Gauss-Bonnet, and it's the thing that's dropped by most people at the end of the semester, because you don't quite have time for it. And there's not going to be a question on the test. But it's one of those things that you could sort of put out there and have a greater awareness of in mathematics. Just as: this is an interesting, beautiful result. I would say, you know, it's one of humanity's greatest achievements to my mind. You don't have to really be able to understand it perfectly in order to appreciate it. You certainly—as I proved you—can appreciate it without being able to state it exactly.

EL: Yeah, well, you've sold me—although, as we've learned to this podcast, I'm extremely open—susceptible to suggestion.

KK: That’s true. Evelyn's favorite theorem has changed multiple times now. That's right.

EL: Yeah. And I think you brought it back to Gauss-Bonnet. Because when when we had Jeanne Clelland earlier, who said Gauss-Bonnet, I was like, “Well, yeah, I guess the uniformization theorem is trash now”—my previous favorite theorem, but now—it had been pulled over to Cantor again, but you’ve brought it back.

KK: Excellent. All right, so that's another thing we do on this podcast is ask our guest to pair their theorem with something. So Edmund, what pairs well with Gauss-Bonnet?

EH: Well, I have to go with a walnut and pear salad.

KK: Okay.

EL: All right.

KK: I’m intrigued.

EH: Well, I think I've already mentioned lettuce.

EL: Yes.

EH: Lettuce is an incredibly interesting curved surface. Yeah. And then you've got pears, which gives you—

KK: Spheres.

EH: A nice positively curved thing. But they're not just boring spheres.

EL: Yeah.

EH: They have some nice interesting changes of curvature. And then walnuts are also something with very interesting changing curvature. They have very sharply positively curved pieces where they're sort of coming in but then they've got all these sort of wrinkly saddley parts. In fact, one of the applications of the Gauss-Bonnet theorem in nature is how do you create a surface that sort of fits onto itself and fills a lot of space—or doesn't fill that much space but gives you a very high surface area to volume ratio. So walnut is an example—or brains or coral—you see the same forms coming up. And the way many of those things grow is by basically giving more turning as you grow to your boundary.

KK: Right.

EH: And that naturally sort of forces this negatively-curved thing. So I think the salad really shows you different ways in which this surface can—the theorem can affect the behaviors of the surfaces.

EL: Yeah, well, what I want now is something completely flat to put in the salad. Do you have any suggestions?

KK: Usually you put goat cheese in such a thing, but that doesn't really work.

EL: That’s—well, parmesan. You could shave paremesan.

EH: Yeah, shavings of parmesan. Or maybe some thin-cut salami.

EL: Okay.

EH: And so even though those things would bend over—I mean, we’re now on to a different theorem of Gauss, and I don’t meant to corrupt Evelyn away—but you know, when you thinly cut the salami, it can it can bend but it doesn't actually change its curvature.

KK: Right.

EH: Your loops on that salami are going to have the same behavior that they had before. And I guess I should also say that I did create a toy that makes that paper model that I talked about easier to use. You don't have to use tape. You can hook together pieces. And so the toy is called Curvahedra.

KK: I was going to say, you should promote your toy. Yeah.

EH: I’m terrible at self-promotion, yes.

EL: We will help you. Yes, this is a very fun toy. I actually got to play with it for first time a few weeks ago when you did a little short thing and I think when I had seen pictures of it before I thought it was not going to be as sturdy as it is. But this is—yeah, it's called Curvahedra—look it up. It’s these quite sturdy—you know, you don't need to worry about ripping the pieces as you put them together—but you can create these things that look really intricate, and you can create positive curvature, or flat things, or negative curvature in all these different conformations. It's a very fun thing to play with.

EH: And it is a sort of physical version of exactly the Gauss-Bonnet theorem. As you hook together pieces, you're controlling what happens on a loop. And then as you put more of those loops together, you can get a variety of different surfaces, from hyperbolic planes to spheres to—of course, kids have made animals and creatures with it. So you get this sort of control. In fact, it's one of those things that, you put it into the hands of kids, and they do things that you didn't think were really possible with it because their ability to play with these ideas and be free is always so inspiring. So that's what I said, this is a theorem that you can—people can understand as something in the real world. And then you can tell the story of how this understanding of the world is linked directly back to abstract, esoteric mathematics, of the most advanced sort.

KK: Right. One of my favorite things about Curvahedra, though, is the video that you put online somewhere—I think was on Twitter—of it popping out of your suitcase, like you compressed it down into your suitcase to travel home one time?

EH: Yes, I have a model that's about to a two-foot cube. And so you can’t travel with that easily, but it can compress very small. And that same object has been in my suitcase and other things several times, and it's now sitting in my office here.

KK: That’s great fun. And also you've made similar models out of metal, correct?

EH: Yes. So the basic system—not the big one you can crush down to put into suitcases.

KK: No, certainly not.

EH: I’ve made a couple of the spheres. And we're currently working on a proposal to go outside the Honors College at the University of Arkansas. That grew out of a course—it was a design that was created from Curvahedra and other inspirations—by a course I taught with Carl Smith, who is a landscape architect in our landscape architecture school. And so there's going to be—hopefully at some point there's going to be a 12-foot tall Curvahedra-style model outside the Honors College at University of Arkansas.

KK: Very nice.

EL: Nice.

KK: Yeah, this has been great fun. Anything else we want to talk about?

EL: Yeah, well, do you want to say a website or Twitter account or anything where people can find you online?

EH: So I’m actually @Gelada on Twitter, and there is @Curvahedra, and my blog, which is very rarely updated, but has some nice stuff, is called Maxwell Demon.

EL: Yeah, and can you spell your Twitter?

EH: Yes, so Gelada is spelled G-E-L-A-D-A. They are baboons in Ethiopia, or it’s a cold beer in Brazil. I discovered that latter one after being on Twitter, and I regularly get @-ed by people in Brazil, who were not wanting to talk to me at all, but they're asking each other out for beers.

EL: Ah.

EH: And yeah, so then there's also curvahedra.com, where you can get that toy.

EL: Cool. Thanks for joining us.

KK: Yeah, thanks Edmund.

EH: Thank you.

[outro]

On today’s episode, we were pleased to talk with Edmund Harris, a mathematician and mathematical artist at the University of Arkansas, who is our second guest to sing the praises of the Gauss-Bonnet theorem. Below are some links you might find useful as you listen to the episode.  

Edmund’s Twitter account, @Gelada 

His blog, Maxwell’s Demon 

The website and Twitter account for Curvahedra, the toys he makes that help you explore the Gauss-Bonnet theorem and just have a lot of good fun with geometry 

Our episode with Jeanne Clelland, who also chose the Gauss-Bonnet theorem 

Edmund and Evelyn both attended the Illustrating Mathematics program at the Institute of Computational and Applied Mathematics (ICERM). The program website, which includes videos of some interesting talks at the intersection of math and art, is here.

Episode 48 - Sophie Carr

14 november 2019 | 23 min

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm one of your hosts, Kevin Knudson. I'm a professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance writer, usually based in Salt Lake City, but currently coming to you from Providence, Rhode Island.

KK: Hooray! Yeah, you're at ICERM.

EL: Yes. The Institute for computational and experimental research in mathematics, an acronym that I am now good at remembering.

KK: I’m glad you told me. I was trying to remember what it stood for this morning because I'm going next week. We'll be in the same place for, like, only the second time ever.

EL: Yeah.

KK: And the universe didn't implode the first time. So I think we're safe.

EL: Yeah.

KK: So the ICERM thing is visualizing mathematics, I mean, we're sort of doing like—next week is about geometry and topology, which since both of us are nominally that, that's just the right place for us to be.

EL: Yeah, it's it's going to be a fun semester. I'm also very excited because I recently turned in—it feels weird to call it a manuscript, but it is being published by a place that publishes books. It is the final draft of a page-a-day calendar about math. And I hope that by the time we air this, I will be able to have a link where people can purchase this and give it to give it to themselves or to their favorite mathematician.

KK: Yeah.

EL: So that's just, every day you can have a little morsel of math to start your morning.

KK: I’m looking forward to that. That’s really exciting. Yeah, that's that's great. All right, so we're continuing a tradition in this episode.

EL: Yes.

KK: So Christian Lawson-Perfect organizes this thing through the Aperiodical called the Great Internet Math-Off [Editor’s note: Whoops, it’s called the Big Internet Math-Off!] of which you were a participant in the first one but not this one, not the second go-around. And we had the first winner on. The winner gets named the World's Most Interesting Mathematician (among those people who Christian could round up and who were free in July). And so we wanted to keep this trend going of getting the most interesting mathematicians in the world on this podcast. And we are pleased to welcome this year's winner, Sophie Carr. Sophie, you want to introduce yourself, please?

SC: Oh, hello, thank you very much. Yeah, I'm Sophie Carr. I studied Bayesian networks at university, and now I own and run a data analytics company.

EL: Yeah, and you’re the most interesting mathematician!

SC: I am! For this year, I am the most interesting mathematician in the world. It's entirely Nira’s fault that I entered because he suggested, and put me forward.

KK: That’s right. Nira Chamberlain was last year's winner. And so when we interviewed him he was sitting in his attic wearing a winter coat. It was wintertime and it seemed very cold where he was. You look very comfortable. It looks like you have a very lovely home in the background.

SC: Yes, I mean, I am in two jumpers. Autumn has definitely arrived. Summer has gone, and it's a little chilly at the moment.

KK: I can only dare to dream. Yeah.

EL: Yeah, Florida and UK have slightly different seasons.

KK: Just a little bit. So you own a consulting company? That’s correct?

SC: Yeah, I do. I set it up 10 years ago now. There’s me and two other people who work with me. We just have an awful lot of fun finding patterns in numbers. I still find it amazing that we're still going. It's just the best fun ever. We get to go and work on all sorts of different problems with all sorts of different people. It's fantastic.

KK: Yeah, that's great. I mean, I'm glad companies are starting to come around to the idea that mathematicians might actually have something to tell them. Right?

SC: Yes. It really is. When you explain to them, you're not going to do magic and it's not a black box, and you can tell them how it works and how it can really make a difference, they are coming around to that.

KK: That’s fantastic. All right, so we're here to talk about theorems.

EL: Yeah. What is your favorite theorem?

SC: My favorite theorem in the whole world is Bayes’ Theorem.

EL: Yay, I'm so glad that someone will be talking about this! Because I know that this is a great theorem and—confession: I just, I don't appreciate it that much.

KK: You know, same.

EL: I need to be told why it's great.

KK: Yeah, I taught probability one time and I said, “Okay, here's Bayes’ theorem.” I kind of went all right. Fine, but of course the question is what's the prior, Mr. Bates? So tell us. Tell us, please.

EL: Yeah, preach!

KK: Preach for Reverend Bayes.

SC: You know, I don't think there's any any preaching needed. Because I always say this. I mean, there are two bits of statistics, there’s the frequentist and the Bayesian. And I always liken it to rugby union, and rugby league, which are two types of rugby in England. It's different codes, but it's the same thing. So to me, Bayes’ theorem, it's just the way that we naturally think. And it's beautifully simple, and all it does is let you take everything that you know and every piece of information that you have, and use that to update the overall outcome. And you're right, that the really big arguments come about from what the prior is. What is the background information that we have, and can we have actually genuinely have a true prior? And some people say no, because you might not have any information. But that's the great bit! Because then you can go and find out what the prior is. You have to be absolutely open about what you're putting in there. I think the really big debate comes around whether people are happy with uncertainty. Are they happy for you to not give an exact answer? If you go and you say, well, this is the prior, this is what we think the information is as well. And we combine these all, combine these priors, and this is the answer. Let's have a debate. Let's start talking about what we can have. Because at its simplest, you've got two things you’re timesing together. Just two numbers. Something that runs your mobile phone. I mean, that’s quite nifty.

KK: So can we can we remind our listeners what Bayes’ theorem actually says?

SC: Okay, so Bayes’ theorem takes two things. It takes the initial, or the prior distribution. Okay, and that's the bit where the argument is. And that might be just, what's the chance of something happening? What do you think the probability is of something happening? And you combine that with something called likelihood ratio. And it's real simple. The likelihood ratio is just a ratio of the probability of the information, or the evidence you have, assuming one hypothesis,divided by the probability of that information assuming another hypothesis. So you just have to have those two values. [And I say you just have to keep it.

And then all you have to do is times them together! That really is it, and when you start to say to people, it's just two numbers—Now, you can turn that into three numbers if you want. You can turn the likelihood ratio bit into its two separate parts. And you can show Bayes’ theorem very, very simply with decision trees, and that was part of the reason I used decision trees in the Math-Off, was just to show the power of something that is really quite simple, that can drive so, so far. And that's what I love about Bayes’ theorem. I always describe it as something that is stunningly elegant, but unbelievably powerful. And I always liken it to Audrey Hepburn. I think if it were to be a person, it would be Audrey Hepburn. Quite small! I'd say it's, it's this amazing little thing that has two simple numbers. But goodness me, getting those numbers, well, I mean, you can just have so much fun! I think you can.

And maybe it's just me that likes finding the patterns in the numbers and finding those distributions. Coming up with the priors. So come on, Kevin, you said, you sat there and your class said, “Well, what's the prior?”

KK: Yeah.

SC: What do you say? How would you tell people to go about finding a prior? Are they going to use their subjective opinion? Are they going to try and find it from data?

KK: Well, that that is the question, isn't it? Right? So, I mean, often, the problem with probability sometimes is that—at least, like, in political forecasting, right—people tend to round up probabilities to 1 or lop them off to zero. Right? So for example, when, you know, when Trump won the election in 2016, everybody thought it was a huge shock. But you know, 538 had it as, you know, Hillary Clinton was a two-to-one favorite. But two-to-one favorites lose all the time, right?

SC: Yeah.

KK: And and so the question then is, yeah, people like to think about one-off events. And then the question is, how do you estimate the probability of a one time event? And you have to make some guess, right, at the prior. And that’s—I think that's where people get suspicious of Bayes’ theorem, or Bayesian statistics, because how you make this estimate? So how do you make estimates in your daily work as a consultant?

SC: Okay, so we do it in a variety of different ways. So if we're really lucky, there’s some historical data we can go looking at.

KK: Sure.

SC: And often just mining that historical data gives you a good starting point. I always get slightly suspicious of flat distributions. Because if we really, really don't know anything other than that, I think maybe a bit of research before where you find the prior is always a good thing. My favorite priors are when we go and talk to people and start to get out of them their subjective opinion. Because I like statistics, I genuinely love statistics, because of the debate that goes on around it. And I think one of the things that people forget about math is that it's such a living subject. And there are so many brilliant debates—and you can call some of them arguments— people are prepared to go and say, “Look, this is my opinion and this is what I think the shape is.” And then we can do the analysis. Inevitably somebody will stand up and go, “Well, that bit is wrong.” Okay, so tell me why!

EL: Yeah.

SC: What evidence have you got for us to change the shape, or why do you think it should be skewed, or Poisson, or whatever we're using? And sometimes, if we haven't got time to do that we can start to put in flat distributions. We can say, “Well, we think it's about normal.” Or “We think on average, it'll be shoved a little bit to the right or a little bit to the left.” That's the three main ways we go about doing it. And I think the ability to be absolutely open and up front about what you know and what you don’t know helps you find that prior. And I don't really understand why people would be scared of running away from that. Why you would not want to say what the uncertainty is or what you're not sure about. But that might go a long way when people think that math is certain.

EL: Yeah.

SC: That when you say the answer is 12, well it’s 12. And not, “Well, it’s 12 because we kind of do it like this, and actually if something changes, that number might change.” And I think getting comfortable with uncertainty and being uncomfortable, is really the crux for developing those priors.

EL: Yeah. Well, I guess for me, it's hard to reason about statistics in a non frequentist way. Meaning—you know, I'm comfortable with non frequentist statistics to a certain degree. But just like what, as you were, saying, like, what does a 30% chance mean if it's not that we could do this 10 times that have it happen three times. But you can't have a presidential election—the same election—10 times, or you can't run Monday’s weather 10 times, or something like that. But it's just hard for me to interpret what does it mean if there isn't a frequentist interpretation?

SC: Yeah. One of the things we found that works really well is if you start showing patterns—and that's why I always talk about patterns, that we find patterns. It's when you're doing Bayesian stats with priors if you start to show the changes as curves, and I don't mean the distribution, but I mean, just as that rising and falling of numbers, people start to understand what's driving the priors, what assumptions are changing those priors. And then you start to see the impact of that, how the final answer changes. That can be incredibly powerful. Often people don't want that set answer. They want to know what the range is, they want to understand how that changes. And showing that impact as a shape—because I think most people are visual. When you show somebody a surface or, you know, a graph, or whatever it is, that's something you can really get a grip with. And actually I come from a Bayesian belief network. So I kind of found out about Bayes’ theorem by chance. I never set off to learn Bayes’ theorem. I set off to design [unintelligible]. That’s what I grew up wanting to do. But I ended up working on Bayesian networks. That’s the short version of what happened.

EL: So, how—was this a “love at first sight” theorem? Or what was your initial encounter with this theorem? And how did you feel about it? Since this is all about subjective feelings anyway!

SC: Well, my PhD was part-time. I spent eight years collecting subjective opinions. So I started a PhD in Bayesian networks, and there was this brilliant representation of a great big probability table. And this is a while ago now. And I’ve moved on a lot into [unintelligible]. But I've got this Bayesian network and supervisor said, “Here we go,” and I went, “Ah, it’s just lots of ovals connected with arrows”

And I went, “There must be something more to this.” And he went, “There’s this thing called Bayes’ theorem that underpins it and look at how it flows. It’s how the information affects it.” And I went, “Okay!” And so, as with all PhDs, you have this pile of reading, which is apparently going to be really, really good for you.

So I got my pile of reading. I went, “Okay.” And genuinely I just thought, “Yeah, it's just kind of how we all work, isn't it?” And I really had not liked statistics at university at all because I’d only really done frequentist statistics. And it’s not like I dislike frequentist statistics. I just didn’t fall in love with it. But when there was something I could see—and I genuinely think it’s because it's visual. I see the shapes move, I could see the numbers flow, I could see the information flow. I thought, “Oh, this is cool stuff. I understand this. I can get my head around this.” And I could start to see how to put things in and how they changed. And I think also I've got at times a very short attention span. So running millions of replicates never really did it for me.

EL: Yeah.

SC: So I had a bit of an issue with frequentist, where we just have to run lots and lots and lots lots of replicates.

EL: Right.

SC: Can we not assume it's kind of like this shape and see what happens? Then change that shape. Look, that’s great. That's much better for me.

EL: Yeah. So it was kind of a conversion experience there.

SC: I think, for people my age, probably. Because I don’t think Bayesian statistics, years ago, was taught that commonly. it's only really in the past sort of maybe decade that I think it's become really mainstream and been taught in the way it is now. Certainly with its its wide applications. That's what I think people just go, something that they've never heard of is now all in the AI world and it’s in your mobile phone, and it's in your medicine, and it's in your spam filters. And when it suddenly becomes really popular, people start to see what it can do. That's when it's taught more. And then you get all these other debates.

KK: So the other fun thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with Bayes’ theorem?

SC: So this caused a lot of debate in our household.

KK: It always does.

SC: Yeah. And I am going to pair Bayes’ theorem with my favorite food, which is risotto, because risotto only takes three things. It only needs rice and onions and a good stock.

KK: Yes.

SC: And Bayes’ theorem is classically thought with three numbers. And it’s really powerful and gorgeous. And risotto only takes three ingredients, and it’s really gorgeous.

KK: And also, the outcome is uncertain sometimes, right?

SC: Oh, frequently uncertain. And if you change those prior proportions, you will get a very different outcome.

KK: That’s right. You might get soup, or it might might burn.

SC: So, I am going to say that Bayes' theorem is like a risotto.

EL: And you mentioned Audrey Hepburn earlier so maybe it’s even more like sharing a risotto with Audrey Hepburn.

SC: That would be brilliant. How cool would that be?

EL: I know!

SC: I will have my Bayes’ theorem discussion with Audrey Hepburn over risotto. That would be a pretty good day.

EL: Yeah, you could probably get a cardboard cutout. Just, like, invite her to dinner.

SC: Yeah, I'll do that. I'll try and set up a photo, superimpose them.

EL: Yeah.

KK: But Audrey Hepburn should be breakfast somewhere right?

EL: But you can eat risotto for breakfast.

SC: Yeah, you can eat risotto any time of the day.

KK: Sure.

SC: There’s never a bad time for risotto.

KK: No, there isn't. Yeah. My wife actually doesn't like risotto very much, so I never make it.

EL: So is that one of your restaurant foods? So we have this whole like foods that you you tend to order at a restaurant because your partner doesn't like them. And so it's like something that you can—like I don't really like mushrooms, so my partner often will order a mushroom thing at a restaurant.

KK: Yeah, so for me, I don't go out for Italian food because I can make it at home.

EL: Okay.

KK: So I just have a generic I don't I don't eat Italian out. There’s kind of no point, I think.

SC: So you’re right that risotto is my restaurant food because my husband doesn't like it.

KK: Oh.

EL: Aw.

SC: It's my most favorite thing in the world, so yeah, every time we go out, the kids go, “Mom, just don't get the menu. There’s no point. We know what you’re getting.

EL: Yeah. So you said this caused a debate. Did he have a different opinion about what your pairing should be?

SC: Well, there were discussions about whether it was my favorite drink with [a bag of crisps?], and what things could be combined together. And I said, “No, it just has to be risotto.”

KK: Okay. Excellent.

EL: Yeah, we do make that at home. And actually the funny thing is I don't really like mushrooms, but I do like the mushroom risotto that we make.

SC: Oh.

EL: Yeah.

SC: So you've not got a flat prior. You've actually got a little bit of a skew on there.

EL: Yeah, I guess. I’m trying to figure out how to quantify this. Yeah, like my prior distribution for mushroom preference is going to depend on whether it is cooked with arborio rice or not.

SC: See, there we go and you don’t have to worry about numbers you just draw a shape.

EL: Yeah, nice.

KK: Cool. So we also like to give our guests a chance to plug anything they want to plug. Do you have things out there in the world that you want people to know about?

SC: So the only thing I think that's worth mentioning is I do some Royal Institution maths master classes, where we go out and we take our favorite bit of math, and we go and take it to students who are between the ages of about 14 to 17. And that's really what I'm doing coming up in the near future, and they are a brilliant way for lots of people to engage with maths.

EL: Oh, nice.

KK: That’s very cool.

SC: Yeah. They are really good fun.

KK: Have you been doing that for very long?

SC: I’ve been doing them for about two years now. And the first one I ever did was on Bayes’ theorem. And I've never been so terrified, because I don’t teach. And then you have this group of students, and they come up with just the best and most fantastic questions. Every time you do it, you go, “I hadn’t thought of that.”

KK: Yeah.

SC: “And I don't know how to answer that question straight away.” So it's brilliant, and I love doing them. So that's kind of what we've got coming up. And you know, work is just going to be keeping me nicely busy.

EL: Nice.

SC: Yeah.

KK: Well, this has been great fun. Thank you for joining us, and congratulations on being the world's most interesting mathematician for this year.

EL: Yes. Yeah, thanks a lot.

SC: Thank you. I’ve been so excited to do this. I've been listening to your podcast for quite a long time, and I couldn't believe it when you emailed.

okay, thank you very much.

Okay. Thanks.

On this episode, we had the pleasure of talking with Sophie Carr, a statistics consultant and winner of Christian Lawson-Perfect’s Big Internet Math-Off last summer. Here are some links you may enjoy as you listen to this episode.

As we mentioned at the top of the show, Evelyn’s math page-a-day calendar is available for purchase in the AMS bookstore!
Sophie Carr’s twitter account
The Big Internet Math-Off at the Aperiodical
Royal Institution Masterclasses
Sophie Carr is this year’s World’s Most Interesting Mathematician. We also had last year’s World’s Most Interesting Mathematician, Nira Chamberlain, on the show in January. Find his episode here.

Episode 47 - Judy Walker

10 oktober 2019 | 32 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics and all kinds of crazy stuff, and I have no idea what it's going to be today. It is a tale of two very different weather formats today. So I am Kevin Knudson, professor of mathematics at the University of Florida. Here's your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah, where I am using the heater on May 28.

KK: Yes, and it's 100 degrees in Gainesville today, and I'm miserable. So this is bad news. Anyway, so today, we are pleased to welcome Judy Walker. Judy, why don't you introduce yourself?

Judy Walker: Hello. Thank you for having me. I'm Judy Walker. I'm a professor of mathematics at the University of Nebraska.

KK: And what else? You're like—

JW: And I am Associate Vice Chancellor for faculty and academic affairs, so that’s, like, Vice Provost for faculty.

KK: That sounds—

EL: Yeah, that does sound very official!

JW: It does sound very official, doesn't it?

KK: That’s right. Like you're weighing T & P decisions in your hands. It's like, you're like Caesar, right? With the thumbs up and the—

JW: I have no official power whatsoever.

KK: Right.

JW: So yes.

KK: But, well, your power is to make sure procedures get followed, right?

JW: Yes. And I have a lot of I have a lot of influence on other things.

KK: Yeah. Right. Yeah. That sounds like a very challenging job.

JW: And for what it's worth, I will add that it is cloudy and windy today. But I think we're supposed to be, like, 67 degrees. So right in the middle.

KK: All right. Great.

EL: Okay, perfect.

KK: So if we could see the map of the US, there'd be these nice isoclines. And here we are. Right. So we're, my mine is very hot. Mine's red. So we're good. Anyway, we came to talk about math. You’re excited to talk about math for once, right?

JW: Exactly. I guess I'm kind of going to be talking about engineering, too. So—

EL: Great.

KK: That’s cool. We like it all here. So what's your favorite theorem?

JW: So my favorite theorem is the Tsfasman-Vladut-Zink theorem.

KK: Okay, that's a lot of words.

JW: It is—well, it’s a lot of names. It's three names. And it's a theorem that is in error-correcting codes, algebraic coding theory. And it's my favorite theorem, because it solves a problem, or maybe not solves a problem, but shows that something's possible that people didn't think necessarily was possible. And the way that it shows that it's possible is by using some pretty high-powered techniques from algebraic geometry, which had not previously been brought into the field at all.

EL: So what is the basic setting? Like what kind of codes can you correct with this theorem?

JW: Right. So the codes are what does the correcting. We don't correct the codes, we use the codes to correct. So I used to tell my — actually, my advisor told me and then I've told all my PhD students — that you have to have a sentence that you start everything with. And so my sentence is: whenever information is transmitted across a channel, errors are bound to occur. So that is the setting for coding theory. You've got information that you're transmitting. Maybe it's pictures from a satellite, or maybe it's just storing things on a computer, or whatever, but you're storing this information. Or you're transmitting this information, and then on the other end, or when you retrieve it, there's going to be some mistakes. And so it's the goal of coding theory to add redundancy in such a way that you can find those mistakes and fix them. Okay?

And we don't actually consider it an error if you fix the mistake. So an error is when so many mistakes happened in the transmission or in the storage and retrieval, that what you think was sent was not what was actually sent, if that makes sense.

KK: Sure. Okay.

JW: So that's the basic setting for coding theory, and coding theory kind of started in 1948 with Shannon's theorem.

KK: Right.

JW: So Shannon's theorem says that reliable communication is possible. So what it says really, is that whatever your channel is, whether it's transmitting satellite pictures, or storing data, or whatever—whatever your channel is, there is a kind of maximum efficiency that's possible on the channel. And so what Shannon’s theorem says is that for any efficiency up to that maximum, and for any epsilon greater than zero, you can find a code that is that efficient has less than epsilon probability of error, meaning the probability that what you sent is not what you think was sent at the end. Okay?

So that's Shannon's theorem. Right? So that's a great theorem.

EL: Yeah.

JW: It’s not my favorite theorem. It’s not my favorite theorem because it actually kind of bothers me.

KK: Why does it bother you?

JW: Yeah, so the reason that bothers me are — there are two reasons it bothers me. One is that it doesn't tell us how to find these codes. It says good codes exist, but it doesn't tell us how to find them, which is kind of useless if you're actually trying to transmit data in a reliable way. But it's actually even worse than that. It's a probabilistic proof. And so it doesn't just say that good codes exists, it says they're everywhere, but you can't find them. Right? So it's like it's taunting us. So I just—. So yeah. So that's Shannon's theorem. And that's why it's not my favorite theorem. But why it's a really great theorem is that it started this whole field. So the whole field of coding theory has been — or of channel coding, at least, which is what we've been talking about is to find those codes, and not just find them, but find them along with efficient decoding algorithms for them. And so that's Shannon's challenge is to find the good codes with efficient decoding algorithms for those good codes. That's 1948, that that started. Right? Okay.

So just as a digression, let me say that most mathematicians and engineers will agree that at this point in time — so a little more than 70 years after Shannon's theorem, that Shannon's challenge has been met, so that we can find these good codes. They're not going to agree on how it's been met. But they'll all agree that it has been met. So on the one hand, in the late ‘90s — mid-to-late 90s — engineers found turbo codes, and they rediscovered low-density parity check codes. And these are codes that in simulations come very, very close to meeting Shannon's challenge. The theory around these codes is still being developed. So the understanding of why they meet Shannon challenge is still try to be solved. But the engineers will say that it's solved, that Shannon's challenge is met because they've got these simulations, and they're so confident about it, that these codes are actually being used in practice now.

EL: So I have a naive question, which is, like, does the existence of us talking over the internet on on this call, sort of demonstrate that this has been met? Like we we are hearing each other — I mean, not with perfect fidelity, but we're able to transmit messages. Is that? Or is that just not even in the same realm?

JW: No, that's exactly exactly what we're talking about, exactly what we're talking about. And not only that, but I don't know if you've noticed, but every once in a while, Kevin gets a little glitchy, and he doesn't move for a while. That's the code catching up and fixing the errors.

KK: Yeah, that's the irony is this this call has been very glitchy for me.

JW: Right.

KK: Which is why we each record our own channel.

EL: Yeah.

JW: Exactly. So in fact, low-density parity-check codes and turbo codes are being used now in mobile phones, in satellite communications, in digital TV, and in Wi-Fi. So that's exactly what we're using.

EL: Okay.

JW: But the mathematicians will say, “Well, it's not really—we’re not really done. Because we don't know why. We don't really understand these things. We don't have all the theoretical underpinnings of what's going on.” A lot of work has been done a lot, and a lot of that is there. But it's still a work in progress. About 10 years ago, kind of on the flip side, polar codes were discovered. And polar codes are the first family of codes to provably achieve capacity. So they actually provably meet Shannon's challenge. But at this moment, they are unusable. There's just still a lot of work to understand how we can actually use polar codes. So the mathematicians say, “We've met the challenge, because we've got polar codes, but we can't use them.” And the engineers say, “We've met the challenge because we've got turbo codes and LDPC codes, but we don't know why.” Right? And that's an oversimplification, but that's kind of the current state. And so different people are working on different things now. And of course, there are other kinds of coding that that aren’t — that isn't really channel coding. There are still all kinds of unsolved problems. So if anybody tells you that coding theory is dead, tell them they're wrong.

EL: Okay!

JW: It’s still very much alive. Okay, so we talked about Shannon's theorem from 1948. And we talked about the current status of coding theory. And my favorite theorem, this Tsfasman-Vladut-Zink, is from 1982. So in the middle.

EL: Halfway in between.

JW: Yes, yes. Just like my weather being halfway in between. Yes. So around this time, in the early ‘80s, and and preceding that, the way that mathematicians were approaching Shannon's challenge was through the study of linear codes. So linear codes are just subspaces, and we might as well think of—in a lot of applications, the data is zeros and ones. But let's go to Fq instead of F2, so q is any prime power.

KK: Okay, so we're doing algebraic geometry now, right?

JW: We’re not yet. Right now, we’re just talking about finite fields.

KK: Okay.

JW: We will soon be be doing algebraic geometry, but not yet. Is that okay?

EL: You’re just trying to transmit some finite set of characters.

JW: Yes, some finite string of characters. Order matters, right? So it's a string. And so the way that we think about it, we can think about it as a systematic code. So the first k characters are information, and then we're adding on n−k redundancy characters that are computed based on the first k.

KK: Okay.

JW: So if we're in a linear setting, then this collection of code words that include the information and the redundancy, that collection of code words is a subspace, say it's a k-dimensional subspace, of Fqn. So that's a linear code. And we can think about that ratio, k/n, as a measure of how efficient the code is.

KK: Right.

JW: Because it's the number of information bits divided by the total number of bits, or symbols, or characters. So, let's call that ratio, R for rate, right? k/n, we’ll call it R. And then how many errors can the code correct? Well, if you look at that Hamming distance—so that's the number of characters and number of positions in which to code words differ—then the bigger that distance, the more errors you can make and still be closest to the code word that was sent. So then that's not really an error. Right? So maybe we say the number of mistakes goes up.

EL: Yeah. So again, let's normalize that minimum distance of the code by dividing by the length of the code. So we have a ratio, let's call that ∂. So that's our relative minimum distance for the code. So one way to phrase this is if we want a certain error-correcting capability, so a certain ∂, how efficient can the code be? How big can R be? Okay, so there are a lot of bounds relating R and ∂, our information rate and our error-correcting capability, or our relative minimum distance. So one that I want I tell you about is that Gilbert-Varshamov bound.

So the Gilbert-Varshamov bound is from 1952. And it says that there's a sequence of codes, or a family of codes if you want, of increasing length, increasing dimension, increasing minimum distance, so that the rate converges to R and the minimum distance to converges to ∂. And R is at least 1−Hq(∂), where Hq is this entropy function. So you may have heard of the binary entropy function, there's a q-ary entropy function, that's what Hq(∂) is. So one such sequence is the so-called classical Goppa codes, and I want to say that that's from, 1956, so just a little bit later. And those codes were the best-known codes from this point of view for about 30 years. Okay, so let me just say that again. So the Gilbert-Varshamov bounds says that there's a sequence of codes with R at least 1−Hq(∂). The Goppa codes satisfy r=1−Hq(∂). And for 30 years, we couldn't find any codes with R greater than.

EL: That were better than that.

JW: Right. That were greater than this 1−Hq(∂).

KK: Okay.

JW: So people at this point were starting to think that maybe the Gilbert-Varshamov bound wasn't a bound as much as it was the true value of how good can R be given ∂, how efficient can codes be given given their relative minimum distance. So this is where this Tsfasman-Vladut-Zink theorem comes in. So in 1978—and Kevin, now we can talk about algebraic geometry. I know you’ve been waiting for that.

KK: All right, awesome.

JW: Yes. Right. So in 1978, Goppa defined algebraic geometry codes. So the way that definition works: remember, a code is just a subspace of Fqn, right? So how are we going to get a set of space of Fqn? Well, what we're going to do is we're going to take a curve defined over Fq that has a lot of rational points, Fq-rational points, right? So we're going to take one of those points and take a multiple of it and call that our divisor on the curve. And then we're going to take the rest of them. And we're going to take the rational functions in this space L(D). D is our divisor, right? So these are the functions that only have poles at this chosen point of multiplicity, at most the degree that we've chosen.

KK: Okay.

JW: And we're going to evaluate all those functions at all the rest of those points. So remember, those functions form a vector space, and evaluation is a linear map. So what we get out is a vector space. So that's our code. And if we make some assumptions, so if we assume that that degree of that divisor, so that multiplicity that we've chosen, is at least twice the genus minus 2, twice the genus of the curve minus 2, then Riemann-Roch kicks in, and we can compute the dimension of L(D). But if we also assume that that degree is less than the number of points that we're evaluating at, then the map is injective. And so we have exactly what the dimension of the code is. The dimension of the code is the degree of the divisor, so that multiplicity that we chose, plus 1 minus the genus. And the minimum distance, it turns out, is at least n minus the degree of the divisor. So lots of symbols, lots of everything.

EL Yeah, trying to hold this all in my mind, without you writing it on the board for me!

JW: I know, I’m sorry. But when you put it all together, and you normalize out by dividing by the length, what you get is that if you have a family of curves with increasing genus, and an increasing number of rational points, then we can end up with a family of codes, so that in the limit, R, our information rate, is at least 1−∂—that’s that relative minimum distance—minus the limit of the genus divided by the number of rational points. Okay. So g [the genus] and n are both growing. And so what's that limit? So that's that was Goppa’s contribution. I mean, not his only contribution. But that's the contribution of Goppa I want to talk about, just that definition of algebraic geometry code. So it's a pretty cool definition. It’s a pretty cool construction. It’s kind of brand new in the sense that nobody was using algebraic geometry in this very engineering-motivated piece of mathematics.

EL: Right.

JW: So here is algebraic geometry, here is a way of defining codes, and the question is, are they any good? And it really depends on what—how fast can the number of points grow, given how fast the genus is growing? So what Drinfeld and Vladut proved—so this is not the TVZ theorem, not my favorite theorem, but one more theorem to get there—Drinfeld and Vladut proved that if you take, if you define Nq(g) to be the maximum number of Fq-rational points on any curve over Fq of genus g, then as you let g go to go to infinity, and for a fixed q, the limit superior, the lim sup, of the ratio g/Nq(g), is at most 1/√(q−1). Okay, fine. Why do we care? Well, the reason we care is that the Tsfasman-Vladut-Zink theorem, which is again my favorite theorem, it says—so actually, my favorite theorem is a corollary of the Tsfasman-Vladut-Zink theorem. So the Tsfasman-Vladut-Zink theorem says that if q is a square prime power, then there's a sequence of curves over Fq of increasing genus that meets the Drinfeld-Vladut bound.

EL: Okay.

JW: Okay, so the Drinfeld-Vladut bound said you can be at most this good. And Tsfasman-Vladut-Zink says, hey, you can do that.

EL: Yeah, it's sharp.

JW: So if we put it all together, then the Gilbert-Varshamov bound gave us this curve, right? So it was a concave-up curve that intersects the vertical axis, which is the R-axis, at 1 and the horizontal axis, which is the ∂-axis, at 1−1/q. So it's this concave-up thing that's just kind of curving out. Then the Tsfasman-Vladut-Zink line—the theorem gives you a line that looks like R=1−∂−1/√(q−1). Right? So it's just a line of slope −1, right, with y-intercept 1−1/√(q−1). So the question is, does that line intersect that curve? And it turns out that if you have q, a square prime power q at least 49, then the line intersects the curve in two points.

EL: Okay.

JW: So what that is really doing for us is it's telling us that in that interval between those two points, we have an improvement on the Gilbert-Varshamov bound. We have better codes than we thought were possible for 30 years.

EL: Wow!

JW: Yes. So that's my, that's my favorite theorem.

KK: I learned a lot.

EL: And where did you first encounter this theorem?

JW: In graduate school? Okay, in graduate school, which was not in 1982. It was substantially after that, but it was said to me by my advisor, “I think there's a connection between algebraic geometry and coding theory, go learn about that.”

KK: Oh.

JW: And I said, “Okay.”

KK: And so two years later.

JW: Right. Right, right. Actually, two years later, I graduated.

KK: Okay. All right. So you’re much faster than I am.

JW: Well, there was four years before that of doing other things.

EL: So was it kind of love at first sight theorem?

JW: Very much so. Because I mean, it's just so beautiful, right? Because here's this problem that nobody knew how to solve, or maybe everybody thought was solved. Because nobody had any techniques that could get any better than the Gilbert-Varshamov bound. And then here's this idea, just way out of left field saying, hey, let's use algebraic geometry to find some codes. And then, hey, let's look at curves with many points. And hey, that ends up giving us better codes than we thought were possible. It's really, really pretty. Right? It's why mathematicians are better than electrical engineers.

EL: Ooh, shots fired!

JW: Gauntlet thrown. I know.

EL: But it does make you wonder how many other things in math will eventually find something like this, like, will will find for these problems—you know, factoring integers or things like this— that we think are difficult, will someone swoop in with some completely new thing and throw it on its head?

JW: Yes. Exactly. I mean, I don't know anything about it. Maybe you do. But the idea that algebraic topology, right, is useful in big data.

KK: Yeah, sure. That's what I've been working on lately. Yeah. Right.

JW: I love that.

KK: Yeah. Sure.

JW: I love that. I don't know anything about it. But I love it.

KK: Well, the mantra is data has shape. Right? So let me just, you know, smack the statisticians here. So they want to put everything on a straight line, right? But a circle isn't a straight line. So what if your data’s a circle? So topology is very good at finding circles.

JW: Nice.

KK: Well, that's the mantra, at least. So yeah. All these unexpected connections really do come up. I mean, it's really—that’s part of why we keep doing what we're doing, right? I mean, we love it. But we never know what's out there. It's, you know, to boldly go where no one has gone before. Right?

JW: Exactly. And Evelyn, it's funny that you should bring up factoring integers, because you know that the form of cryptography that we use today to make it safe to use our credit cards on the internet, that’s very much at risk when quantum computers are developed.

EL: Right.

JW: And so, it turns out that algebraic geometry codes are not being used in practice, because LDPC codes and turbo codes are much more easily implementable. However, one of the very few known so far unbreakable methods for post-quantum cryptography is based on algebraic geometry codes.

KK: Excellent.

EL: Nice.

JW: So even if we can factor integers,

KK: I can still buy dog food at Amazon. Right?

JW: You can still shop at Amazon because of algebraic geometry codes.

EL: Yeah, the important things.

KK: That’s right.

EL: Well, so another thing we like to do on this podcast is invite our guests to pair their theorem with something, the way we would pair food with fine wines. So what have you chosen for this theorem?

JW: So that was very hard. Yeah. I mean, it's just kind of the most bizarre request.

EL: Yeah.

JW: So I mean, I guess the way that I think about this Tsfasman-Vladut-Zink theorem, I was looking for something that was just, you know, unexpected and exciting and beautiful. But I couldn't come up with anything. And so instead, what I'm going with is lemon zest.

KK: Okay.

EL: Okay.

JW: Which I guess can be unexpected and exciting in a dessert, but also because of the way that you just kind of scrape it off that curve of the lemon. And that's what the Tsfasman-Vladut-Zink theorem is doing, is it’s scraping off a little bit of that Gilbert-Varshamov curve.

KK: This is an excellent visual. I've got it. I zest lemons all the time. I understand now. This is it.

EL: Yeah.

JW: There you go.

KK: So all right. Well, we also like to give our guests a chance to plug anything. You wrote a book once. Is that still right? I have it on my shelf.

JW: Yeah. I did write a book once. So that book actually was—Yeah, so I wasn't going to plug anything, but I will plug the book a little bit, but more I'm going to plug a suite of programs. So the book is called, I think, Codes and Curves.

KK: That sounds right.

JW: You would think I would know that.

KK: I’d have to find it. But it is on my shelf.

JW: Yes. It's on mine too, surprisingly, which is right behind me, actually, if you have the video on.

So that book really just a grew out of lecture notes from lectures I gave at the program for women and mathematics at the Institute for Advanced Study. Okay, so I will take my opportunity to plug something to plug that program, to plug EDGE, to plug the Carleton program, and to plug the Smith post-bac program, and to plug the Nebraska conference for undergraduate women in mathematics. So what do all these programs have in common they have in common? They have in common two things that are closely related. One is that they are all programs for women in mathematics. And the other is that they were all the subject of study of a recent NSF grant that I had with Ami Radunskaya and Deanna Haunsperger and Ruth Haas that studied what are the most important or effective aspects of these programs and how can we scale them?

EL: Oh, nice.

JW: Yes. And some of the results of that study, along with a lot of other information, are on our website. That is women do math.org?

EL: I will be visting it as soon as we get off this phone call.

JW: Right. Awesome. I hope it's functioning

KK: And because Judy won't promote herself, I will say, you know, she's been a significant leader in promoting programs for women in mathematics through the University of Nebraska’s math department there. There's a picture of her shaking Bill Clinton's hand somewhere.

JW: Well, that's also on my shelf. Okay. Yeah, I think it's online somewhere, too.

KK: Right. Their program won a national excellence award from the President. Really excellent stuff there at the University of Nebraska. Really a model nationally.

EL: Yeah, I’m familiar with that as one of the best graduate math programs for women.

JW: Thank you.

EL: Yeah. Great job!

EL: Yeah, well, we'll have links to all of those programs on the website. So if you didn't catch one, and you're listening, you can to the website for the podcast and find all those. Yeah. Well, thank you so much for joining us, Judy.

JW: Thank you for the opportunity.

KK: Yeah, this has been great fun. Thanks.

JW: All right. Thank you.

On this episode, we were happy to talk with Judy Walker, who studies coding theory at the University of Nebraska. She told us about her favorite theorem, the Tsfasman-Vladut-Zink theorem. Here are some links to more information about topics we mentioned in the episode.

Goppa (algebraic geometry) code

Hamming distance

Gilbert-Varshamov bound

Judy Walker’s book Codes and Curves

The Program for Women and Mathematics at the Institute for Advanced Study

EDGE

The Carleton Summer Mathematics Program for women undergraduates

The Smith College post-baccalaureate program for women in math

The Nebraska Conference for Undergraduate Women in Mathematics (Evelyn will be speaking at the conference in 2020)

WomenDoMath.org

Episode 46 - Adriana Salerno

12 september 2019 | 31 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcasts where there's no quiz at the end. I’m coming up with a new tagline for it.

Kevin Knudson: Good.

EL: I just thought I'd throw that in. Yeah, so I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer from Salt Lake City—or in Salt Lake City, Utah, not originally from here. And here's your other host.

KK: I’m Kevin Knudson, professor of mathematics at the University of Florida in Gainesville, but not from Gainesville. This is part of being a mathematician, right? No one lives where they're from.

EL: Yeah, I guess probably a lot of professions could say this, too.

KK: Yeah, I don’t know. It’s also a sort of a generational thing, right? I think people used to just tend to, you know, live where they grew up, but now not so much. But anyway.

EL: Yeah.

KK: Oh, well, it's okay. I like it here.

EL: Yeah. I mean, it's great here right now it's spring, and I've been doing a ton of gardening, which always seems like such a chore and then I'm out smelling the dirt and looking at earthworms and stuff, and it's very nice.

KK: I’m bird watching like crazy these days. Yesterday, we went out and we saw the bobolinks were migrating through. They're not native here, they just come through for, like, a week, and then they're gone.

EL: The what?

KK: Bobolinks, B-O-B-O-L-I-N-K. They kind of fool you, they look a little bit like an oriole, but the orange is on the wrong side. It's on the back of the neck instead of underneath.

EL: Okay, I'll have to look up a picture of that later.

KK: And then this morning for the first time ever, we had a rose-breasted grosbeak at our feeder. Never seen one before and they're not native around here, they just migrate through. So this is

EL: Very nice. Yes.

KK: This is what I'm doing in my late middle age. This is what I do. I just took up bird watching, you know?

EL: Yeah. Well, I can see the appeal.

KK: Yeah, it's great.

EL: Yes. But we are excited today to be talking with Adriana Salerno. Do you want to introduce yourself?

Adriana Salerno: Hi. Yeah, I'm Adriana Salerno. Now I am an associate professor of math at Bates College in Maine. And I am also not from Maine. I live in Maine. I'm originally from Caracas, Venezuela, so quite a ways away.

EL: Yeah.

AS: Again, you don't choose where you live, but maybe you get to choose where you work. So that's nice.

EL: Yeah. And you're you're not only a professor there, but you're also the department chair right now, right?

AS: Oh, yeah. Yeah, I'm trying to forget. No, I’m kidding.

EL: Sorry!

KK: You know, speaking of, before we we started recording here, I spent my afternoon writing annual faculty evaluations. I’m in the first year as chair. I have 58 of them to write.

AS: Oh, I don't have to do those, which I'm very happy about. But we are hiring a staff position, and I'm in charge of that. And that's been a lot.

EL: And we actually met because both of us have done this mass media fellowship for people interested in math or science and writing. And so you've done a lot of writing not for mathematicians as well, throughout your career path.

AS: Yeah, yeah. I mean, I did the mass media fellowship in 2007. And since then, I've been trying to write more and more about mathematics for a general audience. These days, I mostly spend time writing for blogs for the AMS. And right now I'm editing and writing for inclusion/exclusion. I wish I had more time to write than I do. It's one of those things that I really like to do, and I don't think I do enough of, but these opportunities are great because I get to use those—or scratch that itch, I guess, by talking to you all.

EL: Yes.

KK: Well, so speaking of, we assume you have a favorite theorem that you want to tell us about. What is it?

AS: Well, so it's always hard to decide, right? But I guess I was inspired by a conversation I had with Evelyn at the Joint Math Meetings. So I've decided my favorite theorem is Cantor's diagonalization argument that the real numbers do not have the same cardinality as the natural numbers.

EL: Yes, and I’m so excited about this! Ever since we talked at the Joint Meetings, I’ve been very excited about getting you to talk about this.

AS: Good. Good.

EL: Because really, it’s such a great theorem.

AS: Yeah. Well, I was thinking about it today. And I'm like, how am I going to explain this? But I have chosen that, and I'm sticking with it. Yeah.

EL: Yes.

KK: Good.

AS: So yeah, it’s—one of the coolest things about it is sort of it’s this first experience that you have, as a math student—at least it was for me—where you realize that there are different sizes of infinity. And so another way of saying that is that this theorem shows, without a doubt, I believe—although some students still doubt me after we go over it—it shows that you can have different sizes of infinity. And so the first step, even, is to say, “How do you decide if two things have the same size of infinity?” Right? And so it's a very, very lovely sort of succession of ideas. And so the first thing is, how do you decide that two things are the same size? Well, if they're finite, you count them, and you see that you have the same number of things. But even when things are finite—and say, you're a little kid, and you don't know how to count—another way of saying there's the same number of things is if you can match them up in pairs, right? So you know, if you want to say I have the same number of crayons as I have apples, you can match a crayon to an apple and see that you don't have anything left over, right?

EL: Yeah.

AS: And so it's just a very natural idea. And so when you think about infinite sets—or not even infinite sets—but you can think of this idea of size by saying two things are the same size if I can match every element in one set to every element in another set, just one by one. And so I really like, I'm borrowing from Kelsey Houston-Edwards’ PBS show, but what I really like that she said that you have two sets, and every element has a buddy, right? And so then I love that language, and so I'm borrowing from from her. But then that works for finite sets, but you can extend it to an infinite set.You can say, for example, that two infinite sets are the same size if I can find a matching between every element in the first set and every element in the second set. It’s very hard to picture in your head, I think, but we're going to try to do this. So for example, you can say that the natural numbers, the counting numbers, 1, 2, 3, 4, etc, have the same size as the even numbers, because you can make a matching where you say, “Match the number 1 with the number 2 on the other side. And then the number 2, with the number 4 on the other side.” And you have all the counting numbers, and for every counting number, you have two times that number as the even buddy.

EL: Yeah. And I think this is, it's a simple example that you started with, but it even hints at the weirdness of infinity.

AS: Yeah.

EL: You’ve got this matching, but the even numbers are also a subset of the natural numbers. Ooh, things are going to get a little weird here.

KK: Clearly, there aren’t as many even numbers, right?

AS: Yeah.

KK: This is where you fight with your students all the time.

AS: That’s exactly—so when you're teaching this, the first thing you do is talk about things that have the same cardinality. And then everybody, it can take a while, you know, like, infinity is so weird that you can actually do these matching. So Hilbert’s infinite hotel is a really great way of doing this sort of more conceptually. So you have infinitely many rooms. And so for example, suppose that rooms numbers from 1, 2, 3, to infinity, mean, and so on. Yes, you have to be careful because infinity is not a number. You have to be careful with that. But say that all the rooms are occupied. And so then, you know, say someone shows up in the middle of the night, and they say, “I need a room.” And so what you do if you're the hotel manager is you tell everyone to move one room over. And so everyone moves one room over and you put this person in, and room number one. And so that's another way of seeing that. So the one-to-one pairing, or the matching here is every person has a room. And so the number of rooms and the number of people are the same—the word is cardinality because you don't want to say number because you can't count that.

KK: Right.

AS: And so you you say cardinality instead. But it's really weird, right? Because the first time you think about this, you say, “Well, you know, there's infinity, and there's infinity plus one.” That's like the kind of thing that you would say as a kid, right? And they're the same! When you have the natural numbers and the natural numbers and one extra thing, or like with zero, for example—unless you're in the camp that says zero is an actual number—but we're not going to get to that discussion right now.

KK: I’m camp zero is a natural number.

AS: Okay. I feel like I know maybe half people who say zero is a natural number and the other half say it's not. And I don't think anyone has good arguments other than, ah, it must be true! And so then the cool thing is, once you start doing that, then you start seeing, for example—and these are, these are kind of tricky examples, it can get tricky. Like, you can say that the integers like the positive whole numbers, negative whole numbers and zero, that also has the same cardinality as the natural numbers. Because you can just start with zero—I mean, basically, when you want to say that something has the same cardinality as the natural numbers, what you're really trying to do is to find a buddy, so you're trying to pair someone with one or two, or three. But really, what you can do is just list them in order, right? Like you can have like the first one, the second one, the third one, the fourth one, and you know that that's a good matching. It's like the hotel. You can put everyone in a room. And then you know they're the same number. Everyone has a room. So with the integers, for example, the whole numbers, positive, negative and zero, then you can say, “Okay, put zero first, then one, then negative one, and two, then negative two then three, then negative three,” and then they're the same size, right? And so once you start thinking about this—I remember this pretty clearly from from college—once you start thinking about this, then you're like, “Well, obviously, because infinity is infinity.” That’s the next step. So the first step is like, well no, infinity plus one and infinity are different. But then you get convinced that there is a way of matching things that where you can get things that seem pretty different, or a subset of a set, and they have the same cardinality. And then you go the other direction, which is “Well, of course, anything infinite is going to be the same size as anything else that's infinite.” And so then it turns out that even the rationals are the same size as the natural numbers. And that's way more complicated than we have time for. But if you add real numbers, meaning irrationals as well, then you have a whole different situation.

KK: You do indeed.

AS: It’s mind blowing, right? And so if you just think about the real numbers between zero and one, so just get go real simple. I mean, small, relatively. So you're just looking at decimal expansions. And so if those numbers had the same cardinality as the natural numbers, then you should be able to have a first one and a second one and a third one, and a fourth one. Or you can pair one number was the number 1, one number with the number 2, etc. And that list should be complete, and in the words of Kelsey Houston-Edwards, everyone should have a buddy. And so then, here's the cool thing, this is a proof that these two sizes of infinity are not the same, and it's a proof by contradiction, which is, again, your favorite proof when you are learning how to prove things. I mean, when I was learning proofs, I wanted to do everything by contradiction. So proving something by contradiction means you want to assume, “Well, what if we can list all the all the real numbers?” There’s a first one, a second one, a third one, etc. So Cantor’s amazing insight was that you can always find a number that was not on that list. Every time you make this list: a first one, a second one, a third one… there is some missing element.

And so you line up all your decimals. So you have the first number in decimal. And so you have like, you know, 0.12345… or something like that. And then you have the next one. And the next one. And like, I mean, this is really hard to do verbally, but we're going to do it. And so you sort of line them up, and you have infinite decimals. So you have point, a whole bunch of decimals, point, a whole bunch of decimals. And so you can make a missing number by taking that first number in the first decimal place a just changing that number. Okay, so if it was a 1, you write down a 2. And so you know, because we’ve known how to compare decimals since we were little kids, that what you need to compare is decimal place by decimal place. So these are different because they're different in this one spot, right? And then you go to the second number, and the second decimal point. And then you say, “Well, whatever number I see there, I'm going to make the second decimal point of my new number different.” So if you had a 3, you change it to a 4, whatever it is, as long as it's not the original number. And and this is why it's called the diagonalization argument, or the diagonal argument, because you have lined all those numbers up, and you can go through the diagonal, and for each one of those decimal points, at each decimal place, you just change the value. And what you're going to get is a number, another real number, infinitely many decimals, and it's going to be different from every number on your list, just by virtue of how you made it. And so then, what that shows is that the answer to the “what if” is: you can’t. The “what if” is, if you have a list of all real numbers, it's not complete. So there is never going to be a way that you can make that list complete. And this is the part where every time I tell my students, at some point, they're like, “Wait, there are different sizes of infinity? What?” Then—and that’s sort of lovely, because it's just this this mind-blowing moment where you've convinced yourself, by the way, that you were to infinity is infinity, and then you realize that there's something bigger than the cardinality of the natural numbers. And and then it's really fun when you tell them, “Well, is there something in between?” They’re like, “Of course! There must be!” And then you're like, “Wait, no one knows.”

KK: Maybe not.

EL: Yeah.

AS: So yeah, I just love that argument. And I love how simple it is. And at the same time, it's, simple, but it's very, very deep, right? You really have to understand how these numbers match up with each other. And it requires a big leap of imagination to just think of doing this and realizing that you could make a number that was not on this infinite list by just doing that simple trick.

EL: Yeah.

AS: And so I just think it's a really, really beautiful theorem. And then I also have a really personal connection to this theorem. But it's one of my favorite things to teach. And I'm going to be teaching at this term, and I’m really looking forward to seeing how that how that lands. Sometimes it lands really well. Sometimes people are like, “Eh, you’re just making stuff up.” Yeah.

EL: Yeah.

KK: Well, then you can really blow their minds then when you show them the Cantor set, right?

AS: Yeah, yeah.

KK: And say, “Well, look, I mean, here's this subset of the reels that has the same cardinality, but it's nothing.”

AS: Exactly. Yeah, there's nothing there. Yeah.

EL: Yeah. I remember, then, when I first saw this argument, really carefully talking myself through, “Like, okay, but what if I just added that number I just made to the end of the list? Why wouldn't that work?” And trying to go through, like, “Why can't I—Oh, and then there must be other numbers that don’t fit on the list either.” It's not like we got within 1 of being the right cardinality.

AS: Right.

EL: For these infinite number. So yeah, it's a really cool idea. But you said you had some personal connections to this. So do you want to talk more about those?

AS Sure. So I am from Venezuela, and I went to college there. And I liked college, it was fine. I knew—Well, one thing that you do have to decide when you're a student in high school is, you don't really apply to college, you apply to a major within the college. And so then I knew I wanted to do math. And I signed up for math at a specific university. And so then the first year was very similar to what you would do in the States, which is sort of this general year where everybody's thinking calculus, or everybody's taking—you have some subset of things that everybody takes. And then your second year, you start really going into the math major. And so this was my first real analysis class. This was my first serious proof-y class in my university. And we learned Cantor’s diagonalization argument, which was pretty early. But I loved this argument. I felt so mind-blown. You know, I was like, “This is why I want to do math,” you know, I was just so excited. And I knew I understood everything. And so I took the exam, and I got horrible grade. And in particular, I got zero points on the “prove that the real and the natural numbers don't have the same cardinality.” And so I went to the professor, and I saw my exam, and I was really confused. And I went to the professor, and I said, “I really don't understand what's wrong with this problem. Could you help me understand?” Because I thought I understood this. And then—you know, that's a typical thing. I probably said it in a more obnoxious way than I remember now. But I felt like I was being pretty reasonable. I was not the kind of kid that would go up to my professors too often to ask for points. I really was like, “I don't know what I did wrong.” And especially because I felt like I really got it.

EL: Right.

AS: And so then he just looked at me and said, “If you don't understand what's wrong with this problem, you should not be a math major.” And that was it. That was the end of that conversation. Well, I still don't know what's wrong with this problem, and now you just told me I need to do something else. Just go do something else at a different school. Right? And I mean, I don't know that that was particularly sexist. But I do know that I was the only woman in that class, and I know that I felt it a lot. I think he probably would have said that—I really do think that he in particular would have said that to any student. I don’t think it was just me being female that affected that at all. But I do think that if I had been less stubborn about my math identity, I might have taken him up on that. But I was just like, “No, I'm going to show you!” And eventually I got an A in his class. He taught real analysis every semester, so I had to take the class with him every time and at some point, I cracked his code. And he at some point respected me, and thought I deserved to be there. But he was just very old-fashioned. You know, I don't think it's even sexism. It's just very, very, like, this is how we do things. And then I went—eventually, I did talk to someone. I think it was a teaching assistant. And I was like, “I don't know what's wrong with this problem.” And he looked at it. And he said, “Well, here's the problem. When you were listing—so you needed to list all these generic numbers and their decimal expansion. And I did, “Okay, the first number is point A1, A2, A3, etc. The second number is point B1, B2, B3, etc. The third one is point C1, C2, C3, etc, dot dot dot, right? And he said, “You have listed 26 numbers. And that's not going to be an infinite list.” Right?

KK: That’s cheap.

AS: And I was just like, “Okay, but I got the idea, right?” I was like, “Okay, it's true.” He’s like, “The way you wrote it is incorrect.” And I'm like, sure.

EL: Sort of.

KK: I’ve written that same thing on a chalkboard.

AS: You know, this shows you—like, fine, you can be more careful, you can be more precise, but from this, you shouldn't be a math major? That’s pretty intense.

EL: Yeah.

AS: And I knew the mechanics, I knew what was supposed to be happening, I knew how to make the missing number, right? Like you just need A1: you change it to some other number, B2: you change it to some other number, C3: you change it to some other number. And so, I just thought—I mean, that was a moment where I was just literally told I should not be in math because I made a silly mistake. And it was a moment where I realized that—now looking back, I realize my math identity was pretty strong, because I just said, “Well, ask someone else to see what was wrong, and I'm not going to ask this guy anymore because it's clear what he thinks.”

EL: Yeah.

AS: And sort of the stubbornness of, “Well, I’ll show him that I do deserve to be here.” But I think of all the students who might have taken classes with him, who would have heard that and then been like, “Yeah, maybe I need to do something else.” I mean, it just makes me really sad to hear, especially now that I'm a professor, and teaching these kinds of things. It just makes me sad to see which people were just scared away by someone like that, you know?

EL: Yeah.

AS: So that was a big moment for me. Yeah.

EL: Yeah. Quite a disproportionate response to, what’s basically a bookkeeping difficulty.

AS: Yeah.

EL: So, you know, we like to get our mathematicians to pair their theorems, with something on this show. And what have you chosen as your pairing for Cantor's diagonalization argument?

AS: Well, now that you suggested, music and other things, I'm maybe changing my mind.

EL: You could pair more than one thing.

AS: I was trying to find something that was just like—I need to sort of express the sort of mind-blowing nature of this, right? And so I was like, a tequila shot! You know, really just strong. And like, “Whoa, what just happened?” And so that was one thing that I thought about. And then—I don't know, just mind-blowing experiences, like, when I saw the Himalayas from an airplane, or when—you know, there are some moments where you're just like, “I can't believe this exists.” I can't believe this is a thing that I get to experience. So I guess, you know, there's been—most of these have been with traveling, where you just see something that you're just like, “I can't believe that I get to experience this.” And so I think Cantor's diagonalization argument is something like that, like seeing this amazing landscape where you're just like, “How does this even exist?”

EL: Yeah, I like that. I mean, I've had that experience looking out of airplane windows too. One time we were just flying by the coast of Greenland. And these fjords there. Of course, an airplane window is tiny and it's not exactly high-definition picture quality out of the thick plastic there, but it just took my breath away.

AS: Yeah.

EL: Yeah, I like that. And we can even invite our listeners to think of their own mind-blowing favorite experiences that they've they've had. Hopefully legal experiences in their jurisdiction.

KK: Well, oh wait, it's not 4/20 anymore. Oh, well. So we also like to invite our guests to plug anything they want to plug. So you write for the AMS, the inclusion/exclusion blog, are there other places where we might find your mathematical writing for the general public?

AS: Well, that's my main plug and outlet right now. But I I do write for the MAA Focus magazine sometimes. That's sort of my main, and sometimes the AWM newsletter. So you might find some of my writing there. And the blog. I mean, again, now that I'm chair and doing a lot of other things, I'm not writing as much, but I definitely like to—I’ve gotten really into maybe this is a weird plug, but I've gotten really into storytelling.

EL: Oh yeah, you’ve been on Story Collider?

AS: Yeah, I was on one Story Collider. I've done some of the local stuff. But you can find me on the internet telling stories about being a mathematician. Some of them about some pretty fantastic experiences, and some not so great experiences.

EL: Yeah. Okay. Yeah. Well, we'll link to your Twitter, and that can help people find you too.

AS: Oh, yeah. Cool.

EL: Thanks a lot for joining us.

AS: Yeah. Thanks for having me and listen to me ramble about infinity.

EL: Oh, I just love this theorem so much.

KK: Yeah, we could talk about infinity all day. Thanks, Adriana.

AS: Yeah. Thank you so much.

We were excited to have Bates College mathematician Adriana Salerno on the show. She is also the chair of the department at Bates and a former Mass Media Fellow (just like Evelyn). Here are some links you might enjoy along with this episode.

Salerno's website

Salerno on Twitter
AAAS Mass Media Fellowship for graduate students in math and science who are interested in writing about math and science for non-experts
Hilbert’s Infinite Hotel
Evelyn’s blog post about the Cantor set
Salerno’s StoryCollider episode
The inclusion/exclusion blog, an AMS blog about diversity, inclusion, race, gender, biases, and all that fun stuff

Episode 45 - Your Flash Favorite Theorems

8 augusti 2019 | 36 min

Kevin Knudson: 1-2-3

Kevin Knudson and Evelyn Lamb: Welcome to My Favorite Theorem!

KK: Okay, good.

EL: Yeah.

[Theme music]

KK: So we’re at the JMM.

EL: Yeah, we’re here at the Joint Math Meetings. They’re in Baltimore this year. The last time I was at the Joint Meetings in Baltimore I got really sick, but so far I seem to not be sick.

KK: That’s good. You’ve only been here a couple of days, though.

EL: Yeah. There’s still time.

KK: Yeah, so I’ve only been to the Joint Meetings one other time in my life, 20 years ago as a postdoc in Baltimore. I’ve just got a thing for Baltimore, I guess.

EL: Yeah, I guess so.

KK: So people may have seen this on Twitter. Fun fact: this is our first time meeting in person.

EL: Yeah.

KK: And you’re every bit as charming in real life as you are over video.

EL: And you’re taller than I expected because my first approximation of all humans is that they are my height, and you are not my height.

KK: But you’re not exceptionally short.

EL: No.

KK: You’re actually above average height, right?

EL: I’m about average for a woman, which makes me below average for humans.

KK: Well, if we’re going to the Netherlands, for example, I’m below average for the Netherlands.

EL: Yes.

KK: So I’m actually leaving today. I was only here for a couple of days. I was here for the department chairs workshop. You’re here through when?

EL: I’m leaving on Friday, tomorrow. Yeah, while we’ve been here we’ve been collecting flash favorite theorems where people have been telling us about their favorite theorem in a small amount of time. So yeah, we’re excited to share those with you.

KK: Yeah, this is going to be a good compilation. I’m going to try to get a couple more before I leave town. We’ll see what happens.

EL: Yeah. All right.

KK: Enjoy.

EL: I am here with Eric Sullivan. Can you tell us a little bit about yourself?

Eric Sullivan: Yeah, I'm an associate professor at Carroll College in Helena, Montana, lover of all things mathematics.

EL: And here with me in the Salt Lake City Airport, I assume catching a connecting flight to the Joint Math Meetings.

ES: You got it.

EL: All right, and what is your favorite theorem, or the favorite theorem you'd like to tell me about right now?

ES: Oh, I have many favorite theorems, but the one that's really coming to mind right now, especially since I'm teaching complex analysis this semester, are the Cauchy-Riemann equations.

EL: Very nice.

ES: Giving us a beautiful connection between analytic functions, and ultimately, harmonic functions. Really lovely. And it seems like a mystery to my students when they first see it, but it's beautiful math.

EL: Yeah, it is. They are kind of mysterious, even after you've seen them for a while. It's like, why does this balance so beautifully?

ES: Right? And the way you get there with the limit, so I'm just going to take the limit going one way, then I’ll take the limit going the other way and voila, out comes these beautiful partial differential equations.

EL: Yeah, very lovely. And I know I'm putting you on the spot. But do you have a pairing for this theorem?

ES: Ooh, a pairing? Oh boy, something was a very complex taste. Maybe chili.

EL: Okay.

ES: I’ll say chili because there's all sorts of flavors mixed in with chili, and complex analysis seems to mix all sorts of flavors together.

EL: All right, I like it. Well, thank you. This is the first lightning My Favorite Theorem I'm recording so far at the joing meetings, or even before, on the way, so yeah, thanks for joining me.

Courtney Gibbons: I'm Courtney Gibbons. I'm a professor at Hamilton College in upstate New York. And my favorite theorem is Hilbert’s Nullstellensatz, which translates to zero point theorem, but if you run it through Google Translate, it's actually quite beautiful. It's like the “empty star theorem” or something like that. It's very astronomical. And I love this theorem because it's one of those magical theorems that connect one area that I love, algebra, to another area that I don't really understand, but would like to love, geometry. And I find that in my classes, when I ask someone, “What's a parabola?” I have a handful of students who do some sort of interpretive dance. And I have a handful of students were like, “Oh, it's like y equals some x squared stuff.” And I'm like, “I'm with you.” I think of the equation. And some people think of the curve, the plot, and that's the geometric object, and the Nullstellensatz tells you how to take ideals and relate them to varieties. So it connects algebra and geometry. And it's just gorgeous, and the proof is gorgeous, and everything about it is wonderful, and David Hilbert was wonderful. And if I were going to pair it with something, I’d probably pair it with a trip to an observatory, so that you could go appreciate the beauty of the stars, and think about the wonderful connectedness of all of mathematics and the universe. And maybe you should have, like, a beer or something too.

EL: Why not?

CG: Yeah. Why not? Exactly.

EL: Good. Well, thank you. Absolutely.

KK: All right, JMM flash theorem time. Introduce yourself, please.

Shelley Kandola: Hi. My name is Shelly Kandola. I'm a grad student at the University of Minnesota.

KK: And it’s warmer here than where you are usually.

SK: Yeah, it's 15 degrees in Minnesota right now.

KK: That’s awful.

SK: Yeah.

KK: Well anyway, we’ve got to be quick here. What's your favorite theorem?

SK: The Banach-Tarski paradox.

KK: This is an amazing result that I still don't really understand and I can't wrap my head around.

SK: Yeah, you've got a solid sphere, a filled-in S2, and you can cut it into four pieces using rigid motions, and then put them back together and get two solid spheres that are the same size as the original.

KK: Well, theoretically, you can do this, right? This isn't something you can actually do, is it?

SK: Physically no, but with the power of group theory, yes.

KK: With the power of group theory.

SK: The free group on two generators.

KK: Why do you like this theorem so much?

SK: So I like it because it was the basis of my senior research project in college.

KK: It just seems so weird it was something you should think about?

SK: Yeah, it intrigued me. It's a paradox. And it's the first theorem I dove really deep into, and we found a way to generalize it to arbitrarily many dimensions with one tweak added.

KK: Cool. So what does one pair with the Banach-Tarski paradox?

SK: One of my favorite Futurama episodes. There's this one episode where there's a Banach-Tarski duplicator, and Bender jumps into the duplicator, and he makes two more, and he wants to build an army of himself.

KK: Sure.

SK: But every time he jumps in, the two copies that come out, are half the size of the original. He ends up with an army of nanobots. It contradicts the whole statement of the paradox that you're getting two things back that are the same size as the original.

KK: Although an army of Benders might be fun.

SK: Yeah, they certainly wreak havoc.

KK: Don’t we all have a little inner Bender?

SK: Oh yeah. He's powered by beer.

KK: Well, thanks for joining us. You gave a really good talk this morning.

SK: Thanks.

KK: Good luck.

SK: Thank you for having me.

KK: Sure.

David Plaxco: My name is David Plaxco. I'm a math education researcher at Clayton State University. And my favorite theorem is really more of an exercise, I think most people would think. It's proving that the set of all elements in a group that conjugate with a fixed element is a subgroup of the group. I'll tell you why. Because in my dissertation, that exercise was the linchpin in understanding how students can learn by proving.

EL: Okay.

DP: So I was working with a student. He had read ahead in the textbook and knew that not all groups are commutative, so you can't always commute any two elements you feel like. And he generalized this to thinking about inverses. He didn’t think that every inverse was necessarily two-sided, which in a group you are. Anyway, so he was trying to prove that that set was a subgroup and came to this impasse because he wanted to left cancel and right cancel with inverses and could only do them on one side. And then he started to question, like, maybe I'm just crazy, like maybe you can use the same inverse on both sides. And then he proved it himself using associativity. So he made, like, I call it John’s lemma, he came up with this kind of side proof to show that, well, if you're associative and you have a right inverse and a left inverse, then those have to be the same. And then he came back and was able to left and right cancel at free will any inverse, and then proved that it was a subgroup, so through his own proof activity, he was able to change his own conceptual understanding about what it means to be an inverse, like how groups work, all these things, and it gave him so much more power moving forward. So that's how that theorem became my favorite theorem because it gave me insight into how individuals can learn.

EL: Nice. And do you have a pairing for this theorem?

DP: My diploma. Because it helped me get it.

EL: That seems appropriate. Thanks.

DP: Thanks.

Terence Tsui: So I'm Terrance, and I'm currently a final year undergraduate studying in Oxford. My favorite theorem is actually a really elegant proof of Euler’s identity on the Riemann zeta function. We all know that the Riemann zeta function is defined in a way of the sum of 1/ks where k runs across all the natural numbers. But at the same time, Euler has given a really good other formulation: we say status is the same as the infinite product of 1-1/ps, where p runs across the primes. And then it's really interesting, because if you look at you see, on one hand, an infinite some, and on the other hand, you have an infinite product. And it’s very rare that we see that infinite sums and infinite products actually coincide. And they’re only there because it is a function that actually works on nearly every s larger than 1. And that means that this beautiful, elegant identity actually runs correct for infinitely many values. And the most interesting thing about this theorem is that the proof to it could be done probabilistically, where we consider some certain particular events, and we realize that the Riemann zeta series sum is actually equivalent to finding a certain intersection of infinitely many independent events. And first it is just an infinite product of certain events. And first we have the Riemann zeta function equalling a particular infinite product. And I think that is something that is really out of out of our imagination, because not only does it link two things—a sum to an infinite product, but at the same time, the way that it proves it comes from somewhere we could not even imagine, which is from probability. So if I need to pair this theorem with something, I would say it’s like a spider web, because you can see that there's very intricate connections and that things connect to each other, but in the most mysterious ways.

ELL: Cool. Well, thanks.

TT: Thank you.

Courtney Davis: So Hi, I'm Courtney Davis. I am an associate professor at Pepperdine University out in LA.

EL: Okay. And I hear that we have a favorite model, not a favorite theorem from you.

CD: Yes. So I'm a math biologist. So I'm going to say the obvious one, which is SIR modeling, because it is the entry way into getting to do this cool stuff. It’s the way that I get to show students how to write models. It's the first model I ever saw that had biology in it. And it's something that is ubiquitous and used widely. And so despite being the first thing everyone learns, it's still the first thing everyone learns. And that's what makes it interesting to me.

EL: Yeah. And and can you kind of just sum up in a couple sentences what this model is, what SIR means?

CD: Yeah. So SIR is you are modeling the spread of disease through a susceptible (S) population through infected and into recovered or immune, and you can change that up quite a lot. There are a lot of different ways to do it. It's not one fixed model. And it's all founded on the very simple premise that when two individuals run into each other in a population, that looks like multiplication. And so you can take multiplication, and with that build all the interactions that you really need, in order to capture what's actually happening in a population that at least is well mixed, so that you have a big room of people moving around about it, for instance.

EL: Okay. And I'm going to spring something on you, which is that usually we pair something with our theorem, or in this case model, so we have our guests, you know, choose a food, beverage, piece of art, or anything. Is there anything that you would suggest that pairs well with SIR?

CD: With an SIR model, I would say, a paint gun.

EL: Okay.

CD: I don't know that that's what you're looking for.

EL: That’s great.

CD: Simply because running around and doing pandemic games or other such things is also a common way to get data on college campuses so that you can introduce students, and they can parameterize their models by paint guns or water guns or something like that.

EL: Oh, cool. I like it. Thank you.

CD: Absolutely. Thank you.

Jenny Kenkel: I’m Jenny Kenkel. I'm a graduate student at the University of Utah. I study commutative algebra. My favorite theorem is this isomorphism between a particular local cohomology module and an injective module: The top local cohomology of a Gorenstein ring is isomorphic to the injective hull of its residue field. But I was thinking that maybe it would pair really well with like, a dark chocolate and a sharp cheddar, because these two things are isomorphic, and you would never expect that. But then they go really well together, just in the same way that I think a dark chocolate and a sharp cheddar seem kind of like a weird pairing, but then it's amazing. Also, they're both beautiful.

EL: Nice, thank you.

JK: Thank you.

Dan Daly: My name is Dan Daly. And I am the interim chair of the Department of Mathematics at Southeast Missouri State University.

KK: Southeast—is that in the boot?

DD: That is close to the boot heel. It's about two hours south of St. Louis.

KK: Okay. I'm a Cardinals fan. So I'm ready, we’ve got something here. So what's your favorite theorem?

DD: So my favorite theorem is actually the classification of finite simple groups.

KK: That’s a big theorem.

DD: That is a very big, big,

KK: Like 10,000 pages of theorem.

DD: At least

KK: Yeah. So what draws you to this? Is it your area?

DD: So I am interested in algebraic and combinatorics, and I am generally interested in all things related to permutations.

KK: Okay.

DD: And one of the things that drew me to this theorem is that it's such an amazing, collaborative effort and one of the landmarks of 20th century mathematics.

KK: Big deal. Yeah.

DD: And, you know, it just to me, it seems such a such an amazing result that we can classify these building blocks of finite groups.

KK: Right. So what does one pair this with?

DD: So I think since it's such a collaborative effort, I'm going to pair it with Louvre museum.

KK: The Louvre, okay.

DD: Because it's a collection of all of these different results that are paired together to create something that is really, truly one of a kind.

KK: I’ve never been. Have you?

DD: I have. It’s a wonderful place. Yeah. It’s a fabulous place. One of my favorite places.

KK: I’m going to wait until I can afford to rent it out like Beyonce and Jay Z.

DD: Yeah, right.

KK: All right, well thanks, Dan. Enjoy your time at the Joint Math Meetings.

DD: All right, thank you much.

Charlie Cunningham: My name's Charlie Cunningham. I'm visiting assistant professor at Haverford College. And my area of research originally is, or still is, geometric group theory. But the theorem that I want to talk about was a little bit closer to set theory, which is I want to talk about the existence of solutions to Cauchy’s functional equation.

EL: Okay. And what is Cauchy’s functional equation?

CC: So Cauchy’s functional equation is a really basic sort of thing you can ask about a function. It's asking, all right, you take the real numbers, and you ask is there—what are the functions from the real numbers to the real numbers where if you add two numbers together, and then apply the function, it's the same thing as applying the function to both of those numbers and then adding them together?

EL: Okay. So kind of like you're naive student and wanting to—how a function should behave.

CC: Yes. Right. So this would come up in a couple of places. So if you’ve taken linear algebra, that's the first axiom of a linear function. It doesn't ask about the scaling part. It's just the additive part. And if you've done group theory, it's a fancy way, is it's all the homomorphisms from the real numbers to themselves, an additive group. So the theorem basically, is that well, well, first of all, the question is, well, there are some obvious ones. There are all the functions where you just multiply by a fixed number, all the linear functions you’d know from linear algebra, like 2 times x, 3 times x, or π times x, any real number times x. So the question is, are there any others? Or are those the only functions that exists at all that satisfy this equation? And the theorem turns out that the answer depends on the fundamental axioms you take for mathematics.

EL: Wow. Okay.

CC: Right. So the answer is just to use a little bit of set theory, that if you are working in a set theory, which most mathematicians do, that has something called the axiom of choice in it, then the answer is no, there are lots and lots and lots of other functions that satisfy this equation, other than those obvious ones, but they're almost impossible to think about or write down. They're not continuous anywhere, they are not differentiable anywhere. They're not measurable, if anyone knows what that means. Their graph, if you tried to draw them, are dense in the entire plane, which means any little circle you draw on the plane intersects of the graph somewhere. They still pass the vertical line test. They’re still functions that are well-defined. And I really like this theorem. One reason is because it's a really great place for math students to learn that there isn't always one right answer in math. Sometimes the answer to very reasonably posed questions isn't true or false. It depends on the fundamental universe we’re working in. It depends on the what we all sit down and agree are the starting rules of our system. And it's a sort of question where you wouldn't realize that those sorts of considerations would come up. It also comes up—When I've asked linear algebra students, it's equivalent to the statement are both parts of the definition of a linear function actually necessary? We usually give them to you as two pieces: one, it satisfies this, and the other is scalars pull out. Do we actually need that second part? Can we prove that scalars pull out just from the first part? And this is the only way to prove the answer's no. It's a good exercise to try yourself to prove just from this axiom, that rational scalars pull out, any rational number has to pull out of that function. But real numbers, not necessarily. And these are the counter examples. So it's a good place at that level when you're first learning math, to realize that there are really subtle issues of what we really think truth means when we're beginning to have these conversations

EL: Nice. And what is your theorem pairing?

CC: My theorem pairing, I'm going to pair it with artichokes.

EL: Okay.

CC: I think that artichokes also had a bad rap for a lot of time, for a long time. You should also look at the artichoke war, if you've never heard of it, a great piece of history of New York City, and it took a long time for people to really understand that these prickly, weird looking vegetables can actually be delicious if approached from the right perspective.

EL: Nice. Well, thank you.

Ellie Dannenburg: So I'm Ellie Dannenberg, and I am visiting assistant professor at Pomona College in Claremont, California. And my favorite theorem is the Koebe-Andreev-Thurston circle packing theorem, which says that if you give me a triangulation of a surface, that I can find you exactly one circle packing where the vertices of your triangulation correspond to circles, and an edge between two vertices says that those circles are tangent.

EL: Okay, so this seems site kind of related to Voronoi things? Maybe I'm totally going in a wrong direction.

ED: So, I know that these are—so I don't think they're exactly related.

EL: Okay. Nevermind. Continue!

ED: Okay. But, right, it’s cool because the theorem says you can find a circle packing if I hand you a triangulation. But what is more exciting is you can only find one. So that's it.

EL: Oh, huh. Cool. All right. And do you have something that you would like to pair with this theorem?

ED: So I will pair this theorem with muhammara, which is this excellent Middle Eastern dip made from walnuts and red peppers and pomegranate molasses that is delicious and goes well with anything.

EL: Okay. Well, it's a good pairing. My husband makes a very good version. Yeah. Thank you.

ED: Thank you.

Manuel González Villa: This is Manuel González Villa. I'm a researcher in CIMAT [Centro de Investigación en Matemáticas] in Guanajuato, Mexico, and my favorite theorem is the Newton-Puiseux theorem. This is a generalization of implicit function theorem but for singular points of algebraic curves. That means you can parameterize a neighborhood of a singular point on an algebraic curve with a power series expansion, but with rational exponents, and the denominators of those exponents are bounded. The amazing thing about this theorem is that it’s very old. It comes back from Newton. But some people will still use it in research. I learned this theorem in Madrid where I made my PhD from a professor call Antonia Díaz-Cano. And also I learned with the topologist José María Montesinos to apply this theorem. It has some high-dimensional generalizations for some type of singularities, which are called quasi-ordinary.

The exponents—so you get a power series, so you get an infinite number of exponents. But there is a finite subset of those exponents which are the important ones, because they codify all the topology around the singular point of the algebraic curve. And this is why this theorem is very important. And the book I learned it from is Robert Walker’s Algebraic Curves. And if you want a more recent reference, I recommend you to look at Eduardo Casas-Alvero’s book on singularities of plane curves. Thank you very much.

EL: Okay.

EL: Yeah. So can you introduce yourself?

JoAnne Growney: My name is JoAnne Growney. I'm a retired math professor and a poet.

EL: And what is your favorite theorem?

JG: Well, the last talk I went to has had me debating about it. What I was prepared to say an hour ago was that it was the proof by contradiction that the real numbers are countable, and Cantor's diagonal proof. I like proofs by contradiction because I kind of like to think that way: on the one hand, and then the opposite. But I just returned from listening to a program on math and art. And I thought, wow, the Pythagorean theorem is something that I use every day. And maybe I'm being unfair to take something about infinity instead of something practical, but I like both of them.

EL: Okay, so we've got a tie there. And have you chosen something to pair with either of your theorems? We like to do, like, a wine and food pairing or, you know, but with theorems, you know, is there something that you think goes especially well, for example a poem, if you’ve got one.

JG: Well, actually, I was thinking of—the Pythagorean theorem, and it's probably a sound thing, made me think of a carrot.

EL: Okay.

JG: And oh, the theorem about infinity, it truly should make me think of a poem, but I don't have a pairing in mind.

EL: Okay. Well, thank you.

JG: Thank you.

Mikael Vejdemo-Johansson: I’m Michael Vejdemo-Johansson. I'm from the City University of New York.

KK: City University of New York. Which one?

MVJ: College of Staten Island and the Graduate Center.

KK: Excellent. All right, so we're sitting in an Afghan restaurant at the JMM. And what is your favorite theorem?

MVJ: My favorite theorem is the nerve lemma.

KK: Okay, so remind everyone what this is.

MVJ: So the nerve lemma says—well, it’s basically a family of theorems, but the original one as I understand it says that if you have a covering of a topological space where all the cover elements and all arbitrary intersections of cover elements are simple enough, then the intersection complex, the nerve complex of the covering that inserts a simplex for each nonlinear intersection is homotopy equivalent to the whole space.

KK: Right. This is extremely important in topology.

MVJ: It fuels most of topological data analysis one way or another.

KK: Absolutely. Very important theorem. So what pairs well, with the nerve lemma?

MVJ: I’m going to go with cotton candy.

KK: Cotton candy. Okay, why is that?

MVJ: Because the way that you end up collapsing a large and fluffy cloud of sugar into just thick, chewy fibers if you handle it right.

KK: That's right. Okay. Right. This pairing makes total sense to me. Of course, I’m a topologist, so that helps. Thanks for joining us, Mikael.

MVJ: Thank you for having me.

Michelle Manes: I’m Michelle Manes. I'm a professor at the University of Hawaii. And my favorite theorem is Sharkovskii’s theorem, which is sometimes called period three implies chaos. So the statement is very simple. You have a weird ordering of the natural numbers. So 3 is bigger than 5 is bigger than 7 is bigger than 9, etc, all the odd numbers. And then those are all bigger than 2 times 3 is bigger than 2 times 5 is bigger than 2 times 7, etc. And then down a row 4 times every odd number, and you get the idea. And then everything with an odd factor is bigger than every power of 2. And the powers of 2 are listed in decreasing order. So 23 is bigger than 22 is bigger than 2 is bigger than 1.

EL: Okay.

MM: So 1 is the smallest, 3 is the biggest, and you have this big weird array. And the statement says that if you have a continuous function on the real line, and it has a point of period n, for n somewhere in the Sharkovskii ordering, so put your finger down on n, it’s got a point of period everything less than n in that ordering. So in particular, if it has a point of period 3, it has points of every period, every integer. So I mean, I like the theorem, because the hypothesis is remarkable. The hypothesis is continuity. It's so minimal.

EL: Yeah.

MM: And you have this crazy ordering. And the conclusion is so strong. And the proof is just really lovely. It basically uses the intermediate value theorem and pretty pictures of folding the real line back on itself and things like that.

EL: Oh, cool.

MM: So yeah, it's my favorite theorem. Absolutely.

EL: Okay. And do you have something that you would suggest pairing with this theorem?

MM: So for me, because when I think of the theorem, I think of the proof of it, which involves this, like stretching and wrapping and stretching and wrapping, and an intermediate value theorem, it feels very kinetic to me. And so I feel like it pairs with one of these kind of moving sculptures that moves in the wind, where things sort of flow around.

EL: Oh, nice.

MM: Yeah, it feels like a kinetic theorem to me. So I'm going to start with the kinetic sculpture.

EL: Okay. Thank you.

MM: Thanks.

John Cobb: Hey there, I’m John Cobb, and I'm going to tell you my favorite theorem.

EL: Yeah. And where are you?

JC: I’m at College of Charleston applying for PhD programs right now.

EL: Okay.

JC: Okay. So I picked one I thought was really important, and I'm surprised it isn't on the podcast already. I have to say it's Gödel’s incompleteness theorems. Partly because for personal reasons. I'm in a logic class right now regarding the mechanics of the actual proof. But when I heard it, I was becoming aware of the power of mathematics, and hearing the power of math to talk about its own limitations, mathematics about mathematics, was something that really solidified my journey into math.

EL: And so what have you chosen to pair with your theorems?

JC: Yeah, I was unprepared for this question. So I’m making up on the spot.

EL: So you would say your your preparation was…incomplete?

JC: [laughing] I would say that! Man. I'll go with the crowd favorite pizza for no reason in particular.

EL: Well pizza is the best food and it's good with everything.

JC: Yeah.

EL: So that's a reason enough.

JC: Awesome. Well, thank you for the opportunity.

EL: Yeah, thanks.

Talia Fernós: My name is Talia Fernós, and I'm an associate professor at the University of North Carolina at Greensboro. My favorite theorem is Riemann’s rearrangement theorem. And basically, what it says is that if you have a conditionally convergent series, you can rearrange the terms in the series that the series converges to your favorite number.

EL: Oh, yeah. Okay, when you said the name of it earlier, I didn't remember, I didn't know that was the name of the theorem. But yes, that's a great theorem!

TF: Yeah. So the proof basically goes as follows. So if you do this with, for example, the series which is 1/n times -1 to, say, the n+1, so that looks like 1-1/2+1/3-1/4, and so on. So when you try to see why this is itself convergent, what you'll see is that you jump forward 1, then back a half, and then forward a third, back a fourth, so if you kind of draw this on the board, you get this spiral. And you see that it very quickly, kind of zooms in or spirals into whatever the limit is.

So now, this is conditionally convergent, because if you sum just 1/n, this diverges. And you can use the integral test to show that. So now, if you have a conditionally convergent series, you will have necessarily that it has infinitely many positive terms and infinitely many negative terms. And that each of those series independently also diverge. So when you want to show that a rearrangement is possible, so that it converges to your favorite number, what you're going to do is, let's say that you're trying to make this converge to 1, okay? So you're going to add up as many positive terms as necessary, until you overshoot 1, and then as many negative terms as necessary until you undershoot, and you continue in this way until you kind of have again, this spiraling effect into 1. And now the reason why this does converge is that the fact that it's conditionally convergent also tells you that the terms go to zero. So you can add sort of smaller and smaller things.

EL: Yeah, and you you don't run out of things to use.

TF: Right.

EL: Yeah. Cool. And what have you chosen to pair with this theorem?

F: For its spiraling behavior, escargot, which I don't eat.

EL: Yeah, I have eaten it. I don't seek it out necessarily. But it is very spiraly.

TF: Okay. What does it taste like?

EL: It tastes like butter and parsley.

TF: Okay. Whatever it’s cooked in.

EL: Basically. It's a little chewy. It's not unpleasant. I don't find a terribly unpleasant, but I don’t

TF: think it's a delicacy.

EL: Yeah. But I'm not very French. So I guess that's fair. Well, thanks.

TF: Sure.

This episode of my favorite theorem is a whirlwind of “flash favorite theorems” we recorded at the Joint Mathematics Meetings in Baltimore in January 2019. We had 16 guests, so we’ll keep this brief. Below is a list of our guests and their theorems with timestamps for each guest in case you want to skip around in the episode. We hope you enjoy this festival of theorem love as much as we enjoyed talking to all of these mathematicians!

1:58 Eric Sullivan from Carroll College in Montana loves the Cauchy-Riemann equations.
3:48 Courtney Gibbons from Hamilton College in New York loves Hilbert’s Nullstellensatz.
5:08 Shelley Kandola from the University of Minnesota loves the Banach-Tarski paradox.
7:20 David Plaxco from Clayton State University in Georgia loves a group theory exercise that helped him with his dissertation.
9:40 Terence Tsui from Oxford University in the UK loves a probabilistic proof of the equivalence of two forms of the Riemann zeta function.
12:14 Courtney Davis from Pepperdine University in California loves the SIR model.
14:25 Jenny Kenkel from the University of Utah loves the isomorphism between the top local cohomology of a Gorenstein ring and the injective hull of its residue field.
15:14 Dan Daly of Southeast Missouri State University loves the classification of finite simple groups.
16:42 Charlie Cunningham of Haverford College in Pennsylvania loves Cauchy’s functional equation.
20:38 Ellie Dannenberg of Pomona College in California loves the Koebe-Andreev-Thurston circle packing theorem.
22:15 Manuel González Villa of CIMAT (Centro de Investigación en Matemáticas) in Guanajuato, Mexico loves the Newton-Puiseux theorem.
24:08 JoAnne Growney, a retired math professor and current math poet, loves the Pythagorean theorem.
25:54 Mikael Vejdemo-Johansson of CUNY loves the nerve lemma.
27:27 Michelle Manes of the University of Hawaii loves Sharkovskii’s theorem.
29:38 John Cobb of the College of Charleston loves Gödel’s incompleteness theorems.
31:04 Talia Fernós of the University of North Carolina at Greensboro loves Riemann’s rearrangement theorem.

Episode 44 - James Propp

11 juli 2019 | 29 min

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast. I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right, yeah. Up early today for me. You know, you’re on the East Coast, and I'm in the Mountain Time Zone. And actually, when my husband is on math trips — sorry, if I'm on math trips on the East Coast, and he's in the mountain time zone, then we have, like, the same schedule, and we can talk to each other before we go to bed. I'm sort of a night owl. So yeah, it's early today. And I always complain about that the whole time.

KK: Sure. Is he a morning person?

EL: Yes, very much.

KK: So Ellen, and I are decidedly not. I mean, I'd still be in bed, really, if I had my way. But you know, now that I'm a responsible adult chair of the department, I have to—even in the summer—get in here to make sure that things are running smoothly.

EL: But yeah, other than the ungodly hour (it’s 8am, so everyone can laugh at me), everything is great.

KK: Right. Cool. All right, I’m excited for this episode.

EL: Yes. And today, we're very happy to have Jim Propp join us. Hi, Jim, can you tell us a little bit about yourself?

Jim Propp: Yeah, I'm a math professor at UMass Lowell. My research is in combinatorics, probability, and dynamical systems. And I also blog and tweet about mathematics.

KK: You do. Your blog’s great, actually.

EL: Yeah.

KK: I really enjoy it, and you know, you're smart. Once a month.

EL: Yes. That that was a wise choice. Most months, I think on the 17th, Jim has an excellent post at Math Enchantments.

KK: Right.

EL: So that’s a big treat. I think somehow I didn't realize that you did some things with dynamical systems too. I feel like I'm familiar with you in, like, the combinatorics kind of world. So I learned something new already.

KK: Yup.

JP: Yeah, I actually did my PhD work in ergodic theory. And after a few years of doing a postdoc in that field, I thought, “No, I'm going to go back to combinatorics," which was sort of my first love. And then some probability mixed into that.

KK: Right. And actually, we had some job candidates this year in combinatorics, and one of them was talking about—you have a list of problems, apparently, that's famous. I don't know.

JP: Oh, yes. Tilings. Enumeration of tilings.

KK: That’s right. It was a talk about tilings. Really interesting stuff.

JP: Yeah, actually, I should say, I have gone back to dynamical systems a little bit, combining it with combinatorics. And that's a big part of what I do these days, but I won't be talking about that at all.

EL: Okay. And what is your favorite theorem?

JP: Ah, well, I've actually been sort of leading you on a bit because I'm not going to tell you my favorite theorem, partly because I don't have a favorite theorem.

KK: Sure.

JP: And if I did, I wouldn't tell you about it on this podcast, because it would probably have a heavy visual component, like most of my favorite things in math, and it probably wouldn't be suited to the purely auditory podcast medium.

KK: Okay, so what are you gonna tell us?

JP: Well, I could tell you about one theorem that I like that doesn't have much geometric content. But I'm not going to do that either.

EL: Okay, so what bottom of the barrel…

JP: I’m going to tell you about two theorems that I like, okay, they’re sort of like twins. One is in continuous mathematics, and one is in discrete mathematics.

KK: Great.

JP: The first one, the one in continuous mathematics, is pretty obscure. And the second one, in discrete mathematics, is incredibly obscure. Like nobody’s named it. And I've only found it actually referred to, stated as a result in the literature once. But I feel it's kind of underneath the surface, making a lot of things work, and also showing resemblances between discrete and continuous mathematics. So these are, like, my two favorite underappreciated theorems.

EL: Okay.

KK: Oh, excellent. Okay, great. So what have we got?

JP: Okay, so for both of these theorems, the underlying principle, and this is going to sound kind of stupid, is if something doesn't change, it’s constant.

EL: Okay. Yes, that is a good principle.

JP: Yeah. Well, it sounds like a tautology, because, you know, doesn't “not changing” and “being constant” mean the same thing? Or it sounds like a garbled version of “change is the only constant.” But no, this is actually a mathematical idea. So in the continuous realm, when I say “something,” what I mean is some differentiable function. And when I say “doesn't change,” I mean, has derivative zero.

KK: Sure.

JP: Derivatives are the way you measure change for differentiable functions. So if you’ve got a differentiable function whose derivative is zero—let’s assume it's a function on the real line, so its derivative is zero everywhere—then it's just a constant function.

KK: Yes. And this is a corollary of the mean value theorem, correct?

JP: Yes! I should mention that the converse is very different. The converse is almost a triviality. The converse says if you've got a constant function, then its derivative is zero.

KK: Sure.

JP: And that just follows immediately from the definition of the derivative. But the constant value theorem, as you say, is a consequence of the mean value theorem, which is not a triviality to prove.

KK: No.

JP: In fact, we'll come back later to the chain of implications that lead you to the constant value theorem, because it's surprisingly long in most developments.

KK: Yes.

JP: But anyway, I want to point out that it's kind of everywhere, this result, at least in log tables— I mean, not log tables, but anti-differentiation tables. If you look up anti-derivatives, you'll always see this “+C” in the anti-derivative in any responsible, mathematically rigorous table of integrals.

EL: Right.

JP: Because for anti-derivatives, there's always this ambiguity of a constant. And those are the only anti-derivatives of a function that's defined on the whole real line. You know, you just add a constant to it, no other way of modifying the function will leave its derivative alone. And more generally, when you've got a theorem that says what all the solutions to some differential equation are, the theorem that guarantees there aren't any other solutions you aren't expecting is usually proved by appealing to the constant value theorem at some level. You show that something has derivative zero, you say, “Oh, it must be constant. “

KK: Right.

JP: Okay. So before I talk about how the constant value theorem gets proved, I want to talk about how it gets used, especially in Newtonian physics, because that's sort of where calculus comes from. So Newtonian physics says that if you know the initial state of a system, you know, of a bunch of objects—you know their positions, you know their velocities—and you know the forces that act on those objects as the system evolves, then you can predict where the objects will be later on, by solving a differential equation. And if you know the initial state and the differential equation, then you can predict exactly what's going to happen, the future of the system is uniquely determined.

KK: Right.

JP: Okay. So for instance, take a simple case: you’ve got an object moving at a constant velocity. And let's say there are no forces acting on it all. Okay? Since there are no forces, the acceleration is zero. The acceleration is the rate of change of the velocity, so the velocity has derivative zero everywhere. So that means the velocity will be constant. And the object will just keep on moving at the same speed. If the constant value theorem were false, you wouldn't really be able to make that assertion that, you know, the object continues traveling at constant velocity just because there are no forces acting on it.

KK: Sure.

JP: So, kind of, pillars of Newtonian physics are that when you know the derivative, then you really know the function up to an ambiguity that can be resolved by appealing to initial conditions.

EL: Yeah.

KK: Sure.

JP: Okay. So this is actually telling us something deep about the real numbers, which Newton didn't realize, but which came out, like, in the 19th century, when people began to try to make rigorous sense of Newton's ideas. And there's actually a kind of deformed version of Newton's physics that's crazy, in the sense that you can't really predict things from their derivatives and from their initial conditions, which no responsible physicist has ever proposed, because it's so unrealistic. But there are some kind of crazy mathematicians who don't like irrational numbers. I won't name names. But they think we should purge mathematics of the real number system and all of these horrible numbers that are in it. And we should just do things with rational numbers. And if these people tried to do physics just using rational numbers, they would run into trouble.

EL: Right.

JP: Because you can have a function from the rational numbers to itself, whose derivative is zero everywhere—with derivative being defined, you know, in the natural way for functions from the rationals to itself—that isn't a constant function.

KK: Okay.

JP: So I don't know if you guys have heard this story before.

KK: This is making my head hurt a little, but okay. Yeah.

EL: Yeah, I feel like I have heard this, but I cannot recall any details. So please tell me.

JP: Okay, so we know that the square root of two is irrational, so every rational number, if you square it, is either going to be less than two, or greater than two.

KK: Yes.

JP: So we could define a function from the rational numbers to itself that takes the value zero if the input value x satisfies the inequality x squared is less than 2 and takes the value 1 if x squared is bigger than two.

EL: Yes.

JP: Okay. So this is not a constant function.

KK: No.

JP: Right. Okay. But it actually is not only continuous, but differentiable as a function of the

EL: Of the rationals…

JP: From the rationals to itself.

KK: Right. The derivative zero but it's not constant. Okay.

JP: Yeah. Because take any rational number, okay, it's going to have a little neighborhood around it avoiding the square—avoiding the hole in the rational number line where the square root of 2 would be. And it's going to be constant on that little interval. So the derivative of that function is going to be zero.

KK: Sure.

JP: At every rational number. So there you have a non-constant function whose derivative is zero everywhere. Okay. And that's not good.

KK: No.

JP: It’s not good. for math. It's terrible for physics. So you really need the completeness property of the reals in a key way to know that the constant value theorem is true. Because it just fails for things like the set of rational numbers.

EL: Right.

KK: Okay.

JP: This is part of the story that Newton didn't know, but people like Cauchy figured it out, you know, long after.

KK: Right.

JP: Okay. So let's go back to the question of how you prove the constant value theorem.

EL: Yeah.

JP: Actually, I wanted to jump back, though, because I feel like I wanted to sell a bit more strongly this idea that the constant value theorem is important. Because if you couldn't predict the motions of particles from forces acting on those particles, no one would be interested in Newton's ideas, because the whole story there is that it is predictive of what things will do. It gives us a sort of clockwork universe.

KK: Sure.

JP: So Newton's laws of motion are kind of like the rails that the Newtonian universe runs on, and the constant value theorem is what keeps the universe from jumping off those rails.

KK: Okay. I like that analogy. That’s good.

JP: That’s the note I want to end on for that part of the discussion. But now getting back to the math of it. So how do you prove the constant value theorem? Well, you told me you prove it from the mean value theorem. Do you remember how you prove the mean value theorem?

KK: You use Rolle’s theorem?

EL: Just the mean value theorem turned sideways!

KK: Sort of, yeah. And then I always joke that it’s the Forrest Gump proof. Right? You draw the mean value theorem, you draw the picture on the board, and then you tilt your head, then you see that it's Rolle’s theorem. Okay, but Rolle’s theorem requires, I guess what we sometimes call in calculus books Fermat’s theorem, that if you have a differentiable function, and you're at a local max or min, the derivative is equal to zero. Right?

JP: Yup. Okay, actually, the fact that there exists even such a point at all is something.

KK: Right.

JP: So I think that's called the Extreme Value Theorem.

KK: Maybe? Well, the Extreme Value Theorem I always think of as as—well, I'm a topologist—that’s the image of a compact set is is compact.

JP: Okay.

KK: Right. Okay. So they need to know what the compact sets of the real line are.

JP: You need to know about boundedness, stuff like that, closedness.

KK: Closed and bounded, right. Okay. You're right. This is an increasingly long chain of things that we never teach in Calculus I, really.

JP: Yeah. I've tried to do this in some honors classes with, you know, varying levels of success.

KK: Sure.

JP: There’s the boundedness theorem, which says that, you know, a continuous function is bounded on a closed interval. But then how do you prove that? Well, you know, Bolzano-Weierstrass would be a natural choice if you're teaching a graduate class, maybe you prove that from the monotone convergence theorem. But ultimately, everything goes back to the least upper bound property, or something like it.

KK: Which is an axiom.

JP: Which is an axiom, that’s right. But it sort of makes sense that you'd have to ultimately appeal to some heavy-duty axiom, because like I said, for the rational numbers, the constant value theorem fails. So at some point, you really need to appeal to the the completeness of the reals.

EL: Yeah, the structure of the real numbers.

KK: This is fascinating. I've never really thought about it in this much detail. This is great.

JP: Okay. Well, I'm going to blow your mind…

KK: Good!

JP: …because this is the really cool part. Okay. The constant value theorem isn't just a consequence of the least upper bound property. It actually implies the least upper bound property.

KK: Wow. Okay.

JP: So all these facts that this, this chain of implications, actually closes up to become a loop.

KK: Okay.

JP: Each of them implies all the others.

KK: Wow. Okay.

JP: So the precise statement is that if you have an ordered field, so that’s a number system that satisfies the field axioms: you've got the four basic operations of pre-college math, as well as inequality, satisfying the usual axioms there. And it has the Archimedean property, which we don't teach at the pre-college level. But informally, it just says that nothing is infinitely bigger or infinitely smaller than anything else in our number system. Take any positive thing, add it to itself enough times, it becomes as big as you like.

KK: Okay.

JP: You know, enough mice added together can outweigh an elephant.

KK: Sure.

EL: Yeah.

JP: That kind of thing. So if you've got an ordered field that satisfies the Archimedean property, then each of those eight propositions is equivalent to all the others.

KK: Okay.

JP: So I really like that because, you know, we tend to think of math as being kind of linear in the sense that you have axioms, and from those you prove theorems, and from those you prove more theorems—it's a kind of a unidirectional flow of the sap of implication. But this is sort of more organic, there's sort of a two-way traffic between the axioms and the theorems. And sometimes the theorems contain the axioms hidden inside them. So I kind of like that.

KK: Excellent.

JP: Yeah.

KK: So math’s a circle, it's not a line.

JP: That’s right. Anyway, I did say I was going to talk about two theorems. So that was the continuous constant value theorem. So I want to tell you about something that I call the discrete constant value theorem that someone else may have given another name to, but I've never seen it. Which also says that if something doesn't change, its constant. But now we're talking about sequences and the something is just going to be some sequence. And when I say doesn't change, it means each term is equal to the next, or the difference between them is zero.

EL: Okay.

JP: So how would you prove that?

EL: Yeah, it really feels like something you don't need to prove.

KK: Yeah.

JP: If you pretend for the moment that it's not obvious, then how would you convince yourself?

KK: So you're trying to show that the sequence is eventually constant?

JP: It’s constant from the get-go, every term is equal to the next.

EL: Yeah. So the definition of your sequence is—or part of the definition of your sequence is—a sub n equals a sub n+1.

JP: That’s right.

EL: Or minus one, right?

JP: Right.

EL: So I guess you'd have to use induction.

KK: Right.

JP: Yeah, you’d use mathematical induction.

KK: Right.

JP: Okay. So you can prove this principle, or theorem, using mathematical induction. But the reverse is also true.

KK: Sure.

JP: You can actually prove the principle of mathematical induction from the discrete constant value theorem.

EL: And maybe we should actually say what the principle of mathematical induction is.

KK: Sure.

JP: Sure.

EL: Yeah. So that would be, you know, if you want to prove that something is true for, you know, the entire set of whole numbers, you prove it for the first one—for 1—and then prove that if it's true for n, then it's true for n+1. So I always have this image in my mind of, like, someone hauling in a chain, or like a big rope on a boat or something. And they're like, you know, each each pull of the of their arm is the is the next number. And you just pull it in, and the whole thing gets into boat. Apparently, that's where you want to be. Yeah, so that's induction.

JP: Yeah. So you can use mathematical induction to prove the discrete constant value theorem, but you can also do the reverse.

EL: Okay.

JP: So just as the continuous constant value theorem could be used as an axiom of completeness for the real number system, the discrete constant value theorem could be used as an axiom for, I don't want to say completeness, but the heavy-duty axiom for doing arithmetic over the counting numbers, to replace the axiom of induction.

EL: Yeah, it has me putting in my mind, like, oh, how could I rephrase, you know, my standard induction proof—that at this point, kind of just runs itself once I decided to try to prove something by induction—like how to make that into a sequence, a statement about sequences?

JP: Yeah, for some applications, it's not so natural. But one of the applications we teach students mathematical induction for is proving formulas, right? Like, the sum of the first n positive integers is n times n+1 over 2.

KK: Right.

JP: And so we do a base case. And then we do an induction step. And that's the format we usually use.

KK: Right.

JP: Okay. Well, proving formulas like that has been more or less automated these days. Not completely, but a lot of it has been. And the way computers actually prove things like that is using something more like the discrete constant value theorem.

EL: Okay.

JP: So for example, say you've got a sequence who's nth term is defined as the sum of the first n positive integers.

KK: Okay.

JP: So it’s 1, 1+2, 1+2+3,…. Then you have another sequence whose nth term is defined by the formula, n times n+1 over 2.

KK: Right.

JP: And you ask a computer to prove that those two sequences are equal to each other term by term. The way these automated systems will work, is they will show that the two sequences differ by a constant,

EL: and then show that the constant is zero.

JP: And then they’ll show that the constant is zero. So you show that the two sequences at each step increase by the same amount. So whatever the initial offset was, it’s going to be the same. And then you see what that offset is.

EL: Yeah.

KK: Okay, sure.

JP: So this is this looking a lot more like what we do in differential equations classes, where, you know, if you try and solve a differential equation, you determine a solution up to some unknown real parameters, and then you solve for them from initial conditions. There's a real strong analogy between solving difference equations in discrete math, and solving differential equations in continuous math. But somehow, the way we teach the subjects hides that.

EL: Yeah.

JP: The way we teach mathematical induction, by sort of having the base case come first, and then the induction step come later, is the reverse order from what we do with differential equations. But there's a way to, you know, change the way we present things so they're both mathematically rigorous, but they're much more similar to each other.

KK: Yeah, we've got this bad habit of compartmentalizing in math, right? I mean, the lower levels of the curriculum, you know, it's like, “Okay, well, in this course, you do derivatives and optimization. And in this course, you learn how to plow through integration techniques. And this course is multi-variable stuff. And in this course, we're going to talk about differential equations.” Only later do you do the more interesting things like induction and things like that. So are you arguing that we should just, you know, scrap it all in and start with induction on day one?

JP: Start with induction? No.

KK: Sure, why not?

JP: I’ve given talks about why we should not teach mathematical induction.

KK: Really?

JP: Yeah. Well, I mean, it’s not entirely serious. But I argue that we should basically teach the difference calculus, as a sort of counterpart to the differential calculus, and give students the chance to see that these ideas of characteristic polynomials and so forth, that work in differential equations, also work with difference equations. And then like, maybe near the end, we can blow their mind with that wonderful result that Robert Ghrist talked about.

KK: Yeah.

JP: Where you say that one of these operators, the difference operator, is e to the power of the derivative operator.

KK: Right.

EL: Yeah.

JP: They’re not just parallel theories. They're linked in a profound way.

EL: Yeah, I was just thinking this episode reminded me a lot of our conversation with him, just linking those two things that, yeah they they are in very different places in in my mental map of how how I think about math.

KK: All right. So what does one pair with these theorems?

JP: Okay, I'm going pair the potato chip.

EL: Okay, great. I love potato chips.

KK: I do too.

JP: So I think potato chips sort of bridge the gap between continuous mathematics and discrete mathematics.

EL: Okay.

JP: So the potato chip as an icon of continuous mathematics comes by way of Stokes’ theorem.

KK: Sure.

JP: So if you’ve ever seen these books like Purcell’s Electromagnetism that sort of illustrate what Stokes’ theorem is telling you, you have a closed loop and a membrane spanning in it,

EL: Right.

JP: …a little like a potato chip.

KK: Sure. Right.

JP: And the potato chip as an icon of discrete mathematics comes from the way it resembles mathematical induction.

KK: You can't eat just one.

JP: That’s right. You eat a potato chip, and then you eat another, and then another, and you keep saying, “This is the last one,” but there is no last potato chip.

EL: Yeah.

JP: And if there’s no last potato chip, you just keep eating them.

KK: That’s right. You need another bag. That's it.

EL: Yeah.

JP: But the other reason I really like the potato chip as sort of a unifying theme of mathematics, is that potato chips are crisp, in the way that mathematics as a whole is crisp. You know, people complain sometimes that math is dry. But that's not really what they're complaining about. Because people love potato chips, which are also dry. What they really mean is that it’s flavorless, that the way it's being taught to them lacks flavor.

KK: That’s valid, actually. Yeah.

JP: So I think what we need to do is, you know, when the math is too flavorless, sometimes we have to dip it into something.

EL: Yeah, get your onion dip.

JP: Yeah, the onion dip of applications, the salsa of biography, you know, but math itself should not be moist, you know?

EL: So, do you prefer like the plain—like, salted, obviously—potato chips, or do you like the flavors?

JP: Yeah, I don't like the flavors so much.

EL: Oh.

JP: I don’t like barbecue or anything like that. I just like salt.

KK: I like the salt and vinegar. That’s…

EL: Yeah, that's a good one. But Kettle Chips makes this salt and pepper flavor.

KK: Oh, yeah. I’ve had those. Those are good.

EL: It’s great. Their Honey Dijon is one of my favorites too. And I love barbecue. I love every—I love a lot of flavors of chips. I shouldn't say “every.”

KK: Well yeah, because Lay's always has this deal every year with the competition, like, with these crazy flavors. So, they had the chicken and waffles one year.

EL: Yeah, I think there was a cappuccino one time. I didn’t try that one.

KK: Yeah, no, that’s no good.

JP: I just realized, though, potato chips have even more mathematical content than I was thinking. Because there's the whole idea of negative curvature of surfaces.

EL: Yes, the Pringles is the ur-example of negatively curved surface.

JP: Yeah. And also, there's this wacky idea of varifolds, limits of manifolds, where you have these corrugated surfaces and you make the corregations get smaller and smaller, like, I think it’s Ruffes.

EL: Yeah, right.

JP: So a varifold is, like, the limit of a Ruffles potato chip as the Ruffles shrink, and the angles don't decrease to zero. There’s probably a whole curriculum.

EL: Yeah, we need a spinoff podcast. Make this—tell what this potato chip says about math.

KK: Right.

EL: Just give everyone a potato chip and go for it.

KK: Excellent.

EL: Very nice. I like this pairing a lot.

KK: Yeah.

EL: Even though it's now, like, 8:30-something, I'll probably go eat some potato chips before I have breakfast, or as breakfast.

JP: I want to thank you, Evelyn, because I know it wasn't your choice to do it this early in the morning. I had childcare duties, so thank you for your flexibility.

EL: I dug deep. Well, it was a sunny day today. So actually the light coming in helped wake me up. It's been really rainy this whole month, and that's not great for me getting out of bed before, you know, 10 or 11 in the morning.

KK: Sure. So we also like to give our guests a chance to plug things. You have some stuff coming up, right?

JP: I do. Well, there's always my Mathematical Enchantments essays. And I think my July essay, which will come out on the 17th, as always, will be about the constant value theorems. And I'll include links to stuff I have written on the subject. So anyone who wants to know more, should definitely go first to my blog. And then in early August, I'll be giving some talks in New York City. And they'll be about a theorem with some visual content called the Wall of Fire theorem, which I love and which was actually inspired by an exhibit at the museum. So it's going to be great actually to give a talk right next to the exhibit that inspired it.

EL: Oh, yeah, very nice.

KK: This is at the Museum of Math, the National Museum of Math, right? Okay.

JP: Yeah, I’ll actually give a bunch of talks. So the first talk is going to be, like, a short one, 20 minutes. That's part of a conference called MOVES, which stands for the Mathematics of Various Entertaining Subjects.

KK: Yeah.

JP: It’s held every two years at the museum, and I don't know if my talk will be on the fourth, fifth or sixth of August, but it'll be somewhere in that range. And then the second talk will be a bit longer, quite a bit longer. And it's for the general public. And I'll give it twice on August 7th, first at 4pm, and then at 7pm. And it'll feature a hands-on component for audience members. So it should be fun. And that's part of the museum's Math Encounters series, which is held every month. And for people who aren't able to come to the talk, there'll be a video on the Math Encounters website at some point.

EL: Oh, good. I've been meaning to check that because I'm on their email list, and so I get that, but obviously living in Salt Lake City, I don't end up in New York a whole lot. So yeah, I'm always like, “Oh, that would have been a nice one to go to.”

KK: Yeah.

EL: But I'll have to look for the videos.

KK: So, Jim, thanks for joining us.

JP: Thank you for having me.

KK: Thanks for making me confront that things go backwards in mathematics sometimes.

EL: Yes.

KK: Thanks again.

EL: Yeah, lots of fun.

JP: Thank you very much. Have a great day.

[outro]

In this episode of My Favorite Theorem, we were happy to talk with Jim Propp, a mathematician at the University of Massachusetts Lowell. He told us about the constant value theorem and the way it unites continuous and discrete mathematics.

Here are some links you might find interesting after listening to the episode:

Propp’s companion essay to this episode

Propp’s mathematical homepage

Propp’s blog Math Enchantments (home page, wordpress site)
His list of problems about enumeration of tilings that we mentioned
Our previous My Favorite Theorem episode with guest Robert Ghrist, who also talked about a link between continuous and discrete math
Propp’s article “Real Analysis in Reverse”
Mean Value Theorem
Rolle’s Theorem
Fermat’s Theorem
Varifold
MOVES (Mathematics of Various Entertaining Subjects), a conference he will be speaking at in August
Math Encounters, a series at the Museum of Mathematics (he will be speaking there in August)

Episode 43 - Matilde Lalin

13 juni 2019 | 29 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about theorems and math and all kinds of things. I'm one of your host,s Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City. How are you today?

KK: I have a sunburn.

EL: Yeah. Can’t sympathize.

KK: No, no, I was, you know, Ellen and I went out birdwatching on Saturday, and it didn't seem like it was sunny at all, and I didn't wear a hat. So I got my head a little sunburned. And then yesterday, she was doing a print festival down in St. Pete. And even though I thought we were in the shade—look my, arms. They're like totally red. I don’t know. This is what happens.

EL: You know, March in Florida, you really can’t get away without SPF.

KK: No, you really can't. You would think I would have learned this lesson after 10 years of living here, but it just doesn't work. So anyway. Yeah. How are you?

EL: Oh, I'm all right. Yeah. Not sunburned.

KK: Okay. Good for you. Yeah. I'm on spring break. So, you know, I'm feeling pretty good. I got some time to breathe at least. So anyway, enough about us. This is actually a podcast where we invite guests on instead of boring the world with our chit chat. Today, we're pleased to welcome Matilde Lalín, you want to introduce yourself?

Matilde Lalín: Hi. Okay. Thank you for having me here. So I'm originally from Argentina. I grew up in Buenos Aires, and I did my undergraduate there. And then I moved to the US to do my Ph.D., mostly at the University of Texas at Austin. And then I moved to Canada for postdocs, and I stayed in Canada. So right now, I'm a professor at the University of Montreal, and I work in number theory.

EL: And I'm guessing you do not have a sunburn, being in Montreal in March.

ML: So maybe I should say we are celebrating that we are very close to zero Celsius.

KK: Oh, okay.

EL: Yeah, so exciting times.

ML: Yeah. So some of the snow actually is melting.

KK: Oh, okay. I haven’t seen snow in quite a while. I kind of miss it sometimes. But anyway.

EL: Oh, It is very pretty.

KK: Yeah, it is. It’s lovely. Until you have to shovel it every week for six months. But yeah, so Matilde, what is your favorite theorem?

ML: Okay, so I wanted to talk about a problem more than theorem. Well, it will lead to some theorems eventually, and a conjecture. So my favorite problem, let's say, is the congruent number problem.

KK: Okay.

ML: So okay, so basically, a positive integer number is called congruent if it is the area of a right triangle with rational sides.

EL: All three sides, right?

ML: Exactly, exactly. So the question will be, you know, how can you tell that a particular number is congruent? But more generally, can you give a list of all congruent numbers? So for example, six is congruent, because it is the area of the right triangle with sides three, four, and five. So that's easy, but then seven is congruent because it’s the area of the triangle with sides 24/5, 35/12, and 337/60.

KK: Ah, okay.

EL: So that’s not quite as obvious.

ML: Not quite as obvious, exactly. And in fact, there is an example, due to Zagier: 157 is congruent, and so the size of the triangle, they are fractions that have—okay, so the hypotenuse has 46 and 47 digits, the numerator and denominator. And so it can be very big. Okay, let me clarify for a congruent number, there are actually infinitely many triangles that satisfy this. But the example I'm giving you is the smallest, in a sense.

EL: Okay.

ML: So actually it can be very complicated, a priori, to decide whether a number is congruent or not.

KK: Sure.

ML: So this problem appears for the first time in an Arab manuscript in the 10th century, and then it was—

EL: Oh, wow, that's shocking!

ML: Yes. Well, because triangles—I mean, it's a very natural question. But then it was picked up by Fibonacci, who was actually looking at this question from a different point of view. So he was studying arithmetic sequences. So he posed the question of whether you can have a three-term arithmetic sequence whose terms are all squares. So basically, let me give you an example. So 1-25-49. Okay? So those are three squares, and 25−1 is 24. And 49−25 is 24. So that makes it an arithmetic sequence. And each of the three members are squares.

EL: Yeah.

ML: And he said that the difference—so in this case it would be 24, okay? 25−1 is 24, 49−25 is 24—so the difference is called a congruum, if you can build a sequence with this difference, basically. So it turns out that this this problem is essentially equivalent to the congruent number problem, so that's where the name, the word congruent, comes from. Fibonacci was calling this congruum. So congruent has to do with things that sort of congregate.

EL: Okay.

ML: And so kind of this difference of the arithmetic sequence. And you can prove that from such a sequence you can build your triangle. So in the example I gave you, this is a sequence that shows that six is congruent. Well, technically it shows that 24 is congruent, but 24 is a square times six. And so if you have a triangle, you can always multiply the size by the constant, and that would be equivalent to multiplying the area by some square.

KK: Sure, yeah,

EL: Right. Right, and so if it has a square in it, then there's a rational relationship that will still be preserved.

ML: Exactly. So Fibonacci actually managed to prove that seven is congruent. And then he posed as a question, as a conjecture, that one wasn't congruent. So when you say that one is not congruent, you are also saying that the squares are not congruent. The square of any rational number.

EL: Oh.

KK: Okay.

ML: It’s actually kind of a nicer statement, in a sense. It's like a very special case. And then, like 400 years after, Fermat came, and so he actually managed to solve Fibonacci’s question. So he actually proved, using his famous descent, he proved that one is not congruent. And also that two and three are not congruent. So basically, he settled the question for those. And five is known to be congruent, also six and seven. So well, that takes care of the first few numbers. Because four is one in this case.

KK: Four is one, that’s right.

ML: Yeah, exactly. And well, one thing that happens with this problem is that actually, if you go in the direction that Fibonacci was looking, okay, so this sequence of three squares, actually, if you can think of them as—say you call the middle square x, and then one is x−n and the other is x+n. So when you multiply these three together, it gives you a square. And what this is telling you, is that actually giving you a solution to an equation that you could write as, say, y2=x(x−n)(x+n). And that's what is called an elliptic curve.

EL: Oh, okay.

ML: Yes. So basically, an elliptic curve in this context is more general. You could think of it as y2 equals a cubic polynomial in x. And so basically, the congruent numbers problem is asking whether, for such an equation, you have a solution such that y is different from 0. So so then you can study the problem from that point of view. There is a lot, there is a big theory about elliptic curves.

EL: Right. And so I've been wondering, like, is this where people got the idea to bring elliptic curves into number theory? That's always seemed mysterious to me—like, when you first learn about Fermat’s Last Theorem, and you learn there's all this elliptic curve stuff involved in proving that, like, how do people think to bring elliptic curves in this way?

ML: As a matter of fact, okay, so elliptic curves in general, it’s actually a very natural object to study. So I don't know if it came exactly via the congruent number problem, because essentially—okay, so essentially, a natural problem more generally is Diophantine equations. So basically, I give you a polynomial with integer coefficients, and I am asking you about solutions that are either integers or rational. And we understand very well what happens when the degree is one, say an equation of a form ax+by=c. Okay? So those we understand completely. We actually understand very well what happens when the degree is two and actually, degree three is elliptic curves. So it's a very natural progression.

EL: Okay.

ML: So it doesn't necessarily have to come with congruent numbers. However, it is true that many people chose choose to introduce elliptic curves via numbers, because it's such a natural question, such a natural problem. But of course, it leads you to a very specific family of elliptic curves. I mean, not just the whole story. So what is known about elliptic curves that can help understand this question of the congruent number problem: So in 1922 Mordell actually proved that the solutions of an elliptic curve—Actually, I should have said this before. So the solutions of an elliptic curve, say, over the rationals, so if you look at all the rational numbers that are solutions to an equation like that, y2 equals some cubic polynomial in x, they form a group. And actually an abelian group.

And as I was saying, Mordell proved that this group actually is finally-generated. So you can actually give a finite list of elements in the group, and then every element in the group is a combination of those. Okay? So basically, it's very tempting to say, “Well, I mean, if you give me an elliptic curve, I want to find what the group is. So I just give the generators. So this should be very easy, okay? Yeah. [laughter] But actually, it's not easy. So there's no systematic way to find all the generators to determine what the group is. And even—so, you will always have, you may have, points of finite order. So, elements that if you, take some multiple, you get back to 0. So those are easy to find. But the question of whether they are elements of infinite order, and if there are, how many there are, or how many generators you need, all these questions are difficult in generals for an elliptic curve. And so, my favorite theorem actually—so the way I ended up coming up with the idea of talking about the congruent number problem, is actually Mordell’s theorem. So I really like Mordell’s theorem.

KK: And that theorem’s not at all obvious. I mean, so you sort of, I'm not sure if I’ve ever even seen a proof. I mean, I remember, this is one of the first things I learned in algebraic geometry, you draw the picture, you know, of the elliptic curve. And the group law, of course, is given by: take two points, draw the line, and where it intersects the curve in the third point is the sum of those things, right—actually, then you reflect, it’s minus that, right? Yeah. Those three points add to zero. That's right.

EL: We’ll put a picture of this up. Because Kevin's helpful air drawing is not obvious to our listeners.

ML: That’s right.

KK: Yeah. And from that, somehow, the idea that this is a finitely-generated group is really pretty remarkable. But the picture gives you no clue of where the where to find these generators, right?

ML: Well, their first issue, actually, is to prove that this is an associative law. So that statement is annoyingly complicated to prove in elementary ways.

KK: Yeah, commutativity is kind of obvious, right?

ML: But, yes, already to prove that it's a group in the sense that associativity, yeah. And then Mordell’s theorem, actually, it follows, it does some descent. So it follows in the spirit of Fermat’s descent, actually. But I mean, in a more complicated context. But it's very beautiful, yeah.

So as I was saying, the number of generators that have infinite order, that's called the rank, and already knowing whether the rank is zero, or what the value is, that's a very difficult question. And so in 1965, Birch and Swinnerton-Dyer came up with a conjecture that relates the rank to the order of vanishing of a certain function that you build from the elliptic curve. It’s called the L-function. So, in principle, with this conjecture, one can predict the value of the rank. That doesn't mean that we can find easily the generators, but at least we can answer, for example, whether there are infinitely many solutions or not and say that.

EL: Yeah.

ML: So basically, that's kind of the most exciting conjecture associated to this question. And I mean, it goes well beyond this question, and it's one of the Millennium Problems from the Clay.

EL: Right. Yeah. So it’s a high dollar-value question.

ML Yes. And it's interesting, because for this question, it is known that—if the L-function doesn’t vanish, then the rank is zero. So it's known for R[ank] zero, one direction, and the same for R[ank] 1. But not much more is known on average. So this very recent result, relatively recent result by Bhargava and Shankar, where they prove that the rank for, if you take all the elliptic, curves and order them in a certain way, the rank on average is bounded by 7/6. And so that means that there is a positive proportion of elliptic curves that actually satisfy BSD. Okay. But I mean, the question would be what BSD tells us about the original question that I posed.

EL: Right, yeah, so when we were chatting earlier, you said that a lot of questions or theorems about congruent numbers were basically—the theorems were proved as partial solutions to BSD. Am I getting that right?

ML: Yeah, okay. Some progress that is being done nowadays has to do with proving BSD for some particular families, I mean for these elliptic curves are attached to congruent numbers. But if I go back to the first connection, so there is this famous theorem by Tunnell that was published in 1983 where he basically ties the property of being a congruent number to two quadratic equations in three variables having one having double the solutions as the other somehow. So Tunnell’s result came, obviously, in ’83, so much earlier than most advances in BSD.

EL: Okay.

ML: And basically what Tunnell gives is like an algorithm to decide whether a number is congruent or not. And for the case where it’s non-congruent, actually it is conclusive, because this is a case, okay, so it depends on BSD, but this is a case where we know. And then the problem is the case where it will tell you that the number is congruent. So that is assuming BSD. So for now, like I said, many cases will just be the cases that—for example, there is some very recent result by Tian, where basically he proved that BSD applies to certain curves. And so for example, it is known that for primes congruent to five, six, or seven modulo eight, they are congruent. So this is a result that goes back to Heegner and Monsky in ’52, for Heegner. So that's for primes. So that's an infinite family of numbers that satisfy that they are congruent. But every question attached to this problem has to do with, okay, can you generalize this for all natural numbers that are congruent to six, five, or seven modulo eight. For example, that’s some direction of research going on now.

EL: So you could disprove the BSD conjecture, if you could find some number that Turner’s [ed. note: Evelyn misremembered the name of the theorem; this should be “Tunnell’s”] theorem said was congruent, but was actually not congruent?

ML: Yeah, yeah. So you could disprove—say you find a number that is congruent to six mod eight that is not a congruent number, you disprove BSD, yes.

EL: All right. Yeah, so our listeners, I'm sure they'll go—I’m sure no one has ever searched a lot of numbers to check on this. So yeah, that's our assignment for you. So something we like to do on this show then is to ask our guest to pair their theorem with something. So what do you think enhances enjoyment of congruent numbers, the congruent number problem, the Birch and Swinnerton-Dyer Conjecture, all of these things?

ML: Well, for me it is really how I pair my mathematics with things, right? And so I would pair it with chocolate because I am a machine of transforming chocolate into theorems instead of coffee. I will also pair it with mate, which is an infusion from South America. That's my source of caffeine instead of coffee. So it's a very interesting drink that we drink a lot in Argentina, but especially in Uruguay.

KK: Do you have the special straw with the filter and everything?

ML: Yeah, yeah, I have the metal straw. So you you put the metal straw and then you put just the leaves in in your special cup, and you drink from the straw that filters the leaves. So yeah, that's right. And you share it with friends. So it's a very collaborative thing, like mathematics.

EL: So I’ve never tried this. Does this—what does it taste like it? I mean, I know it's hard to describe tastes that you've never actually tasted before. But does it taste kind of like tea, kind of like coffee, kind of like something else entirely?

ML: I would say it tastes like tea. You could think was a special tea.

KK: Okay.

EL: There’s a coffee shop near us that has that. But I haven't tried it yet.

KK: Oh, come on. Give it a shot, Evelyn.

EL: Yeah, I will.

KK: You have to report back in a future episode. Actually, I going to hold you to it the next time we meet. Before then, have some mate. Do you have a chocolate preference? Are you a dark chocolate, milk chocolate?

ML: Milk chocolate, I would say. I'm not super gourmet with chocolate. But I do have my favorite place in Montreal to go drink a good cup of hot chocolate.

KK: All right, I've learned a lot.

EL: Yeah.

KK: This is very informative. In fact, while you were describing the congruent number problem, I was sort of sitting here sketching out equations that I might try to actually solve. Of course, it wasn't an elliptic curves, it was sort of the naive things that you might try. But this is a fascinating problem. And I could see how you could get hooked.

EL: Yeah, well, it does seem like it just has all these different branches. And all these weird dependencies where you can follow these lines around.

KK: I mean, the best mathematics is like that, right? I mean, it's sort of kind of simple, it’s a simple question to ask, you could explain this to a kid. And then the mathematics is so deep, it goes in so many directions. Yeah, it's really, really interesting.

EL: Yeah. Thanks a lot. Are there any places people can find you online? Your website, other other things you'd like to share?

ML: Well, yeah, my website. Shall I say the address?

EL: We can just put a link to that.

ML: Yeah, definitely my website. I actually will be giving a talk in the math club at my university on congruent numbers in a couple of weeks. So I’m going to try to post the slides online, but they are going to be in French.

EL: Okay, well, that'll be good. Our Francophone listeners can can check that out.

ML: I really like some notes that Keith Conrad wrote. And actually, I have to say, he has a bunch of expository papers in different areas that I always find super useful for, you know, going a little bit beyond my classes. And so in general, I recommend that his website for that, and in particular, the notes on the congruent number problem, if you're more interested. And then of course, there are some, some books that discuss congruent numbers and elliptic curves. So for example, a classic reference is Koblitz’s book on, I guess it’s called [Introduction to] Elliptic Curves and Modular Forms.

EL: Oh, yeah, I actually have that book. Because as a grad student, my second or third year, I, for some reason—I was not interested in number theory at all, but I think I liked this professor, so I took this class. So I have this book. And I remember, I just felt like I was swimming in that class.

KK: I have this book too, sitting on my shelf.

EL: The one number theory book two topologists have.

ML: So for me, I got this book before knowing I was going to be a number theorist.

EL: Yeah. No, but it is a nice book. Yeah. But yeah, well, we'll link to those will have. Make sure to get those all in the show notes so people can find those easily.

ML: Yeah.

EL: Well, thanks so much for joining me.

KK: Yeah.

EL: Us. Sorry, Kevin!

ML: Thank you for having us—for having me, now I’m confused! Thanks a lot. It's such a pleasure to be here.

EL: Yeah.

KK: Thanks.

On this episode, we were excited to welcome Matlide Lalín, a math professor at the University of Montreal. She talked about the congruent number problem. A congruent number is a positive integer that is the area of a right triangle with rational side lengths.

Our discussion took us from integers to elliptic curves, which are defined by equations of the form y2=x3+ax+b. As we mention in the episode, solutions to equations of this form satisfy what is known as a group law. That is a fancy way of saying there is a way to “add” two points on the curve to get another point. The diagram Kevin mentioned is here:

Credit: SuperManu (CC BY-SA 3.0) https://commons.wikimedia.org/wiki/File:ECClines-2.svg

Here are links to some other things we talked about on the podcast and resources for diving deeper:

Matilde Lalín’s website

Slides (in French) for her talk about congruent numbers

Keith Conrad’s notes about the congruent number problem (pdf)

John Coates has written about Tian’s recent work on the congruent number problem for Acta Mathematica Vietnamica

Introduction to Elliptic Curves and Modular Forms by Neal Koblitz

For more on Mordell’s Theorem, try Elliptic Curves by Anthony Knapp

[Bhargava and Shankar]

Andrew Wiles’ expository article about the Birch and Swinnerton-Dyer conjecture

To go even deeper with BSD, try The Arithmetic of Elliptic Curves by Joseph Silverman

Bhargava and Shankar’s recent work on BSD is here.

Episode 42 - Moon Duchin

9 maj 2019 | 25 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about math and…I don't even know what it's going to be today. We'll find out. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. So yeah, how are you today?

KK: I’ve had a busy week. Both of my PhD students defended on Monday.

EL: Wow, congratulations.

KK: Yeah, and so, through some weird quirk of my career, these are my first two PhD students. And it was a nice time, slightly nerve-wracking here and there. But everybody went through, everything's good.

EL: Great.

KK: So we have two new professors out there. Well, one guy's going to go into industry. But yeah, how about you?

EL: I’m actually—once we're done with this, I need to go pack for a trip I'm leaving on today. I'm teaching a math writing workshop at Ohio State.

KK: Right. I saw that, yeah.

EL: I mean, if it goes well, then we'll leave this part in the thing. And if it doesn't, no one else will know. But yeah, I'm looking forward to it.

KK: Good. Well, Ellen and I are going to Seattle this weekend.

EL: Fun.

KK: She got invited to be on a panel at the Bainbridge Island Art Museum. And I thought. “I'm going along” because I like Seattle.

EL: Yeah, it's beautiful there.

KK: Love it there. Anyway, enough about us. Today, we are pleased to welcome Moon Duchin to the show. Moon, why don’t you introduce yourself?

Moon Duchin: Hi, I'm Moon. I am a mathematician at Tufts University, where I'm also affiliated with the College of Civic Life. It’s a cool thing Tufts has that not everybody has. And in math, my specialties are geometric group theory, topology (especially low-dimensional topology), and dynamics.

KK Very cool.

EL: Yeah, and so how does this civil life thing work?

MD: Civic life, yeah.

EL: Sorry.

MD: So that's sort of because in the last couple years, I've become really interested in politics and in applications—I think of it as applications of math to civil rights. So that's that's sort of mathematics engaging with civics, it’s kind of how we do government. So that's become a pretty strong locus of my energy in the last couple years.

EL: Yeah.

KK: And I'll vouch for Moon's work here. I mean, I've gone to a couple of the workshops that she's put together. Big one at Tufts in 2017, I guess it was, and then last December, this meeting at Radcliffe. Really cool stuff. Really important work. And I've gotten interested in it too. And let's hope we can begin to turn some tides here. But anyway, enough about that. So, Moon, what's your favorite theorem?

MD: Alright, so I want to tell you about what I think is a really beautiful theorem that is known to some as Gromov’s gap.

KK: Okay.

EL: Which also sounds like it could be the name of a mountain pass in the Urals or something.

MD: I was thinking it sounds like it could be, you know, in there with the Mines of Moria in Middle Earth.

EL: Oh, yeah, that too.

MD: Just make sure you toss the dwarf across the gap. Right, it does sound like that. But of course, it's Mischa Gromov, who is the very prolific Russian-French mathematician who works in all kinds of geometry, differential geometry, groups, and so on.

So what the theorem is about is, what kinds of shapes can you see in groups? So let me set that up a little with—you know, let me set the stage, and then I'll tell you the result.

EL: Okay.

MD: So here's the setting. Suppose you want to understand—the central objects in geometric group theory are, wel,l groups. So what are groups? Of course, those are sets where you can do an operation. So you can think of that as addition, or multiplication, it's just some sort of composition that tells you how to put elements together to get another element. And geometric group theory is the idea that you can get a handle on the way groups work—they’re algebraic objects, but you can study them in terms of shape, geometrically. So there are two basic ways to do that. Either you can look at those spaces that they act on, in other words, spaces where that group tells you how to move around. Or you can look at the group itself as a network, and then try to understand the shape of that network. So let's stick with that second point of view for a moment. So that says, you know, the group has lots of elements and instructions for how to put things together to move around. So I like to think of the network—a really good way to wrap your mind around that is to think about chess pieces. So if I have a chessboard, and I pick a piece—maybe I pick the queen, maybe I pick the knight—there are instructions for how it can move. And then imagine the network where you connect two squares if your piece can get between them in one step. Right?

KK: Okay.

MD: So, of course that's going to make a different network for a knight than it would for a queen and so on, right?

EL: Yeah.

MD: Okay. So that's how to visualize a group, especially an infinite—that works particularly well for infinite groups. That's how to visualize a group as a bunch of points and a bunch of edges. So it's some big graph or network. And then GGT, geometric group theory, says, “What's the shape of that network?” Especially when you view it from a distance, does it look flat? Does it look negatively curved, like a saddle surface? Or does it kind of curve around on itself like a sphere? You know, what's the shape of the group?

And actually, just a cool observation from, you know, a hundred plus years ago, an observation of Felix Klein is that actually the two points of view—the spaces that the group acts on or the group itself—those really are telling the same story. So the shape of the space is about the same as the shape of the group. That's become codified as kind of a fundamental observation in GGT. Okay, great. So that's the space I want to think about. What is the shape—what are the possible shapes of groups? Okay, and that's where Gromov kicks in. So the theorem is about the relationship of area to perimeter. And here's what I mean by that.

Form a loop in your space, in your network. And here, a loop just means you start at a point, you wander around with a path, and you end up back where you started. Okay? And then look at the efficient ways to fill that in with area. So visualize that, like, first you have an outline, and then you try to fill it in with maybe some sort of potato chip-y surface that kind of interpolates around that boundary. Okay, so the question is, if you look at shapes that have the extremal relationship of area to perimeter, then what is the relationship of area to parameter?

So let's do that in Euclidean space first, because it's really familiar. So we know that the extremal shapes there are circles, and you fill those in with discs. And the relationship is that area looks about like perimeter squared, right?

KK: Right.

MD: Okay, great. So now here's the theorem, then. Get ready for it. I love this theorem. In groups, you can find groups where area looks like perimeter to the K power. It can look like perimeter to the 1, it can look like perimeter to the 1, or 3, or 4, and so on. You can build designer groups with any of those exponents. But furthermore, you can also get rational exponents. You can get pretty much any rational exponent you want. You can get 113/5, you can get, you know, 33/10. Pick your favorite exponent, and you can do it.

EL: Can you get less than one?

MD: Well, let's come back to that.

EL: Okay. Sorry.

MD: So let me state Gromov’s theorem in this level of generality. So here's the theorem. You can get pretty much any exponent that you want, as long as it's not between 1 and 2.

KK: Wow.

EL: Oh.

MD: Isn’t that cool? That's Gromov’s gap.

KK: Okay.

EL: Okay.

MD: So there's this wasteland between 1 and 2 that's unachievable.

KK: Wow.

MD: Yeah. And then you can, see—past 2, you can see anything. Um, it actually turns out, it's not just rationals. You can see lots of other kinds of algebraic numbers too.

KK: Sure.

MD: And the closure up there is everything from two to infinity! But nothing between one and two. It's a gap.

EL: Oh, wow. That's so cool!

MD: That’s neat, right? Evelyn, to answer your question under one turns out not to really be well defined for reasons we could talk about. But yeah.

KK: This is remarkable. This sounds like something Gromov would prove, right? I mean, just these weird theorems out of nowhere. I mean—how could that be true? And then there it is. Yeah.

MD: Or that Gromov would state and leave other people to prove.

KK: That—yeah, that's really more accurate. Yeah. So. Okay, so you can't get area to the—I mean, perimeter to the 3/2. I mean, that's, that's really…Okay. Is there any intuition for why you can't get things between one and two?

MD: Yeah, there kind of is, and it's beautiful. It is that the stuff that sits at the exponent 1, in other words, where area is proportional, the perimeter is just really qualitatively different from everything else. Hence the gulf. And what is that stuff? That is hyperbolic groups. So this comes back to Evelyn's wheelhouse, I believe.

EL: It’s been a while since I thought in a research way about this, but yes, vaguely at the distance of my memory.

MD: Let me refresh your memory. Yes, so negatively curved things, things that are saddled shaped, those are the ones where area is proportional to parameter. And everything else is just in a different regime. And that's really what this theorem is telling you.

So that's one beautiful point of view, and kind of intuition, that there's this qualitative difference happening there. But there's something—there’s so many things I love about this theorem. It's just the gateway to lots of beautiful math. But one of the things I love about this theorem is that it fails in higher dimensions, which is really neat. So if you, instead of filling a loop with area, if you were to fill a shell with volume, there would be no gap.

EL: Oh.

MD: Cool, right?

EL: So this is, like, the right way to measure it if you want to find this difference in how these groups behave.

MDL Absolutely. And, you know, another way to say it, is this is an alternative definition of hyperbolic group from the usual one. It's like, the right way to pick out these special groups from everything else is specifically to look at filling loops.

KK: Right. And I might be wrong here, but aren't most groups hyperbolic? Is that?

MD: Yeah, so that's definitely the kind of religious philosophy that’s espoused. But you know, to talk about most groups usually the way people do that is they talk about random constructions of groups. And a lot of that is pretty sensitive to the way you set up what random means. But yeah, that's definitely the, kind of, slogan that you hear a lot in geometric group theory, is that hyperbolic groups are special, but they're also generic.

EL: Yeah.

KK: So are there explicit constructions of groups with say, exponent 33/10, to pick an example?

MD: Yes, there are. Yeah. And actually, if you're going to end up writing this up, I can send you some links to beautiful papers.

EL: Yeah, yeah. But there’s, like, a recipe, kind of, where you're like, “Oh, I like this exponent. I can cook up this group.”

MD: Yeah. And that's why I kind of call them designer groups.

EL: Right, right. Yeah. Your bespoke groups here.

MD: Yeah, there are constructions that do these things.

KK: That’s remarkable. So I was going to guess that your favorite term was the isoperimetric inequality. But I guess this kind of is, right?

MD: I mean, exactly. Right? So the isometric inequality is all about asking, what is the extremal relationship of area to parimeter? And so this is exactly that, but it's in the setting of groups.

KK: Yeah, yeah.

EL: So how did you first come across this theorem?

MD: Well, I guess, in—when you're in the areas of geometric topology, geometric group theory, there's this one book that we sometimes call the Bible—here I'm leaning on this religious metaphor again—which is this this great book by Bridson and Haefliger called Metric Spaces of Non-Positive Curvature. And it really does feel like a Bible. It's this fat volume, you always want it around, you flip to the stuff you need, you don't really read it cover to cover.

KK: Just like the Bible. Yeah,

MD: Exactly. Great. And that's certainly where I first saw it proved. But, yeah, I mean, the ideas that circulate around this theorem are really the fundamental ideas in GGT.

KK: Okay, great. Does this come up in your own work a lot? Do you use this for things you do? Or is this just like, something that you love, you know, for its own sake?

MD: Yeah, no, it does come up in my own work in a couple of ways. But one is I got interested in the relationship between curvature—curvature in the various senses that come from classical geometry—I got interested in the relationship between that and other notions of shape in networks. So this theorem takes you right there. And so for instance, I have a paper of a theorem with Lelièvre, and Mooney, where we look at something really similar, which is, we call it sprawl. It's how spread out do you get when you start at a point and you look at all the different positions you can get to within a certain distance. So you look at a kind of ball around the point. And then you ask how far apart are the points in that ball from each other? So that's actually a pretty fun question. And it turns out, here's another one of these theorems where hyperbolic stuff, there's just a gap between that and everything else.

KK: Right.

MD: So let’s follow that through for a minute. So suppose you start at a base point, and you take the ball of radius R around that base point. And then you ask, “How far apart are the points in that ball from each other?” Well, of course, by the triangle inequality, the farthest apart that could possibly be is 2R because can connect them through the center to each other, right, 2R. Okay, so then you could ask, “Hm, I wonder if there's a space that’s so sprawling, so spread out, so much like, you know, Houston, right, so sprawling that the average distance is actually the maximum?”

EL: Yeah.

MD: Right. What if the average distance between two points is actually equal to 2R?

And that’s, so that's something that we proved. We proved that when you're negatively curved, and you have, you know, a few other mild conditions, basically—but certainly true for negatively curved groups, just like the setting of Gromov’s theorem—so for negatively curved groups, the average is the maximum. You’re as sprawling as you can be. Yeah, isn't that neat? So that's very much in the vein of this kind of result.

KK: Oh, that's very cool. All right.

EL: Yeah. Kind of like the SNCF metric, also, where you have to go to Paris to go anywhere else. Slightly different, but still, that you basically you have to go in to the center to get to the other side.

MD: It’s exactly the same collection of ideas. And I'm just back from Europe, where I can attest that it's really true. You want to get from point to point on the periphery of France, you’d better be going through Paris if you want to do it fast. But yeah, it’s precisely the same idea, right? So the average distance between points on the periphery of France will be: get to Paris and get to the other point. So there's a max there that's also realized.

KK: All right, so France is hyperbolic.

MD: France is hyperbolic. Yup, in terms of travel time.

EL: Very appropriate. It’s such a great country. Why wouldn't it be hyperbolic?

KK: All right, so the other fun thing on this podcast is we ask our guests to pair their theorem with something. So what pairs well with Gromov’s gap theorem?

MD: So I'm actually going to claim that it pairs beautifully with politics. Right? True to form, true to form.

EL: Okay.

KK: Right, yeah, sure.

MD: All right, so let me try and make that connection. So, well, I got really interested in the last few years in gerrymandering in voting districts. And classically, one of the ways that we know that a district is problematic is exactly this same way, that it's built very inefficiently. It has too much perimeter, it has too much boundary, a long, wiggly, winding boundary without enclosing very much area. That's been a longstanding measurement of kind of the fairness or the reasonableness of the district. So I got interested in that through likes of this kind of network curvature stuff, with the idea that maybe the problem is in the relationship between area and perimeter.

And so what does that make you want to do, if you're me? It makes you want to take a state and look at it as a network. And you can do that with census data. You sort of take the little chunks of census geography and connect the ones that are next to each other and presto! You have a network. And it's a pretty big network, but it's finite.

KK: Yeah.

MD: So Pennsylvania's got about 9000 precincts. So you can make a graph out of that. But it's got a whole lot more census blocks. Virginia—we were just looking at Virginia recently—300,000 census blocks. So that's a pretty big network, but you know, still super duper finite, right?

EL: Yeah.

MD: And so you can sort of ask the same question, what's the shape of that network? And does that—you know, maybe the idea is, if the network itself, which is neutral, no one's doing any gerrymandering, that's just where the people are.

EL: Yeah.

MD: If the network network itself is is negatively curved, in some sense, then maybe that explains large perimeters in a reason that isn't due to political malfeasance, you know?

EL: Right.

MD: So I think this is a way of thinking about shape and possibility that lends itself to lots of problems. But I like to pair everything with politics these days.

EL: Yeah, well, I really I think—so I went to your talk at the joint mass meetings a couple of years ago, which I know you've probably given similar talks, with talking about gerrymandering, I think it's really important for people not to take too simplistic a view and just say, “Oh, here's a weird shape.” And you were, you did a really great job of showing, like, there are sometimes good reasons for weird shapes. Obvious things like theres a river here and people end up grouping like this around the river for this reason. But there are a lot of different reasons for this. And if we want to talk about this in a way that can actually be productive, we have to be very nuanced about this and understand all of those subtleties, which are mixing the math—we can't just divorce it from the world. We we mix the math from, you know, the underlying civil rights and, you know, politics, historic inequalities in different groups and things like that.

MD: Yeah, absolutely. That's definitely the point of view that I've been preaching, to stick with the religious metaphor. It’s the one that says, if you want to understand what should be out of bounds, because it's unreasonable when it comes to redistricting, first you have to understand the lay of the land. You have to spec out the landscape of what's possible. And like you're saying, you know, that landscape can have lots of built and structure that districting has to respect. So, yeah, you should really—that could be physical geography, like you mentioned rivers—but it could also be human geography. People distribute themselves in very particular ways. And districting isn't done with respect to like, imaginary people, it’s done with respect to the real, actual people and where they live.

EL: Yeah.

MD: And that's why I really, you know, I think more and more that some of those same tools that we use to study the networks of infinite groups, we can bring those to bear to study the large finite networks of people and how they live and how we want to divide them up.

EL: Yeah, that's, that's a nice pairing, maybe one of the weightier pairings we’ve had.

KK: Yeah, right.

MD: It was either that or a poem, but. I was thinking Gromov’s gap, maybe I could pair that with The Wasteland.

EL: Oh.

MD: Because you can’t get in the wasteland between exponents one and two. Nah, let's go politics.

EL: Well, I've tried to read that poem a few times, and I always feel like I need someone to hold my hand and, like, point everything out to me. It's like, I know there's something there but I haven't quite grabbed on to it yet.

MD: Yeah. Poetry is like math, better with a tour guide.

EL: Yes.

KK: Well, we also like to give our guests a chance to plug things. You want to tell everyone where they can find you online and and maybe about the MGGG?

MD: Sure, yeah, absolutely. So I co-lead a working group called MGGG, the Metric Geometry and Gerrymandering Group, together with a computer scientist named Justin Solomon, who's over at MIT. You can visit us online at mggg.org, where we have lots of cool things to look at, such as the brief we filed with the Supreme Court a couple weeks ago, which just yesterday was actually quoted in oral argument, which was pretty exciting, if quoted in a surprising way.

You can also find cool software tools that we've been developing, like our tool called Districtr lets you draw your own districts and kind of see—try your own hand at either gerrymandering or fair district thing and gives you a sense of how hard that is. We think it's one of the more user-friendly districting tools out there. Lots of different research links and software tools and resources on our site. So that would be that'd be fun if people want to check that out and give us feedback.

Other things I want to mention: Oh, I guess I'm going to do the 538 Politics podcast tomorrow, talking about this new Supreme Court case.

EL: Nice.

MD: Yeah. So I think that'll be fun. Those are some smart folks over there who’ve thought a lot about some of the different ways of measuring gerrymandering, so I think that'll be a pretty high-level conversation.

KK: Yeah, I'm sure. They turn around real fast, like this will be months from now.

MD: Right. I see. Okay, cool. Yeah, by the time this comes out, maybe we'll maybe we'll have yet another Supreme Court decision on gerrymandering that will…

KK: Yeah, fingers crossed.

MD: We’ll all be handling the fallout from.

EL: Yeah.

KK: All right. Well, this has been great fun, Moon. Thanks for joining us.

MD: Oh, it's a pleasure.

[outro]

Episode 41 - Suresh Venkatasubramanian

25 april 2019 | 30 min

Evelyn Lamb: Welcome to My Favorite Theorem, a podcast about theorems where we ask mathematicians to tell us about theorems. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. This is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson. I’m a professor of mathematics at the University of Florida. It seems like I just saw you yesterday.

EL: Yeah, but I looked a little different yesterday.

KK: You did!

EL: In between when I talked with you and this morning, I dyed my hair a new color, so I'm trying out bright Crayola crayon yellow right now.

KK: It looks good. Looks good.

EL: Yeah, it's been fun in this, I don't know, like 18 hours I've had it so far.

KK: Well, you know what, it's what it's sort of dreary winter, right, you feel the need to do something to snap you out of it, although it's sunny in Salt Lake, right, it’s just cold? No big deal.

EL: Yeah, we’ve had some sun. We’ve had a lot of snow recently, as our guest knows, because our guest today also lives in Salt Lake City. So I'm very happy to introduce Suresh Venkatasubramanian. So Hi. Can you tell us a little bit about yourself?

Suresh Venkatasubramanian: Hey, thanks for having me. First of all, I should say if it weren't for your podcast, my dishes would never get done. I put the podcast on, I start doing the dishes and life is good.

EL: Glad to be of service.

SV: So I'm in the computer science department here. I have many names. It depends on who's asking and when. Sometimes I'm a computational geometer, I'm a data miner. Sometimes occasionally a machine learner, though people raise their eyebrows at that. But the name I like the most right now is bias detective, or computational philosopher. These are the two names I like the most right now.

EL: Yeah, yeah. Because you've been doing a lot of work on algorithmic bias. Do you want to talk about that a little bit?

SV: Sure. So one of the things that we're dealing with now as machine learning and associated tools go out into the world is that they're being used for not just, you know, predicting what podcast you should listen to, or what music you should listen to, but they're also being used to decide whether you get a job somewhere, whether you get admission to college, whether you get surveilled by the police, what kind of sentences you might get, whether you get a loan. All these things are where machine learning is now being used just because we have lots of data to collect and seemingly make better decisions. But along with that comes a lot of challenges, because what we're finding is that a lot of the human bias in the way make decisions is being transferred to machine bias. And that causes all kinds of problems, both because of the speed at which these decisions get made and the relative obscurity of automated decision making. So trying to piece together, piece out what's going on, how this changes the way we think about the world, how we—the way we think about knowledge about society, has been taking up most of my time of late.

EL: Yeah, and you've got some interesting papers that you've worked on, right, on how people who design algorithms can help combat some of these biases that can creep in.

SV: Yeah, so there are many, many levels of questions, right? One basic question is how do you even detect whether there’s—so first of all, I mean, I think as a mathematical question, how do you even define what it means for something to be biased? What does that word even mean? These are all loaded terms. And, you know, once you come up with a bunch of different definitions for maybe what is a relatively similar concept, how do they interact? How do you build systems that can sort of avoid these kinds of bias the way you've defined it? And what are the consequences of building systems? What kind of feedback loops do you have in systems that you use? There’s a whole host of questions from the mathematical to the social to the philosophical. So it's very exciting. But it also means everyday, I feel even more dumb than I started the day with so.

EL: Yeah.

KK: So I think the real challenge here is that, you know, people who aren't particularly mathematically inclined just assume that because a computer spit out an answer, it must be valid, it must be correct. And that, in some sense, you know, it's cold, that the machine made this decision, and therefore, it must be right. How do you think we can overcome that idea that, you know, actually, bias can be built into algorithms?

SV: So this is the “math is not racist” argument, basically.

KK: Right.

SV: That comes up time and time again. And yeah, one thing that I think is an encouraging is that we've moved relatively quickly, in a span of, say, three to four years, from “math is not racist” to “Well, duh, of course, algorithms are biased.”

EL: Yeah.

SV: So I guess that's a good thing. But I think the problem is that there’s a lot of commercial and other incentives bound up with the idea that automated systems are more objective. And like most things, there's a kernel of truth to it in the sense that you can avoid certain kinds of obvious biases by automating decision making. But then the problem is you can introduce others, and you can also amplify them. So it's tricky, I think. You're right, it's getting away from the notion where commercially, there's more incentive to argue that way. But also saying, “Look, it's not all bad, you just need more nuance.” You know, arguments for more nuance tend not to go as well as, you know, “Here's exactly how things work,” so it's hard.

KK: Everything’s black or white, we know this, right?

SV: With a 50% probability everything is either true or not, right.

EL: So we invited you on to hear about your favorite theorem. And what is that?

SV: So it's technically an inequality. But I know that in the past, you've allowed this sort of deviation from the rules, so I'm going to propose it anyway.

EL: Yes, we are very flexible.

SV: Okay, good good good. So the inequality is called Fano’s inequality after Robert Fano, and it comes from information theory. And it’s one of those things where, you know, the more I talk about it, the more excited I get about it. I'm not even close to being an expert on the ins and outs of this inequality. But I just love it so much. So I need to tell you about that. So like all good stories, right, this starts with pigeons.

EL: Of course.

SV: Everything starts with pigeons. Yes. So, you may have heard of the pigeonhole principle.

KK: Sure.

SV: Ok. So the pigeonhole principle, for those in the audience who may not, basically, if you have ten pigeons and you have nine pigeonholes, someone's going to get a roommate, right? It’s one of the most obvious statements one can make, but also one of the more powerful ones, because if you unpack a little bit, it's not telling you where to find that pigeonhole with two pigeons, it's merely saying that you are guaranteed that this thing must exist, right, it's a sort of an existence proof that can be stated very simply.

But the pigeonhole principle, which is used in many, many parts of computer science to prove that, you know, some things are impossible or not can be restated, can be used to prove another thing. So we all know that if, okay, I need to store a set of numbers, and if the numbers range from 1 to n, that I need something like log(n) bits to store it. Well, why do I need log(n) bits? One way to think about this is this is the pigeonhole principle in action because you're saying, I have n things. If I have log(n) bits to describe a hole, like an address, then there are 2log(n) different holes, which is n. And if I don't have log(n) bits, then I don't have n holes, and therefore by the pigeonhole principle, two of my things will get the same name. And that's not a good thing. I want to be able to name everything differently.

So immediately, you get the simple statement that if you want to store n things you need log(n) bits, and of course, you know, n could be whatever you want it to be, which means that now—in theoretical computer science, you want to prove lower bounds, you want to say that something is not possible, or some algorithm must take a certain amount of time, or you must need to store so many bits of information to do something. These are typically very hard things to prove because you have to reason about any possible imaginary way of storing something or doing something, and that's very hard. But with things like the pigeonhole principle and the log(n) bit idea, you can do surprisingly many things by saying, “Look, I have to store this many things, I'm going to need at least log of that man bits no matter what you do.” And that's great.

KK: So that's the inequality.

EL: No, not yet.

KK: Okay, all right.

SV: I was stopping because I thought Evelyn was going to say something. I'm building up. It's like, you know, a suspense story here.

KK: Okay, good.

EL: Yes.

KK: Chapter two.

SV: So if you now unpack this log(n) bit thing, what it's really saying is that I can encode elements and numbers in such a way, using a certain number of bits, so that I can decode them perfectly because there aren't two pigeons living in the same pigeonhole. There's no ambiguity. Once I have a pigeonhole, who lives there is very clear, right? It's a perfect decoding. So now you've gone from just talking about storage to talking about an encoding process and a decoding process. And this is where Fano’s inequality comes into play.

So information theory—so going back to Shannon, right—is this whole idea of how you transmit information, how you transmit information efficiently, right, so the typical object of study in information theory is a channel. So you have some input, some sort of bits going into a channel, something happens to it in the channel, maybe some noise, maybe something else, and then something comes out. And one of the things you'd like to do is looking at, so x comes in, y comes out, and given y you'd like to decode and get back to x, right? And it turns out that, you know, you can talk about the ideas of mutual information entropy, they start coming up very naturally, where you start saying if x is stochastic, it's a random variable, and y is random, then the channel capacity, in some sense the amount of information the channel can send through, relates to what is called the mutual information between x and y, which is this quantity that captures, roughly speaking, if I know something about x, what can I say about y and vice versa. This is not quite accurate, but this is more or less what it says.

So information theory at a broader level—and this is where really Fano’s inequality connects to so many things—it’s really about how to quantify the information content that one thing has about another thing, right, through a channel that might do something mysterious to your variable as it goes through. So now what does Fano’s inequality say? So Fano’s inequality you can think of now in the context of bits and decoding and encoding, says something like this. If you have a channel, you take x, you push it into the channel and out comes y, and now you try to reconstruct what x was, right? And let's say there's some error in the process. The error in the process of reconstructing x from y relates to the term that is a function of the mutual information between x and y.

More precisely, if you look at how much, essentially, entropy you have left in x once I tell you y— So for example, let me see if this is a good example for this. So I give you someone's name, let's say it's an American Caucasian name. With a certain degree of probability, you'll be able to predict what their gender is. You won't always get it right, but there will be some probability of this. So you can think of this as saying, okay, there's a certain error in predicting that person's name from the—for predicting the person's gender from the name as you went through the channel. The reason why you have an error is because there's certain one of noise. Some names are sort of gender-ambiguous, and it's not obvious how to tell. And so there's a certain amount of entropy left in the system, even after I've told you the name of the person. There's still an amount of uncertainty. And so your error in predicting that person's gender from the name is related to the amount of entropy left in the system. And this is sort of intuitively reasonable. But what it's doing is connecting two things that you wouldn’t normally expect to be connected. It's connecting a computational metaphor, this process of decoding, right, and the error in decoding, along with a basic information theoretic statement about the relationship between two variables.

And because of that—Fano’s inequality, or even the basic log(n) bits needed to sort n things idea—for me, it's pretty much all of computer science. Because if we want to prove a lower bound on how we compete something, at the very least we can say, look, I need at least this much information to do this computation. I might need other stuff, but I need at least as much information. That clearly will be a lower bound. Can I prove a lower bound? And that has been a surprisingly successful endeavor, in reasoning about the lower bounds for computations, where it would be otherwise very hard to think about, okay, what does this computation do? or what does that computation do? Have I imagined every possible algorithm I can design? I don't care about any of that because Fano’s inequality says it doesn't matter. Just analyze the amount of information content you have in your system. That's going to give you a lower bond on the amount of error you’re going to see.

EL: Okay, so I was reading—you wrote a lovely posts about this, and I was reading that this morning, before we started talking. And I think this is what you just said, but it's one of these things that is very new to me, I'm not used to thinking like a computer scientists or information theorist or anything. So something that was a little bit, I was having trouble understanding, is whether this inequality, how much it depends on the particular relationship you have between the two, x and y that you're looking at.

SV: So one way to answer this question is to say that all you need to know is the conditional entropy of the x given y. That's it. You don't need to know anything else about how y was produced from x. All that you need to know, to put a bound on the decoding error, is the amount of entropy that's left in the system.

KK: Is that effectively computable? I mean, it is that easy to compute?

SV: For the cases where you apply Fano’s inequality, yes. Typically it is. In fact, you will often construct examples where you can show what the conditional entropy is, and therefore be able to reason, directly use Fano’s inequality, to argue for the probability of error. So let me give an example in computer science of how this works.

KK: Okay, great.

SV: Suppose I want to build a—so one of the things we have to do sometimes is build a data structure for a problem, which means you're trying to solve some problem, you want to store information in a convenient way so you can access the information quickly. So you want to build some lookup table or a dictionary so that when someone comes up with the question—“Hey, is this person's name in the dictionary?”—you can quickly give an answer to the question. Ok. So now I want to say, look, I have to process these queries. I want to know how much information I need to store my data structure so that I can answer queries very, very quickly. Okay, and so you have to figure out—so one thing you'd like to do is, okay, I built this awesome data structure, but is it the best possible? I don't know, let me see if I can prove a lower bound on how much information I need to store.

So the way you would use Fano’s inequality to prove that you need a certain amount of information would be to say something like this. You would say, I'm going to design a procedure. I'm going to prove to you that this procedure correctly reconstructs the answer for any query a user might give me. So there's a query, let’s say “Is there an element in the database?” And I have my algorithm, I will prove correctly returns this with some unknown number of bits stored. And given that it correctly returns us answer up to some error probability, I will use Fano’s and equality to say, because of that fact, it must be the case that there is a large amount of mutual information between the original data and the data structure you stored, which is the, essentially, the thing that you've converted the data from through your channel. And so this if that mutual information is large, the amount of bits you need to store this must also be large. And therefore, with small error, you must pay at least these many bits to build this data structure. And so this idea sort of goes through a lot of, in fact, more recent work in lower bounds in data structure design and also communication. So, you know, two parties want to communicate with each other, and they want to decide whether they have the same bit string. And you want to argue that you need these many bits of information for them to communicate, to show that the other same bit string. You use, essentially, either Fano’s inequality or even simpler versions of it to make this argument that you need at least a certain number of bits. This is used in statistics in a very different way. But that's a different story. But in computer science, this is the one of the ways in which the inequality is used.

EL: Okay, getting back to the example that you used, of names and genders. How—can you kind of, I don't know if this is going to work exactly. But can you kind of walk us through how this might help us understand that? Obviously, not having complete information about how, how this does correspond to names, but,

SV: Right, so let's say you have a channel, okay? What’s going into the channel, which is x, is the full information about the person, let's say, including their gender. And the channel, you know, absorbs a lot of this information, and just spits out a name, the name of the person. That's y and now the task is to predict. or reconstruct the person's gender from the name. Okay.

So now you have x, you have y. You want to reconstruct x. And so you have some procedure, some algorithm, some unknown algorithm that you might want to design, that will predict the gender from the name. Okay? And now this procedure will have a certain amount of error, let's call this error p, right, the probability of error, the probability of getting it wrong, basically is p. And so what Fano’s inequality says, roughly speaking—so I mean, I could read out the actual inequality, but on the air, it might not be easy to say—but roughly speaking, it says that this probability p times a few other terms that are not directly relevant to this discussion, is greater than or equal to the entropy of the random variable s, which is you can think of it as drawn from the population of people. So I drew uniformly from the population of people in the US, right, or of Caucasians in the US, because we are limiting ourselves to that thing. So, that's our random variable x. And going through this, I get the value y. So I compute the entropy of the distribution on x conditioned that the gender was, say, female. So I look at how much— and basically that probability of error is greater than or equal to, you know, with some extra constants attached, this entropy.

So in other words, so what it's saying is very intuitive, right? If I tell you, the person is is female— and we're sort of limiting ourselves to this binary choice of gender—but let's just say this person is female, you know—sorry, this person's name is so and so. Right? What is the range of gender? What does the gender distribution look like conditioned on having that name? So let's say the name, let's say is, we think of a name, that would be—let’s say, Dylan, so Dylan is the name. So there's going to be some, you know, probability of the person being male and some property being female. And in the case of a name like Dylan, those probabilities might not be too far apart, right? So you can actually compute this entropy now. You can say, okay, what is the entropy of x given y? You just write the formula for entropy for this distribution. It’s just p1logp(1)+p2log(p2) in this case, where p1+p2=1. And so the probability of error no matter what, how clever your procedure is, is going to be lower bounded by this quantity with some other constants attached to make it all make sense.

EL: Okay.

SV: Does that help?

EL: Yeah.

KK: Right.

SV: I made this completely up on the fly. So I'm not even sure it’s correct. I think it's correct.

KK: It sounds correct. So it's sort of, the noisier your channel, the probability is going to go up, the entropy is going to be higher, right?

SV: Right, yeah.

KK: Well, you're right, that's intuitively obvious, right? Yeah. Right.

SV: Right. And the surprising thing about this is that you don't have to worry about the actual reconstruction procedure, that the amount of information is a limiting factor. No matter what you do, you have to deal with that basic information thing. And you can see why this now connects to my work on algorithmic fairness and bias now, right? Because, for example, one of the things that is often suggested is to say, Oh, you know—like in California they just did a week ago, saying, “You are not allowed to use someone's gender to give them driver’s insurance.”

KK: Okay.

SV: Now, there are many reasons why there are problems with the policy as implemented versus the intent. I understand the intent, but the policy has issues. But one issue is this, well, just removing gender may not be sufficient, because there may be signal in other variables that might allow me, allow my system, to predict gender. I don't know what it's doing, but it could internally be predicting gender. And then it would be doing the thing you're trying to prohibit by saying, just remove the gender variable. So while the intention of the rule is good, it's not clear that it will, as implemented, succeed in achieving the goals it’s set out. But you can't reason about this unless you can reason about information versus computation. And that's why Fano’s inequality turns out to be so important. I sort of used it in spirit in some of my work, and some people in follow-up have used it explicitly in their work to sort of show limits on, you know, to what extent you can actually reverse-engineer, you know, protected variables of this kind, like gender, from other things.

EL: Oh, yeah, that would be really important to understand where you might have bias coming in.

SV: Right. Especially if you don't know what the system is doing. And that's what's so beautiful about Fano’s inequality. It does not care.

EL: Right. Oh, that's so cool. Thanks for telling us about that. So, is this something that you'd kind of learn in one of your first—maybe not first computer science courses, but very early on in in your education, or did you pick this up along the way?

SV: Oh, no, you don’t—I mean, it depends. It is very unlikely that you will hear about Fano’s inequality in any kind of computer science, theoretical computer science class, even in grad school.

EL:Okay.

SV: Usually you pick it up from papers, or if you take a course in information theory, it's a core concept there. So if you take any course in information theory, it'll come up very quickly.

EL: Okay.

SV: But in computer science, it comes up usually when you're reading a paper, and they use a magic lower bound trick. Like, where did they get that from? That's what happened to me. It's like, where do they get this from? And then you go down a rabbit hole, and you come back up three years later with this enlightened understanding of… I mean, Fano’s inequality generalizes in many ways. I mean, there's a beautiful, there's a more geometric interpretation of what Fano’s inequality really is when you go to more advanced versions of it. So there's lots of very beautiful—that’s the thing about a lot of these inequalities, they are stated very simply, but they have these connections to things that are very broad and very deep and can be expressed in different languages, not just in information theory, also in geometry, and that makes them really cool. So there's a nice geometric analog to Fano’s inequality as well that people use in differential privacy and other places.

KK: So what does one pair with Fano’s inequality?

SV: Ah! See, when I first started listening to My Favorite Theorem, I said, “Okay, you know, if one day they ever invite me on, I’m going to talk about Fano’s inequality, and I'm going to think about what to pair with it.” So I spent a lot of time thinking about this.

KK: Good.

SV: So see you have all these fans now, that's a cool thing.

So my choice for the pairing is goat cheese with jalapeño jam spread on top of it on a cracker.

EL: Okay.

KK: Okay.

SV: And the reason I chose this pairing: because they are two things that you wouldn't normally think go together well, but they go together amazingly well on a cracker.

KK: Well of course they do.

SV: And that sort of embodies for me what Fano’s inequality is saying, that two things that you don't expect to go together go together really well.

KK: No, no, no, the tanginess of the cheese and the salty of the olives. Of course, that's good.

SV: Not olives, jalapeño spread, like a spicy spread.

KK: Oh, okay, even better. So this sounds very southern. So in the south what we do, I mean, I grew up in North Carolina, you take cream cheese and pepper jelly, hot pepper jelly, and you put that on.

SV: Perfect. That's exactly it.

KK: Okay. All right. Good. Okay, delicious.

EL: So do you make your own jalapeños for this, or you have a favorite brand?

SV: Oh boy, no, no. I'm a glutton, not a gourmet, so I'll eat it whatever someone gives it to me, but I don't know what to make these things.

EL: Okay. We recently, we had a CSA last fall, and had a surplus of hot peppers. And I am unfortunately not very spice-tolerant. Or spiciness. I love spices, but, you know, can't handle the heat very well. But my spouse loves it. So I've been making these slightly sweet pickled jalapeños. I made those. And since then, he's been asking me, you know, I've just been going out and getting more and making those, so I think I will be serving this to him around the time we air this episode.

KK: Good.

SV: So since you can get everything horrifying on YouTube, one horrifying thing you can watch on YouTube is the world chili-eating competitions.

EL: Oh, no.

SV: Let’s just say it involves lots of milk and barf bags.

KK: I can imagine.

SV: But yeah, I do like my chilies. I like the habañeros and things like that.

EL: Yeah, I just watched the first part of a video where everyone in an orchestra eats this really hot pepper and then they're supposed to play something, but I just couldn't made myself watch the rest. Including the brass and wins and stuff. I was just thinking, “This is so terrible.” I felt too bad for them.

SV: It’s like the Drunk History show.

KK: We’ve often joked about having, you know, drunk favorite theorem.

SV: That would be awesome. That'd be so cool.

KK: We should do that.

EL: Yeah, well, sometimes when I transcribe the episodes, I play the audio slower because then I can kind of keep up with transcribing it. And it really sounds like people are stoned. So we joked about having “Higher Mathematics.”

KK: That’s right.

SV: That’s very good.

EL: Because they’re talking, like, “So…the mean…value…theorem.”

SV: I would subscribe to that show.

EL: Note to all federal authorities: We are not doing this.

KK: No, we’re not doing it.

EL: Yeah. Well, thanks a lot for joining us. So if people want to find you online, they can find you on Twitter, which was I think how we first got introduced to each other. What is your handle on that?

It's @geomblog, so Geometry Blog, that's my first blog. So g e o m, b l o g. I maintain a blog as well. And so that's among the places—I’m a social media butterfly. So you can find me anywhere.

EL: Yeah.

SV: So yeah, but web page is also a good place, my University of Utah page.

EL: We’ll include that, including the link to this post about Fano’s inequality so people can see. You know, it really helped me to read that before talking with you about it, to get some of the actual inequalities of the terms that appear in there straight in my head. So, yeah, thanks a lot for joining us.

KK: Thanks, Suresh.

SV: Thanks for having me.

[outro]

Episode 40 - Ursula Whitcher

11 april 2019 | 28 min

Evelyn Lamb: Hello, and welcome to my favorite theorem, a math podcasts where we ask mathematicians to tell us about their favorite theorems. I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And I am joined as usual by my co host Kevin. Can you introduce yourself?

Kevin Knudson: Sure. I'm still Kevin Knudson, professor of mathematics at the University of Florida. How are things going?

EL: All right. Yeah.

KK: Well, we just talked yesterday, so I doubt much has changed, right? Except I seem to have injured myself between yesterday today. I think it's a function of being—not 50, but not able to say that for much longer.

EL: Yeah, it happens.

KK: It does.

EL: Yeah. Well, hopefully, your podcasting muscles have not been injured.

KK: I just need a few fingers for that.

EL: Alright, so we are very happy today to welcome Ursula Whitcher to the show. Hi, can you tell us a little bit about yourself?

Ursula Whitcher: Hi, my name is Ursula. I am an associate editor at Mathematical Reviews, which, if you've ever used MathSciNet to look up a paper or check your Erdős number or any of those exciting things, there are actually 16 associate editors like me checking all the math that gets posted on MathSciNet and trying to make sure that it makes sense. I got my PhD at the University of Washington in algebraic geometry. I did a postdoc in California and spent a while as a professor at the University of Wisconsin Eau Claire, and then moved here to Ann Arbor where it's a little bit warmer, to start a job here.

EL: Ann Arbor being warmer is just kind of a scary proposition.

UW: It’s barely even snowing. It's kind of weird.

EL: Yeah. Well, and yeah, you mentioned Mathematical Reviews. I—before you got this job, I was not aware that, you know, there were, like, full time employees just of Mathematical Reviews, so that's kind of an interesting thing.

UW: Yeah, it's a really cool operation. We actually go back to sometime in probably the ‘40s.

KK: I think that’s right, yeah.

EL: Oh wow.

UW: So it used to be a paper operation where you could sign up and subscribe to the journal. And at some point, we moved entirely online.

KK: I’m old enough to remember in grad school, when you could get the year’s Math Reviews on CD ROM before MathSciNet was a thing. And you know, I remember pulling the old Math Reviews, physical copies, off the shelf to look up reviews.

UW: We actually have in the basement this set of file cards that our founder, who came from Germany around the Second World War, he had a collection of handwritten cards of all the potential reviewers and their possible interests. And we've still got that floating around. So there's a cool archival project.

KK: I’m ashamed to admit that I'm a lapsed reviewer. I used to review, and then I kind of got busy doing other things and the editors finally wised up and stopped sending me papers.

UW: I try to tell people to just be really picky and only accept the papers that you're really excited to read.

KK: I feel really terrible about this. So maybe I should come back. I owe an an apology to you and the other editors.

UW: Yeah, come back. And then just be really super, super picky and only take things that you are truly overjoyed to read. We don't mind. I—you know, I read apologies for my job for part of my day every day for not reviewing. So I’ve become sort of a connoisseur of the apology letter.

KK: Sure. So is part of your position also that you have some sort of visiting scholar deal at the University of Michigan? Does that come with this?

UW: Yeah, that is. So I get to hang out at the University of Michigan and go to math seminars and learn about all kinds of cool math and use the library card. I'm a really heavy user of my University of Michigan library card. So yeah.

EL: Those are excellent.

KK: It’s a great campus. That’s a great department, a lot of excellent people there.

EL: Yeah. So, what is your favorite theorem, or the favorite theorem you would like to talk about today?

UW: So I decided that I would talk about mirror theorems as a genre.

EL: Okay.

UW: I don't know that I have a single favorite mirror theorem, although I might have a favorite mirror theorem of the past year or two. But as this kind of class of theorems, these are a weird thing, because they run kind of backwards.

Like, typically there's this thing that happens where mathematicians are just hanging out and doing math because math is cool. And then at some point, somebody comes along and is like, “Oh, I see it practical use for this. And maybe I can spin it off into biology or physics or engineering or what have you.” Mirror theorems came the other way. They started with physical observation that there were two ways of phrasing of a theoretical physics idea about possible extra dimensions and string theory and gravity and all kinds of cool things. And then that physical duality, people chewed on and figured out how to turn it into precise mathematical statements. So there are lots of different precise mathematical statements encapsulating maybe something different about about the way these physical theories were phrased, or maybe building then, sort of chaining off of the mathematics and saying something that no longer directly relates to something you could state about a possible physical world. But there is still in like a neat mathematical relationship you wouldn't have figured out without having the underlying physical intuition.

EL: Yeah. And so this is, the general area is called mirror symmetry. And when I first heard that phrase, I assumed it was something about like group theory that was looking at, like, you know, more tangible, things that I would consider symmetric, like what it looks like when you look in a mirror. But that's not what it is, I learned.

UW: So I can tell you why it's called mirror symmetry, although it's kind of a silly reason. The first formulations of mirror symmetry, people were looking at these spaces called Calabi-Yau three-folds, which are—so there are three complex dimensions, six real dimensions, they could maybe be the extra dimensions of the universe, if you're doing string theory. And associated with a Calabi-Yau three-fold, you have a bunch of numbers that tell you about it’s topological information, sort of general stuff about what is this six dimensional shape looking like. And you can arrange those numbers in a diamond that's called the Hodge diamond. And then you can draw a little diagonal line through the Hodge diamond. And some of the mirror theorems predict that if I hand you one Calabi-Yau three-fold with a certain Hodge diamond, there should be somewhere out there in the universe another Calabi-Yau three-fold with another Hodge diamond. And if you flip across this diagonal axis, one is the Hodge diamonds should turn into the other Hodge diamond.

EL: Okay.

UW: So there is a mirror relationship there. And there is a really simple reflection there. But it's like you have to do a whole bunch of topology, and you have to do a whole bunch of geometry and you, like, convince yourself that Hodge diamonds are a thing. And then you have to somehow—like, once you've convinced yourself Hodge diamonds are a thing, you also have to convince yourself that you can go out there and find another space that has the right numbers in the diamond.

EL: So the mirror is, like, the very simplest thing about this. It’s this whole elaborate journey to get to the mirror.

UW: Yeah.

EL: Okay, interesting. I didn't actually know that that was where the mirror came from. So yeah. So can you tell us what these mirror theorems are here?

UW: Sure. So one version of it might be what I said, that given a Calabi-Yau manifold, with this information, that it has a mirror.

Or so then this diamond of information is telling you something about the way that the space changes. And there are different types of information that you could look at. You could look at how it changes algebraically, like if you wrote down an equation with some polynomials, and you changed those coefficients on the polynomials just a little bit, sort of how many different sorts of things, how many possible deformations of that sort could you have? That's one thing that you can measure using, like, one number in this diamond.

EL: Okay.

UW: And then you can also try to measure symplectic structure, which is a related more sort of physics-y information that happens over in a different part of the diamond. And so another type of mirror theorem, maybe a more precise type of mirror theorem, would say, okay, so these deformations measured by this Hodge number on this manifold are isomorphic in some sense to these other sorts of deformations measured by this other Hodge number on this other mirror manifold.

KK: Is there some trick for constructing these mirror manifolds if they exist?

UW: Yeah, there are. There are sort of recipes. And one of the games that people play with mirror symmetry is trying to figure out where the different recipes overlap, when you’ve, like, really found a new mirror construction, and when you’ve found just another way of looking at an old mirror construction. If I hand you one manifold, does it only have a unique mirror or does it have multiple mirrors?

KK: So my advisor tried to teach me Hodge theory once. And I can't even remember exactly what goes on, except there's some sort of bi-grading in the cohomology right?

UW: Right.

KK: And is that where this diamond shows up?

UW: Yeah, exactly. So you think back to when you first learned complex analysis, and there was, like, d/dz direction and there was the d/dz̅ direction.

KK: Right.

UW: And we're working in a setting where we can break up the cohomology really nicely and say, okay, these are the parts of my cohomology that come from a certain number of homomorphic d/dz kind of things. And these are the other pieces of cohomology that can be decomposed and look like, dz̅. And since it’s a Kähler manifold, everything fits together in a nice way.

KK: Right. Okay, there. That's all I needed to know, I think. That's it, you summarized it, you're done.

EL: So, I have a question. When you talk about like mirror theorems, I feel like some amount of mirror symmetry stuff is still conjectural—or “I feel like”—my brief perusal of Wikipedia on this indicates that there are some conjectures involved. And so how much of these theorems are that in different settings, these mirror relationships hold, and how much of them are small steps in this one big conjectural picture. Does that question make sense?

UW: Yeah. So I feel like we know a ton of stuff about Calabi-Yau three-folds that are realized in sort of the nice, natural ways that physicists first started writing down things about Calabi-Yau three-folds.

When you start getting more general on the mathematical side—for instance, there's a whole flavor of mirror symmetry that's called homological mirror symmetry that talks about derived categories and the Fuakaya category—a lot of that stuff has been very conjectural. And it's at the point where people are starting to write down specific theorems about specific classes of examples. And so that's maybe one of the most exciting parts of mirror symmetry right now.

And then there are also generalizations to broader classes of spaces, where it's not just Calabi-Yau three-folds where maybe you're allowing a more general kind of variety or relaxing things, or you're starting to look at, what if we went back to the physics language about potentials, instead of talking about actual geometric spaces? Those start having more conjectural flavor.

EL: Okay, so a lot of this is in the original thing, but then there are different settings where mirror symmetry might be taking place?

UW: Yeah.

EL: Okay. And I assume if you're such a connoisseur of mirror theorems, that this is related to your research also. What kinds of questions are you looking at in mirror symmetry?

UW: Yeah, so I spend some time just playing around with different mirror constructions and seeing if I can match them up, which is always a fun game, trying to see what you know. Lately, what I've been really excited about is taking the sort of classical old-fashioned hands-on mirror constructions where I can hand you a space, and I can take another space, and I can say these two things are mirror manifolds. And then seeing what knowing that tells me, maybe about number theory, about maybe doing something over a finite field in a setting that is less obviously geometry, but where maybe you can still exploit this idea that you have all of this extra structure that you know about because of the mirror and start trying to prove theorems that way.

EL: Oh, wow. I did not know there is this connection in number theory. This is like a whole new tunnel coming out here.

UW: Yeah, no, it's super awesome. We were able to make predictions about zeta functions of K3 surfaces. And in fact we have a theorem about a factor of as zeta function for Calabi-Yau manifolds of any dimensions. And it's a very specific kind of Calabi-Yau manifold, but it's so hard to prove anything about zeta functions! In part because if you're a connoisseur of zeta functions, you know they are controlled by the size of the cohomology, so once your cohomology starts getting really big, it’s really difficult to compute anything directly.

EL: So, like, how tangible are these? Like, here is a manifold and here is its mirror? Are there some manifolds you can really write down and, like, have a visual picture in your mind of what these things actually look like?

UW: Yeah, definitely. So I'm going to tell you about two mirror constructions. I think one of these is maybe more friendly to someone who likes geometry. And one of these is more friendly to someone who likes linear algebra.

EL: Okay.

UW: So the oldest, oldest mirror symmetry construction was, it's due to Greene and Plesser who were physicists. And they knew that they were looking for things with certain symmetries. So they took the diagonal quintic in projective four-space. I have to get to my dimensions right, because I actually often think about four dimensions instead of six.

So you're taking x5+y5+z5 plus, then, v5 and w5, because we ran out of letters, we had to loop around.

EL: Go back.

UW: Yeah. And you say, Okay, well, these are complex numbers, I could multiply any of them by a fifth route of unity, and I would have preserved my total space, right?

Except we're working in projective space, so I have to throw away one of my overall fifth roots of unity because if I multiply by the same fifth root of unity on every coordinate, that doesn't do anything. And then they wanted to maybe fit this into a family where they deformed by the product of all the variables. And if you want to have symmetries of that entire family, you should also make sure that the product of all of your roots of unity, I think multiplies to 1? So anyway, you throw out a couple of fifth roots of unity, because you have these other symmetries from your ambient space and things that you're interested in, and you end up with basically three fifths roots of unity that you can multiply by.

So I've got x5+y5+z5+v5+w5, and I'm modding out by z/5z3. Right? So I’m identifying all of these points in this space, right? I've just like got, like, 125 different things, and I’m shoving all these 125 different things together. So when I do that, this space—which was all nice and smooth and friendly, and it's named after Fermat, because Fermat was interested in equations like that—all of a sudden, I'd made it like, really stuck together and messy, and singular.

KK: Right.

UW: So I go in as a geometer, and I start blowing up, which is what algebraic geometers call this process of going in with your straw and your balloon, and blowing and smoothing out and making everything all nice and shiny again, right?

KK: Right.

UW: And when you do that, you've got a new space, and that's your mirror.

EL: Okay.

KK: So you blow up all the singularities?

UW: Yeah, your resolve the singularity.

KK: That’s a lot.

UW: Yeah. So what you had was, you had something which is floating around in P4. And because we picked a special example, it happens to have a lot of algebraic classes. But a thing in P4, the only algebraic piece you really know about it, in it, is, like, intersecting with a hyper plane.

So it has lots and lots of different ways you can vary all of its different complex parameters on only this one algebraic piece that you know about. And then when you go through this procedure, you end up with something which has very few algebraic ways to modify it. It actually naturally has only a one-parameter algebraic deformation space. But then there are all of these cool new classes that you know about, because you just blew up all of these things. So you're trading around the different types of information you have. You go from lots of deformations on one side to very few deformations on the other.

KK: Okay, so that was the geometry. What's the linear algebra one?

UW: Okay, so the linear algebra one is so much fun. Let's go back to that same space.

EL: I wish our listeners could see how big your smile is right now.

KK: That’s right. It’s really remarkable.

EL: It is truly amazing.

UW: Right. So we've got this polynomial, right, x5+y5+z5+w5+v5. And that thing I was telling you about finding the different fifth roots of unity that we could raise things to, that’s, like, a super tedious algebraic process, right, where you just sit down and you're like, gosh, I can raise different parts of the variables, like fifth roots of unity. And then I throw away some of my fifth roots of unity. So you start with that, the equation and the little algebraic rank that you want to get a group associated with it.

And then you convert your polynomial equation to a matrix. In this case, my matrix is just going to be like all fives down the diagonal.

KK: Okay.

UW: But you can do this more generally with other types of polynomials. The ones that work well for this procedure have all kinds of fancy names, like loops and chains of Fermat’s. So like Fermat’s is just the like different pure powers of variables. Loops would be if I did something like x5+y5+z5+…, and then I looped back around and used an x again.

EL: Okay.

UW: Or, sorry, it should have been like, x4y+y4z, and so on. So you can really see the looping about to happen.

And then chains are a similar thing. Anyway, so given one of these things, you can just read off the powers on your polynomial, and you can stick each one of those into a matrix. And then to get your mirror, you transpose the matrix.

EL: Oh, of course!

UW: And then you run this little crank, to tell you about an associated group.

EL: Okay.

UW: So getting which group goes with your transposed matrix, it's kind of a little bit more work. But I love the fact that you have this, like, huge, complicated physics thing with all this stuff, like the Hodge diamond, and then you're like, oh, and now we transpose a matrix! And, you know it’s a really great duality, right, because if you transpose the matrix again, you get back where you started.

KK: Sure.

EL: Right. Yeah. Well, and it seems like so many questions in math are, “How can we make this question into linear algebra?” It's just, like, one of the biggest hammers mathematicians have.

UW: Yeah.

EL: So another part of this podcast is that we ask our guest to pair their theorem, or in your case, you know, set of theorems, or flavor of theorems, with something. So what have you chosen as you're pairing?

UW: I decided that we should pair the mirror theorems with really fancy ramen.

EL: Okay. So yeah, tell me why.

UW: Okay. So really fancy ramen, like, the good Japanese-style, where you've simmered the broth down for hours and hours, and it's incredibly complex, not the kind that you just go buy in a packet, although that also has its use.

EL: Yeah, no, Top Ramen.

UW: Right. So it's complex. It has, like, a million different variants, right? You can get it with miso, you can get it spicy, you can put different things in it, you can decide whether you want an egg in it that gets a little bit melty or not, all of these different little choices that you get. And yeah, it seems like it's this really simple thing, it’s just noodle soup. And we all know what Top Ramen is. But there's so much more to it. The other reason is that I just personally, historically associate fancy ramen with mirror theorems. Because there was a special semester at the Fields Institute in Toronto, and Toronto has a bunch of amazing ramen. So a lot of the people who were there for that special semester grew to associate the whole thing with fancy ramen, to the point where one of my friends, who's an Italian mathematician, we were some other place in Canada, I think it was Ottawa, and she was like, “Well, why don't we just get ramen for lunch?” And I was like, “Sorry, it turns out that Canada is not a uniform source of amazing ramen.” That was special to Toronto.

KK: Yeah, Ottawa is more about the poutine, I think.

UW: Yeah, I mean, absolutely. There's great stuff in Ottawa. It just like, didn't have this beautiful ramen-mirror symmetry parents that we had all

EL: Right, I really liked this pairing. It works on multiple levels.

KK: Sure. It's personal, but it also works conceptually, it's really good. Yeah. Well, so how long have you been at Math Reviews?

UW: I think I'm in my third year.

KK: Okay.

KK: Do you enjoy it?

UW: I do. It’s a lot of fun.

KK: Is it a permanent gig? Or are these things time limited?

UW: Yeah, it's permanent. And in fact, we are hiring a number theorist. So if you know any number theorists out there who are really interested in, you know, precise editing of mathematics and reading about mathematics and cool stuff like that, tell them to look at our ad on Math Jobs. We're also hiring in analysis and math physics. And we've been hiring in combinatorics as well, although that was a faster hiring process.

EL: Yeah. And we also like to, you know, plug things that you're doing. I know, in addition to math, you have many other creative outlets, including some poetry, right, related to math?

UW: That’s right.

EL: Where can people find that? Ah, well, you can look at my website. Let's see, if you want the poetry you should look at my personal website, which is yarntheory.net.

There's one poem that was just up recently on JoAnne Growney’s blog.

EL: Yeah, that's right.

UW: And I have a poem that's coming out soon, soon, I’m not sure how soon in Journal of Humanistic Math. Yeah, it's a really goofy thing where I made up some form involving the group of units for the multiplicative group associated to the field of seven elements and then played around with that.

EL: I'm really, really looking forward to getting into that. Do you have a little bit of explanation of the mathematical structure in there?

UW: Just the very smallest. I mean, I think what I did was I listed, I found the generators of this group, and then I listed out where they would go as you generated them, and then I looked for the ones that seemed like they were repeating in an order that would make a cool poem structure.

EL: Okay, cool. Yeah. Well, thanks a lot for joining us. We'll be sure to share all that and hopefully people can find some of your work and enjoy it.

UW: Cool.

KK: Thanks, Ursula.

UW: Thanks so much for having me.[outro]

Episode 39 - Fawn Nguyen

28 mars 2019 | 29 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, a math podcast where we get mathematicians to tell us about theorems. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. It's Friday night.

EL: Yeah, yeah, I kind of did things in a weird order today. So there's this concert at the Utah Symphony that I wanted to go to, but I can't go tonight or tomorrow night, which are the only two nights they're doing it. But they had an open rehearsal today. So I went to a symphony concert this morning. And now I'm doing work tonight, so it's kind of a backwards day.

KK: Yeah. Well, I got up super early to meet with my financial advisor.

EL: Oh, aren’t you an adult?

KK: I do want to retire someday. I’ve got 20 years yet, but you know, now it's nighttime and my wife is watching Drag Race and I'm talking about math. So.

EL: Cool. Yeah.

KK: Life is good.

EL: Yes. Well, we're very happy today to have Fawn Nguyen on the show. Hi, Fawn. Could you tell us a little bit about yourself?

Fawn Nguyen: Hi, Evelyn. Hi, Kevin. I was was thinking, “How nerdy can we be? It’s Friday.”

So my name is Fawn Nguyen. I teach at Mesa Union School in Somis, California. And it's about 60 miles north of LA.

EL: Okay.

FN: 30 miles south of Santa Barbara. I teach—the school I'm at is a K-8, one-school district, of about 600 students. Of those 600, about 190 of them are in the junior high, 6-8, but it's a unique one-school district. So we're a family. It's nice. It's my 16th year there but 27th overall.

EL: Okay, and where did you teach before then?

FN: I was in Oregon. And I was actually a science teacher.

KK: Is your current place on the coast or a little more inland?

FN: Coast, yeah, about 10 miles from the coast. I think we have perfect weather, the best weather in the world. So.

KK: It’s beautiful there, it’s really—it’s hard to complain. Yeah.

FN: Yeah. It’s reflected in the mortgage, or rent.

EL: Yeah

KK: Right.

FN: Big time.

EL: So yeah, what theorem have you decided to share with us today?

FN: You mean, not everyone else chose the Pythagorean Theorem?

EL: It is a very good theorem.

FN: Yeah. I chose the Pythagorean Theorem. I have some reasons, actually five reasons.

KK: Five, good.

FN: I was thinking, yeah, that's a lot of reasons considering I don't have, you know, five reasons to do anything else! So I don't know, should I just talk about my reasons?

EL: Yeah, what’s your first reason?

FN: Jump right in?

EL: Well, actually, we should probably actually at least say what the Pythagorean theorem is. It's a very familiar theorem, and everyone should have heard about it in a middle school math class from a teacher as great as Fawn. But yeah, so could you summarize the theorem for us?

FN: Well, that's just it. Yeah, I chose it for one, it’s the first and most obvious reason, because I am at the middle school. And so this is a big one for us, if not the only one. And it's within my pay grade. I can wrap my head around this one. Yeah, it's one of our eighth grade Common Core standards. And the theorem states that when you have a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

KK: Right.

FN: How did I do, Kevin?

KK: Well, yeah, that's perfect. In fact, today I was—it just so happens that today, I was looking through the 1847 Oliver Byrne edition of Euclid’s Elements, this sort of very famous one with the pictures where the colors, shapes, and all of that, and I just I happened to look at that theorem, and the proof of it, which is really very nice.

FN: Yeah. So it being you know, the middle school one for us, and also, when I talk about my students doing this theorem—I just want to make sure that you understand that I no longer teach eighth grade, though. This is the first year actually at Mesa, and I've been there 16 years, that I do not get eighth graders. I'm teaching sixth graders. So when I refer to the lessons, I just want to make sure that you understand these are my former students.

EL: Okay.

FN: Yeah. And once upon a time, we tracked our eighth graders at Mesa. So we had a geometry class for the eighth graders. And so of course, we studied the Pythaogrean theorem then.

KK: So you have reasons.

FN: I have reasons. So that was the first reason, it’s a big one because, yeah. The second reason is there are so many proofs for this theorem, right? It's mainly algebraic or geometric proofs, but it's more than any other theorem. So it's very well known. And, you know, if you ever Google, you get plenty of different proofs. And I had to look this up, but there was a book published in 1940 that already had a 370 proofs in it.

KK: Yeah.

FN: Yeah. Even one of our presidents, I don't know if you know this, but yeah, this is some little nice trivia for the students.

EL: Yeah.

FN: One of our presidents, Garfield, submitted a proof back in 18-something.

EL: Yeah.

FN: He used trapezoids to do that.

KK: He was still in Congress at the time, I think the story is that, you know, that he was in the House of Representatives. And like, it was sort of slow on the floor that day, and he figured out this proof, right.

FN: Yeah. And then people continue to submit, and the latest one that I know was just over a year ago, back in November 2017, was submitted. That's the latest one I know. Maybe there was one just submitted two hours ago, who knows? And his was rearranging the a-squared and b-squared, the smaller squares, into a parallelogram. So I thought that was interesting. Yeah. And what's interesting is Pythagoras, even though it's the Pythagorean Theorem, he was given credit for it, it was a known long before him. And I guess there's evidence to suggest that it was by developed by a Hindu mathematician around 800 BC.

EL: Okay.

FN: And Pythagoras was what? 500-something.

KK: Something like that.

EL: Yeah.

FN: Yeah, something like that. But he was the first, I guess he got credit, because he was the first to submit a proof. He wasn't just talking about it, I guess it was official, it was a formal proof. And his was rearrangement. And I think that's a diagram that a lot of us see. And the kids see it. It's the one where you got the big, the big c square in the middle with the four right triangles around it, four congruent right triangles. Yeah. And then just by rearranging that big c squared became two smaller squares, your a squared and b squared. Yeah.

EL: Yeah. And I think it was known by, or—you know, I'm not a math historian. And I don't want to make up too much history today. But I think it has been known by a lot of different people, even as far back as Egyptians and Babylonians and things, but maybe not presented as a mathematical theorem, in the same kind of way that we might think about theorems now. But yeah, I think this is one of these things that like pretty much every human culture kind of comes up with, figuring out that this is true, this relationship.

KK: Yeah, I think recently wasn’t it? Or last year? There's this Babylonian tablet. And I remember seeing on Twitter or something, there was some controversy about someone claimed that this proved that the the Babylonians knew, all kinds of stuff, but really—

EL: Well, they definitely knew Pythagorean triples.

KK: Yeah, they knew lots of triples. Maybe you wrote about this, Evelyn.

EL: I did write about it, but we won’t derail it this way. We can put a link to that. I’ll get too bothered.

FN: Now that you brought up Pythagorean triples, how many do you know? How many of those can you get the kids to figure out? Of course, including Einstein submitted a proof. And I thought it was funny that people consider Einstein’s proof to be the most elegant. And I'm thinking, “Well, duh, it’s Einstein.” Yeah. And I guess I would have to agree, because there were a lot of rearrangements in the proofs, but Einstein, you know, is like, “Yeah, I don't need no stinking rearrangement.” So he stayed with the right triangle. And what he did was draw in the altitude from the 90 degree angle to the hypotenuse, and used similar triangles. And so there was no rearrangement. He simply made the one triangle into—by drawing in that altitude, he got himself three similar triangles. And yeah, and then he drew squares off of the hypotenuse of each one of those triangles, and then wrote, you know, just wrote up an equation. Okay, now we're just going to divide everything by a factor. The one that was drawn, in you just divide out the triangle, then you just you end up with a squared plus b squared equals c squared. It's hard to do without the the image of it, but yeah.

EL: But yeah, it is really a lovely one.

FN: Yeah. And this is something I didn't know. And it was interesting. I didn't know this until I was teaching it to my eighth graders. And I learned that, I mean, normally, we just see those squares coming off of the right triangle. And then I guess one of the high school students—we were using Geometer’s Sketchpad at the time, and one of the high school students made an animated sketch of the Pythagorean theorem. And, you know, he was literally drawing Harry Potter coming off of the three sides. And you know, and I just, Oh, I said, yeah, yeah. You don't have to have squares as long as they’re similar figures, right, coming off the edges. That would be fine. So that was fun to do. Yeah. So I have my kids just draw circles so that they—just anything but a square coming off of the sides, you know, do other stuff.

EL: Well, now I’m trying to—my brain just went to Harry Potter-nuse.

KK: Boo.

EL: I’m sorry, I’m sorry.

FN: That’s a good one.

KK: So I have forgotten when I actually learned this in life. You know, it's one of those things that you internalize so much that you can't remember what stage of your education you actually learned it in. So this is this is an actual Common Core eighth grade standard?

FN: Yes, yes. In the eighth grade, yeah.

KK: I grew up before the Common Core, so I don't really remember when we learned this.

FN: I don't know. Yeah, prior to Common Core, I was teaching it in geometry. And I don't think it was—it wasn't in algebra, you know, prior to these things we had algebra and then geometry. So yeah.

My third reason—I’m actually keeping track, so that was the second, lots of proofs. So the third reason I love the Pythagorean theorem is one fine day it led me to ask one of the best questions I'd asked of my geometry students. I said to them, “I wonder if you know how to graph and irrational number on the number line.” I mean, the current eighth grade math standard is for the kids to approximate where an irrational numbers is on the number line. That's the extent of the standard. So I went further and just asked my kids to locate it exactly. You know, what the heck?

EL: Nice. Yeah.

FN: And I actually wrote a blog post about it, because it was one of those magical lessons where you didn't want the class to end. And so I titled the post with “The question was mine, but the answer was all his.” And so I just threw it out to the class, I began with just, “Hey, where can we find—how do you construct the square root of seven on the number line?” And so, you know, they did the usual struggle and just playing around with it, but one of the kids towards the end of class, he got it, he came up with a solution. And I think when I saw it, and heard him explain it, it made me tear up, because it's like, so beautiful. And I'm so glad I did, because it was not, you know, a standard at all. And it was just something at the spur of the moment. I wanted to know, because we'd been working a lot with the Pythagorean theorem. And, yeah, so what he did was he drew two concentric circles, one with radius three and one with radius four on the coordinate plane, and the center is at (0,0). If you can imagine two concentric circles. And then he drew in y=-3, a line y=-3. And then you drew a line perpendicular to that line, that horizontal line, so that it intersects the—right, perpendicular to the horizontal line at negative three. And it intersects the larger circle, the one with radius four.

KK: Yeah, okay.

FN: So eventually what he did was he created, yeah, so you would have a right triangle created with one of the corners at (0,0). And the triangle would have legs of—the hypotenuse would be, what, four, the hypotenuse is four. One of the legs is three, and the other leg must be √7.

EL: Oh.

KK: Oh, yeah, okay.

FN: Yeah, yeah. So it's just so beautiful.

EL: That is very clever!

FN: Yeah, it really was. So every time I think about the Pythagorean theorem, I think back on that lesson. The kids really tried. And then from √7, we tried other routes. And we had a great time and continued to the next day.

EL: Oh, nice. I really liked that. That brought a big smile to my face.

KK: Yeah.

FN: The fourth reason I love the Pythagorean theorem is it always makes me think of Fermat’s Last Theorem. You know, it looks familiar, similar enough, where it states that no three positive integers a, b, and c can satisfy the equation of an+bn=cn. So for any integer value of n greater than 2. So it works for the Pythagorean Theorem, but any integer, any exponent greater than two would work. So I love—whenever I can, I love the history of mathematics, and whatever, I try to bring that in with the kids. So I read the book on the Fermat’s Last Theorem, and I kind of bring it up into the students for them to realize, Oh, my gosh, this man, Andrew Wiles, who solved it—and it's, you know, it's an over 300 year old theorem. And yeah, for him to first learn about the theorem when he was 10, and then to spend his life devoted to it. I mean, I can't think of a more beautiful love story than that. And yeah, so bring that to the kids. And I actually showed them the first 10 minutes of the documentary by BBC on Andrew Wiles. And just right, when he tears up, and, you know, I cannot stop tearing up at the same time because it—I don't know, it's just, it's that kind of dedication and perseverance. It's magical, and it's what mathematicians do. And so, you know, hopefully that supports all this productive struggle, and just for the love for mathematics. So, kind of get all geeky on the kids.

EL: Yeah, that is a lovely documentary.

FN: Yeah. Yeah. It's beautiful. All right. My fifth and final reason for loving this theorem is Pythagoras himself. What a nut!

EL: Yeah, I was about to say, he was one weird dude.

FN: Yeah, yeah. So, I mean, he was a mathematician and philosopher, astronomer and who knows what else. And the whole mystery wrapped up in the Pythagorean school, right? He has all these students, devotees. I don't know, it's like a cult! It really is like a cult because they had a strict diet, their clothing, their behaviors are a certain way. They couldn't eat meat or beans, I heard.

EL: Yeah.

FN: Yeah. And something about farting. And they believed that each time you pass gas that part of your soul is gone.

EL: That’s pretty dire for a lot of us, I think.

FN: Yeah. And what's remarkable also was that, the very theorem that he's named, you know, that's where I guess one of his students—I don't remember his name—but apparently he discovered, you know, the hypotenuse of √2 on the simple 1x1 isosceles triangle, and √2 and what did that do to him. The story goes he was thrown overboard for speaking up. He said, hey, there might be this possibility. So that's always fun, right? Death and mathematics, right?

EL: Dire consequences. Give your students a good gory story to go with it.

FN: I always like that. Yeah. But it's the start of irrational numbers. And of course, the Greek geometry—that mathematics is continuous and not as discrete as they had thought.

EL: Well, and it is an interesting irony, then that the Pythagoras theorem is one really easy way to generate examples of irrational numbers, where you find rational sides and a whole lot of them give you irrational hypotenuses.

FN: Yes.

EL: And then, you know, this theorem is the downfall of this idea that all numbers must be rational.

FN: Right. And I mean, the whole cult, I mean, that revelation just completely, you know, turned their their belief upside down, turned the mathematical world at that time upside down. It jeopardized and just humiliated their thinking and their entire belief system. So I can just imagine at that time what that did. So I don't know if any modern story that has that kind of equivalent.

EL: Yeah, no one really based their religion on Fermat’s Last Theorem being untrue. Or something like this.

FN: Right, right. Exactly.

EL: Yeah. I like all of your reasons. And you've touched on some really great—like, I will definitely share some links to some of those proofs of the Pythagorean theorem you mentioned.

So another part of this podcast is that we ask our guests to pair their theorem with something. So what have you chosen for your pairing?

FN: I chose football.

KK; Okay, all right.

FN: I chose football. It's my love. I love all things football. And the reason I chose football is simply because of this one video. And I don't know if you've seen it. I don't know if anyone's mentioned it. But I think a lot of geometry teachers may have shown it. It's by Benjamin Watson doing a touchdown-saving tackle. So again, his name is Benjamin Watson. I don't know how many years ago this was, but he’s a tight end for the New England Patriots. So what happened was, he came out of nowhere. Well, there was an interception. So he came out of nowhere to stop a potential pick-six at the one-yard line.

EL: Oh, wow.

FN: I mean, it's the most beautiful thing! So yeah, so if you look at that clip, even the coach say something to remember for the rest—anybody who sees it, for the rest of their life just because he never gave up, obviously. But you know, the whole point is he ran the diagonal of the field is what happened.

KK: Okay.

EL: Yeah, so you’ve got the hypotenuse.

FN: You’ve got the hypotenuse going. The shortest distance is still that straight line, and he never gave up. Oh, I mean, this guy ran the whole way, 90 yards, whatever he needed from from the very one end to the other. No one saw Ben Watson coming, just because, we say literally out of nowhere. Didn’t expect it. And the camera, what's cool is, you know, the camera is just watching the runner, right, just following the runner. And so the camera didn’t see it until later. Later when they did film, yeah, they zoomed out and said, Oh, my God, that's where he was coming from, the other hypotenuse, I mean, the other end of the hypotenuse. Yeah. But I pair everything, every mathematics activity I do, I try to pair it with a nice Cabernet. How's that?

EL: Okay.

KK: Not during school, I hope.

FN: Absolutely not.

EL: Don’t share it with your students.

FN: I’m a one glass drinker anyway, I'm a very, very lightweight. I talk about drinking, but I'm a wuss, Asian flush. Yeah.

EL: Yeah. Well, so, I’m not really a football person. But my husband is a Patriots fan. And I must admit, I'm a little disappointed that you picked an example with the Patriots because he already has a big enough head about how good the Patriots are, and I take a lot of joy in them not doing well, which unfortunately doesn’t happen very much these days.

KK: Never happens.

EL: There are certain recent Super Bowls that I am not allowed to talk about.

FN: Oh, okay.

KK: I can think of one in particular.

EL: There are few. But I'll say no more. And now I'm just going to say it on this podcast that will be publicly available, and I'll instruct him not to listen to this episode.

FN: Yeah, now my new favorite team actually, pro—well, college is Ducks, of course, but pro would be Dallas Cowboys. Just because that’s the favorite team from my fiance. So we actually, yeah, for Christmas, this past Christmas, I gave him that gift. We flew to Dallas to watch a Cowboys game.

EL: Oh, wow. We might have been in Dallas around the same time. So I grew up in Dallas.

FN: Oh!

EL: And so if I were a football fan, that would be my team because I definitely have a strong association with Dallas Cowboys and my dad being in a good mood.

FN: There we go.

EL: And I grew up in in the Troy Aikman era, so luckily the Cowboys did well a lot.

FN: Well, they’re doing well this year, too. So this Saturday, big game, right? Is it? Yeah.

KK: I feel old. So when I was growing up, I used to, I loved pro football growing up, and I've sort of lost interest now. But growing up in the ‘70s, it was either you're a Cowboys fan or a Steelers fan. That was the big rivalry.

FN: Yeah.

KK: I was not a Cowboys fan, I’m sorry to say.

FN: I never was either until recently.

KK: I was born in Wisconsin, and my mother grew up there, so I’m contractually obligated to be a Green Bay fan. I mean, I’m not allowed to do anything else.

EL: Well, it's very good hearted, big hearted of you, Fawn, to support your fiance's team. I admire that. I, unfortunately, I'm not that good a person.

FN: I definitely benefit because yeah, the stadium. What an experience at AT&T Stadium. Amazing.

EL: Yeah, it is quite something. We went to a game for my late grandfather's birthday a few years before he passed away. My cousins, my husband and I, my dad and uncle a ton of people went to a game there. And that was our first game at that stadium. And yeah, that is quite an experience. I just, I don't even understand—like the screen that they've got so you can watch the game bigger than the game is like the biggest screen I've ever seen in my life. I don't even understand how it works.

FN: Same here, it’s huge. And yet somehow the camera, when you watch the game on television, that screen’s not there, and then you realize that it's really high up. Yeah.

KK: Cool. Well, we learned some stuff, right?

EL: Yeah.

KK: And this has been great fun.

EL: Yeah, we want to make sure to plug your stuff. So Fawn is active on Twitter. You can find her at—what is your handle?

FN: fawnpnguyen. So my middle initial, Fawn P Nguyen.

EL: And Nguyen is spelled N-G-U-Y-E-N?

FN: Very good. Yes.

EL: Okay. And you also have a blog? What's the title of that?

FN: fawnnguyen.com. It’s very original.

EL: But it's just lovely. You're writing on there is so lovely. And it, yeah, it's just such a human picture. Like you really, when you read that you really see the feeling you have for your students and everything, and it's really beautiful.

FN: Thank you. They are my love. And I just want to say, Evelyn, when you asked me to do this, I was freaking out, like oh my god, Evelyn the math queen. I mean, I was thinking God, can you ask me do something else like washing the windows? Make you some pho?

KK: Wait, we could have had pho?

FN: We could have had pho. Because this was terrifying. But you know, it's a joy. Pythagorean theorem, I can take on this one. Because it's just so much fun. I mean, I've been in the classroom for a long time, but I don't see myself leaving it anytime soon because yeah, I don't know what else I would be doing because this is my love. My love is to be with the kids.

KK: Well, bless you. It's hard work. My sister in law teaches eighth grade math in suburban Atlanta, and I know how hard she works, too. It's really—

FN: We’re really saints, I mean—

KK: You are. It’s a real challenge. And middle school especially, because, you know, the material is difficult enough, and then you're dealing with all these raging hormones. And it's really, it's a challenge.

EL: Well, thanks so much for joining us. I really enjoyed it.

KK: Thanks, Fawn.

FN: Thank you so much for asking me. It was a pleasure. Thank you so much.

[outro]

Episode 38 - Robert Ghrist

14 mars 2019 | 24 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast that starts with math and goes all kinds of weird places. I'm Kevin Knudson, professor of mathematics at the University of Florida and here is your other host.

Evelyn Lamb: Hi. I'm Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. Happy New Year, Kevin.

KK: And to you too, Evelyn. So this will happen, this will get out there in the public later. But today is January 1, 2019.

EL: Yes, it is.

KK: And Evelyn tells me that it was it's cold in Utah, and I have my air conditioning on.

EL: Yes.

KK: That seems about right.

EL: Yeah. Our high is supposed to be 20 today. And the low is 6 or 7. So we really, really don't have the air conditioner on.

KK: Yeah, it's going to be 82 here today in Gainesville. [For our listeners outside of the USA: temperatures are in Fahrenheit. Kevin does not live in a thermal vent.] I have flip flops on.

EL: Yeah. Yeah, I’m a little jealous.

KK: This is when it's not so bad to live in Florida, I’ve got to say.

EL: Yeah.

KK: Anyway, Well, today, we are pleased to welcome Robert Ghrist. Rob, you want to introduce yourself?

RG: Hello, this is Robert Ghrist.

KK: And say something about yourself a little, like who you are.

RG: Okay. All right, so I am a professor of mathematics and electrical and systems engineering at the University of Pennsylvania. This is in Philadelphia, Pennsylvania. I've been in this position at this wonderful school for a decade now. Previous to that I had tenured positions at University of Illinois in Urbana Champaign and Georgia Institute of Technology.

KK: So you've been around.

RG: A little bit.

EL: Oh, and so I was just wondering, so it that a joint appointment between two different departments, or is it all in the math department?

RG: This is a split appointment, not only between two different departments, but between two different schools.

EL: Oh, wow.

RG: The math appointment is in the School of Arts and Sciences, and the engineering appointment is in the School of Engineering. This is kind of a tricky sort of position to work out. This is one of the things that I love about the University of Pennsylvania is there are very low walls between the disciplines, and a sort of creative position like this is is very workable. And I love that.

KK: Yeah, and your undergraduate degree was actually in engineering, right?

RG: That’s correct I got turned on to math by my calculus professor, a swell guy by the name of Henry Wente, a geometer.

KK: Excellent. Well, I see you’re continuing the tradition. Well, we’ll talk about that later. And actually, you actually have an endowed chair name for someone rather famous, right?

RG: That’s true. The full title is The Andrea Mitchell PIK professor of mathematics, electrical and systems engineering. This is Andrea Mitchell from NBC News. She and her husband Alan Greenspan funded this position. Did not intend to hire me specifically, or a mathematician. I think she was rather surprised when the chair that she endowed wound up going to a mathematician, but there it is, we get along swell. She's great.

KK: Nice.

EL: Nice. That's really interesting.

KK: Yup. So, Rob, what is your favorite theorem?

RG: My favorite theorem is, I don't know. I don't know the name. I don't know that this theorem has a name, but I love this theorem. I use this theorem all the time in all the classes I teach, it seems. It's a, it's a funny thing about basically Taylor expansion, or Taylor series, but in an operator theoretic language. And the theorem, roughly speaking, goes like this: if you take the differentiation operator on functions, let's say just single input, single output functions, the kind of things you do in basic calculus class. Call the differentiation operator D. Consider the relationship between that operator and the shift operator. I’m going to call the shift operator E. This is the operator that takes a function f, and then just shifts the input by 1. So E(f(x)) is really— pardon me, E(x) is really f evaluated at x+1. We use shift—

EL: I need a pencil and paper.

RG: Yeah, I know, right? We use the shift operator all the time in signal processing, in all kinds of things in both mathematics and engineering. And here's the theorem, here's the theorem. There's this wonderful relationship between these two operators. And it's the following. If you exponentiate the differentiation operator, if you take e to the D, you get the shift operator.

KK: This is remarkable.

RG: What does this mean? What does this mean?

KK: Yeah, what does it mean? I actually did work this out once. So what our listeners don't know is that you and I actually had this conversation once in a bar in Philadelphia, and the audio quality was so bad, we're having to redo this.

EL: Yeah.

KK: So I went home, and I worked this out. And it's true, it does work out. But what does this mean, Rob? Sort of, you know, in a manifestation physically?

RG: Yeah, so let me back up. The first question that I ask students, when they show up in calculus class at my university is, is, what is e to the x? What does that even mean? What does that mean when x is an irrational number, or an imaginary number, or something like a square matrix, or an operator? And, of course, that takes us back to the the interpretation of exponentiation in terms of the Taylor series at zero, that I take that infinite series, and I use that to define what exponentiation means. And because things like operators, things like differentiations, or shifts, you can take powers of those by composition, by iteration, and you can rescale them. Then you can exponentiate them.

So I can talk about what it means to exponentiate the differentiation operator by taking first D to the 0, which of course, is the identity, the do-nothing operator, and then adding to it D, and then adding to that D squared divided by 2 factorial [n factorial is the product of the integers 1x2,…n], that's the second derivative, then D cubed divided by 3 factorial, that's the third derivative. If I can keep going, I’ve exponentiated the differentiation operator. And the theorem is that this is the shift operator in disguise. And the proof is one line. It's Taylor expansion. And there you go.

Now, this isn't your typical sort of my favorite theorem, in that I haven't listed all the hypotheses. I haven't been careful with anything at all. But one of the reasons that this is my favorite theorem is because it's so useful when I'm teaching calculus to students, when I'm teaching basic dynamical systems to students where, you know, in a more advanced class, yeah, we'd have a lot of hypotheses. And, oh, let's be careful. But when you're first starting out, first trying to figure out what is differentiation, what is exponentiation, this is a great little theorem.

EL: Yeah, this conceptual trip going between the Taylor series, or going between the idea of e to the x, or 2 to the x or something where we really have a, you know, a fairly good grasp of what exponentiation means in that case, if we, if we're talking about squares or something like that, and going then to the Taylor series, this very formal thing, I think that's a really hard conceptual shift. I know that was really hard for me.

RG: Agreed.

KK: Yeah. So I, I wonder, though, I mean, so what's a good application of this theorem, like, in a dynamics class, for example? Where does this pop up sort of naturally? And I can see that it works. And I also agree that this idea of—I start calculus there, too, by the way, when I say, you know, what does e to the .1 mean, what does that even mean?

RG: What does that even mean?

KK; Yeah, and that’s a good question that students have never really thought about. They’re just used to punching .1 into a calculator and hitting the e to the x key and calling it a day. So, but where would this actually show up in practice? Do you have a good example?

RG: Right. So when I teach dynamical systems, it's almost exclusively to engineering students. And they're really interested in getting to the practical applications, which is a great way to sneak in a bunch of interesting mathematics and really give them some good mathematics education. When doing dynamical systems from an applied point of view, stability is one of the most important things that you care about. And one of the big ideas that one has to ingest is that of stability criteria for, let's say, equilibria in a dynamical system. Now, there are two types of dynamical systems that people care about, depending on what notion of time you're using— continuous time or discrete time. Most books on the subject are written for one or the other type of system. I like to teach them both at once, but one of the challenges of doing that is that the stability criteria are are different, very different-looking. In continuous time, what characterizes a stable equilibrium is when you look at all of the eigenvalues of the linearization, the real parts are less than zero. When you move to a discrete time dynamical system, that is a mapping, then again, you're looking at eigenvalues of the linearization, but now you want the modulus to be less than 1. And I find that students always struggle with “Why is it different?” “Why is it this way here, and that way there?” And of course, of course, the reason is my favorite little theorem, because if I look at the evolution operator in continuous-time dynamics—that’s the derivative—versus the evolution operator in discrete-time dynamics—that is the shift, move forward one step in time—then if I want to know the relationship between the stability and, pardon me, the stable and unstable regions, it is exponentiation. If I exponentiate the left hand side of the complex plane, what do I get? I get the region in the plane with modulus less than 1.

KK: Right.

RG: I find that students have a real “aha” moment when they see that relationship, and when they can connect it to the relationship between the evolution operators.

EL: I’m having an “aha” moment about this right now, too. This isn't something I had really thought about before. So yeah, this is a really neat observation or theorem.

RG: Yeah, I never really see this written down in books.

KK: That’s—clearly now you should write a book.

RG: Another one?

KK: Well, we'll talk about how you spend your time in a little while here. But, no, Rob, I mean, so Rob has this—I don't know if it's famous, but well known—massive open online course through Coursera where he does calculus, and it's spectacular. If our listeners haven't seen it, is it on YouTube, Rob? Can you actually get it at YouTube now?

RG: Yes, yes. The the University of Pennsylvania has all the lectures posted on a YouTube channel.

KK: Well, I actually downloaded to my machine. I took the MOOC A few years ago, just for fun. And I passed! Remarkably.

RG: With flying colors, with flying colors.

KK: Yeah, I'm sure you graded my exam personally, Rob.

RG: Personally.

KK: And anyway, this is evidence for how lucky are students are, I think. Because, you know, you put so much time into this, and these these little “aha” moments. And the MOOC is full of these things. Just really remarkable stuff, especially that last chapter, which is so next level, the digital calculus stuff, which sort of reminds me of what we're talking about. Is there some connection there?

RG: Oh yes, it was creating that portion of the MOOC that really, really got me to do a deep dive into discrete analogs of continuous calculus, looking at the falling powers notation that is popular in computer science in Knuth’s work and others, thinking in terms of operators. Yeah, that portion of the MOOC really got me thinking a lot about these sorts of things.

KK: Yeah, I really can't recommend this highly enough. It’s really great.

EL: Yeah, so I have not had the benefit of this MOOC yet. So digital calculus, is that meaning, like, calculus for computers? Or what exactly is that? What does that mean?

RG: One of the things that I found students really got confused about in a basic single variable calculus class is, as soon as you hit sequences and series, their heads just explode because they get sequences and series confused with one another, and it all seems unmotivated. And why are we bothering with all these convergence tests? And where’d they come from? All this sort of thing.

EL: And why is it in a calculus class?

RG: Why is it even in a calculus class after all these derivatives and integrals? So the way that I teach it is when we get to sequences and series, you know, in the last quarter of the semester, I say, Okay, we've done calculus for functions with an analog input and an analog output. Now, we want to redo calculus for functions with a digital input and an analog output. And such functions we're going to call sequences. But I'm really just going to think of it as a function. How would you differentiate such a thing? How would you integrate such a thing? That leads one to think about finite differences, which leads to some some nice approaches to numerical methods. That leads one to looking at sums and numerical integration. And when you get to improper integrals over an unbounded domain? Well, that's series, and convergence tests matter.

KK: Yeah, it's super interesting. We will provide links to this. We’ll find the YouTube links and provide them.

EL: Yeah.

KK: So another fun part of this podcast, Rob, is that we ask our guests to pair their theorem with something, and I assume you're going to go with the same pairing from our conversation back in Philadelphia.

RG: Oh yes, that’s right.

KK: What is it?

RG: My work is fueled by a certain liquid beverage.

KK: Yeah.

RG: It’s not wine. It's not beer. It's not whiskey. It's not even coffee, although I drink a whole lot of coffee. What really gets me through to that next-level math is Monster. That's right. Monster Energy Drink, low carb if you please, because sugar is not so good for you. Monster, on the other hand, is pretty great for me, at any rate. I do not recommend it for people who are pregnant or have health problems, problems with hearts, anything like this, people under the age of 18, etc, etc. But for me, yeah, Monster.

KK: Yeah. There's lots of empties in your office, too like, up on the shelf there, which I'm sure have some significance.

RG: The wall of shame, that’s right. All those empty monster cans.

KK: See, I can't get into the energy drinks. I don't know. I mean, I know you're also fond of scotch. But does that does that help bring you down from the Monster, or is it…

RG: That’s a rare treat. That's a rare treat.

KK: Yeah, it should be. So when did your obsession with Monster start? Does this go back to grad school, or did it even exist when we were in grad school? Rob and I are roughly the same age. Were energy drinks a thing when we were in grad school? I don't remember.

RG: No, no. I didn't have them until, gosh, what is it, sometime within the past decade? I think it was when I was first working on that old calculus MOOC, like, what was that, six years ago? Six, seven years ago, is when I was doing that.

KK: Yup.

RG: That was difficult. That was difficult work. I had to make a lot of videos in a short amount of time. And, yup, the Monster was great. I would love to get some corporate sponsorship from them. You know, maybe, maybe try to pitch extreme math? I don't know. I don't think that's going to work.

KK: I don't know. I think it's a good angle, right? I mean, you know, they have this monster truck business, right? So there is this sort of whole extreme sports kind of thing. So why not? You know?

EL: Yeah, I'm sure they're just looking for a math podcast to sponsor. That's definitely next on their branding strategy.

KK: That’s right. Yeah. But not us. They should sponsor you, Rob. Because you're the true consumer.

RG: You know, fortune favors the bold. I'd be willing to hold up a can and say, “If you're not drinking Monster, you're only proving lemmas,” or something like that.

EL: You’ve thought this through. You've got their pitch already, or their slogan already made.

RG: That’s right. Yup.

KK: All right. Excellent. So we always like to give our guests a chance to to pitch their projects. Would you like to tell us about Calculus Blue?

RG: Oh, absolutely! This is—the thing that I am currently working on is a set of videos for multivariable calculus. I'm viewing this as something like a video text, a v-text instead of an e-text, where I have a bunch of videos explaining topics in multivariable calculus that are arranged in chapters. They’re broken up into small chunks, you know, roughly five minutes per video. These are up on my YouTube channel.There's another, I don't know, five or six hours worth of videos that are going to drop some time in the next week covering multivariate integration. This is a lot of fun. I'm having a ton of fun doing some 3d drawing, 3d animation. Multivariable calculus is just great for that kind of visualization. This semester, I'm going to use the videos to teach multivariable calculus at Penn in a flipped manner and experiment with how well that works. And then it'll be available for anyone to use.

KK; Yeah, I'm looking forward to these. I see the previews on Twitter, and they really are spectacular. How long does any one of those videos take you? It seems like, I mean, I know you've gotten really good at at the graphics packages that you need to create those things. But, you know, like a 10 minute video. How long does one of those things take to produce?

RG: I don't even want to say,

KK: Okay.

RG: I do not even want to say, no. I've been up since four o'clock this morning rendering video and compositing. Yeah, this is my day, pretty much. It's not easy. But it is worthwhile. Yeah.

KK: Well, I agree. I mean, I think, you know, so many of our colleagues, I think, kind of view calculus as this drudgery. But I still love teaching it. And I know you do, too.

RG: Absolutely.

KK: And I think it's important, because this is really a lot of what our job is, as academics, as professional mathematicians. Yes, we're proving theorems, all that stuff, that's great. But, you know, day in, day out, we're teaching undergraduates introductory mathematics. That's a lot of what we do. And I think it's really important to do it well.

EL: Well, and it can help, you know, bring people into math like it did for Rob.

KK: That’s right.

RG: Exactly. That's exactly right. Controversial opinion, but, you know, you get these people out there who say, oh, calculus, this is outdated, we don't need that anymore, just teach people data analysis or statistics. I think that's a colossal error. And that it's possible to take all of these classical ideas in calculus and just make them current, make them relevant, connect them to modern applications, and really reinvigorate the subject that you need to have a strong foundation in in order to proceed.

KK: Absolutely. And I, you know, I try to mix the two, I try to bring data into calculus and say, you know, look, engineering students, you’re mostly going to have data, but this stuff still applies. You know, calculus for me is a lot about approximation, right? That's what the whole Taylor Series business, that's what it's for.

RG: Definitely.

KK: And really trying to get students to understand that is one of my main goals. Well, this has been great fun. Thanks for taking time out from rendering video.

EL: Yeah, video rendering.

RG: Yes. I'm going to turn around and go right back to rendering as soon as we're done.

KK: That’s right, you basically have a professional quality studio in your in your basement, right?Is this how this works?

RG: This is how it works. Been renovating, oh, I don't know. It's about a year ago I started renovations and got a nice little studio up and running.

KK: Excellent. Do you have, like, foam on the walls and stuff like that?

RG: Yes, I'm touching the foam right now.

KK: All right. Yeah. So Evelyn I aren't that high-tech. We've just now gotten to the sort of like, multi-channel recording kind of thing.

RG: Ooh.

KK: Well, yeah, well, we're doing this now, right, where we’re each recording our own audio. I'm pleased with the results so far. Well, Rob, thanks again, and we appreciate your joining.

EL: Thanks for joining us.

RG: Thank you. It's been a pleasure chatting.

Episode 37 - Cynthia Flores

28 februari 2019 | 25 min

Evelyn Lamb: Hello and welcome to My Favorite Theorem, a podcast where we ask mathematicians to tell us about their favorite theorems. I'm Evelyn Lamb. I'm one of your hosts. I am a freelance math and science writer in Salt Lake City, Utah. Here's your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going, Evelyn?

All right. I am making some bread right now, and it smells really great in my house. So slightly torturous because I won't be able to eat it for a while.

KK: Sure. So I make my own pizza dough. But I always stop at bread. I never that extra step. I don't. I don't know why. Are you making baguettes? Are you doing the whole...

EL: No. I do it in the bread machine.

KK: Oh.

EL: Because I'm not going to make, yeah, I'm not going to knead and shape a loaf. So that's the compromise.

KK: Oh. That that's the fun part. So I've had a sourdough starter for, the same one for at least three or four years now. I've kept this thing going. And I make my own pizza crust, but I'm just lazy with bread. I don't eat a lot of bread.

EL: So yeah. And joining us to talk about--on BreadCast today--I'm very delighted to welcome Cynthia Flores. So hi. Tell us a little bit about yourself.

Cynthia Flores. Oh, thanks for having me on the show. I'm so grateful to join you today, Kevin and Evelyn. Well, I'm an assistant professor of mathematics and applied physics at California State University Channel Islands in the Department of Math and Physics. The main campus is not located on the Channel Islands.

EL: Oh, that's a bummer.

CF: It's actually located in Camarillo, California. It's one hour south of Santa Barbara, one hour north of downtown Los Angeles, roughly.

But the math department does get to have an annual research retreat at the research station located in the Santa Rosa Island. So that's kind of neat.

KK: Oh, how terrible for you.

EL: Yeah. That must be so beautiful.

KK: I was in Laguna Beach about a week and a half ago, which is, of course, further south from there, but still just spectacularly beautiful. Really nice.

CF: Yeah, I feel really fortunate to have the opportunity to stay in the Southern California area. I did my PhD at UC Santa Barbara, where I studied the intersections of mathematical physics, partial differential equations, and harmonic analysis and has motivated what I'm going to talk about today.

KK: Good, good, good.

EL: Yeah. Well, and Cynthia was on another podcast I host, the Lathisms podcast. And I really enjoyed talking with her then about the some of the research that she does. And she had some fun stories. So yeah, what is your favorite theorem? What do you want to share with us today?

CF: I'm glad you asked. I have several favorite theorems, and it was really hard to pick, and my students have heard me say repeatedly that my favorite theorem is the fundamental theorem of calculus.

EL: Great theorem.

KK: Sure.

CF: It's also a very, I find, intimidating theorem to talk about on this series, especially with so many creative individuals pairing their favorite theorems with awesome foods and activities. And so I just thought that one was maybe something to live up to. And I wanted to start with something that's a little closer to to my research area. So I found myself thinking of other favorites, and there was one in particular that does happen to lie at the intersection of my research area, which is mathematical physics, PDEs and harmonic analysis. And it's known as Heisenberg's Uncertainty Principle. That's how it's really known by the physics community. And in mathematics, it's most often referred to as Heisenberg's Uncertainty Inequality.

EL: Okay.

CF: So, is it familiar? I don't know.

EL: I feel like I've heard of it, but I don't--I feel like I've only heard of it from kind of the pop science point of view, not from the more technical point of view. So I'm very excited to learn more about it.

KK: So I actually have a story here. I taught a course in mathematics and literature a couple years back with a friend of mine in the in the foreign languages department. And we watched A Serious Man, this Coen Brothers movie, which, if you haven't seen is really interesting. But anyway, one of the things I made sure to talk about was Heisenberg's Uncertainty Principle, because that's sort of one of the themes, and of course now I forgotten what the inequality is. But I mean, I remember it involves Planck's constant, and there's some probability distribution, so let's hear it.

CF: Mm hmm. Yeah, yeah. So I was, I was like, this is what I'm going to pair it with. Like, I'm going to pair the conversation, like the mathematics description, physical description, with, basically I was thinking of pairing it with something Netflix and chill-like. I'm really glad that you brought that up, and I'll tell you more in a little bit about what I'm pairing it with. But first, I'll start mathematically. Mathematically, the theorem could be stated as follows. Given a function with sufficient regularity and decay assumptions, the L2 norm of the function is less than or equal to 2 over the dimension the function's defined on, multiplied by the product of the L2 norm of its first moment and the L2 norm of its gradient. And so mathematically, that's the inequality.

This wasn't stated this way by Heisenberg in the 1920s, which I believe he was recognized with a Nobel Prize for later on. Physically, Heisenberg described this in different ways it could be understood. Uncertainty might be understood as the lack of knowledge of a quantity taken by an observer, for example, or to the uncertainty due to experimental inaccuracy, or ambiguity in some definition, or statistical spread, as, as Kevin mentioned.

And actually, I'm going to recommend to the listeners to go to YouTube. There's a YouTuber named Veritasium, I'm not sure if I'm pronouncing that correctly.

EL: Oh, yeah, yeah.

CF: Yeah, he has a four minute demonstration of the original thought by Heisenberg and an experiment having to do with lasers that basically tells us it's impossible to simultaneously measure the position and momentum of a particle with infinite precision. The infinite precision part would be referring to something that we might call certainty. So in the experiment, that the YouTuber is recreating, a laser is shown through two plates that form a slit, and the split is becoming narrower and narrower. The laser is shone through the slit and then projected onto a screen. And as the slit is made narrower, the spot on the screen, as expected, is also becoming narrower and narrower. And at some point--you know, Veritasium does a really good job of creating this sort of like little "what's going to happen" excitement-- just when the slit seems to completely disappear and become infinitesimally small, the expectation might be that the that the laser projecting onto the screen would disappear too, but actually at a certain point, when the slit is so narrow, it's about to close, the spot on the screen becomes wider. We see spread. And this is because the photons of light have become so localized that the slit and their horizontal momentum has to become less well defined. And this is a demonstration of Heisenberg's Uncertainty Principle. And so according to Heisenberg--and this is from one of his manuscripts, and I wish I would have written down which one, I'm just going to read it-- "at the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum, and this change is the greater the smaller the wavelength of the light employed, in other words, the more exact determination of the position. At the instant at which the position of the electron is known, its momentum, therefore, can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely the momentum is known, and conversely." And there's also, so momentum and position were sort of the original context in which Heisenberg's Uncertainty Principle was stated, mainly for quantum mechanics. The inequality theorem that I presented was really from an introductory book to nonlinear PDEs, which is really what I study, nonlinear dispersive PDEs specifically. So you could use a lot of Fourier transforms and stuff like that.

But it has multiple variations, one of which is Heisenberg's Uncertainty Principle of energy and time, which more or less is going to tell you the same thing, which is going to bring me to my pairing. Can I can I share my pairing?

KK: Sure.

CF: My pairing--I want to pair this with a Netflix and chill evening with friends who enjoy the Adult Swim animated show Rick and Morty.

KK: Okay.

CF: And an evening where you also have the opportunity to discuss sort of deep philosophical questions about uncertainty and chaos. And so really the show's on Hulu, so you could watch the show. I don't know if either one of you are familiar with the show.

KK: Oh, yeah. Yeah. I have a 19 year old son, how could I not be?

CF: And actually my students were the ones that brought this show and this specific episode to my attention, and I watched it, and so I'll see a little bit about the show. It's a bit, it's somewhat inspired by The Back to the Future movies. It's a comedy about total reckless and epic space adventures with lots of dark humor and a lot of almost real science. I mean, Rick is a mad scientist and Morty is in many ways opposite to Rick. Morty is Rick's grandson and sidekick. Rick uses a portal gun that he created. It allows him and Morty to travel to different realities where they go on some fun adventures. And there are several references to formulas and theorems describing our everyday life. And so particularly the season premiere of season two, Rick and Morty pays homage to Heisenberg's Uncertainty Principle as well as Schrodinger's cat paradox and includes a mathematical proof of Rick's impression of his grandkids.

I won't spoil what he proves about them. I'll let the reader, the listeners, go check that out. But basically season one ended with Rick freezing time for everyone except for him and his two grandkids Morty and Summer. In this premiere, Rick unfreezes time and causes a disturbance in their reference timeline, and any uncertainty introduced by the three individuals gets them removed entirely from time and causes a split in reality into multiple simultaneous realities. The entire episode is following Heisenberg's Uncertainty Principle for energy and time and alluding to the concept from the quantum world that chaos is found in the distribution of energy levels of certain atomic systems. So I'm going to back it up a little bit. We talked about Heisenberg's Uncertainty Principle in terms of momentum and position. Heisenberg's Uncertainty Principle for energy and time is for simultaneous measurements of energy and time, and specifically that the distribution of energy levels and uncertainty and its measurement is a metaphor to chaos within the system. So within a time interval, it's not possible to measure chaos precisely. There has to be uncertainty in the measure, so that the product of the uncertainty and energy and the uncertainty and the time remains larger than h over 4π, the Planck's constant. In other words, you cannot have both, you cannot simultaneously have both small uncertainty in both measurements. In other words, lots of certainty, right? You have small uncertainty, you have lots of certainty.

So you can't have that both happening. So less chaos leads to more uncertainty and vice versa. Less than certainty (or more certainty) leads to more chaos. And so this, you know, this episode, if you watch it, seems to present the common misconception that more uncertainty leads to more chaos. And this is where I've thought about this really hard and even tried to find someone who put it nicely, maybe on a video, I couldn't. But I think--this is just my opinion--I think the writers really got it right on this episode, because the moment that the timeline merges in the episode is the moment when the main character Rick has given up on his chances of fixing a broken tool he was counting on for fixing the timeline. So in fact, in this episode, he's shown doing something which is unlike him. He's shown praying and asking God, or his maker, for forgiveness, you know, as the timeline is, as all of these realities are collapsing. And in my opinion of Rick, this is the largest amount of uncertainty he's ever displayed throughout this series. And this happens right at the moment that he restores the timeline and therefore reduces the chaos. So I really think the writers got it right on that.

EL: That's really neat.

CF: Yeah, yeah, I loved it. And so for me, I also find this as a perfect time to, you know, hang out with friends, Netflix and chill it up, and then afterwards talk, you know. I would really like to challenge the listeners to observe this phenomenon in their real life. And for some people, it might be a stretch. But to some extent, I think we observe Heisenberg's Uncertainty Principle in our daily lives, like in the sense that the more sure that we are about something, or the more plan that we've made for something, the more likely were observe chaos, right? The more things we've gotten planned out, the more things that are actually likely to go wrong. I get to see this at the university, right, with so many young minds planning out their future. And I really see that the more certain a student feels about their plan, the more likely they're going to feel chaos in their life, if things don't go according to plan. So I really enjoy Heisenberg's Uncertainty Principle on so many levels, mathematically, physically, maybe even philosophically, and observing it in our real lives.

EL: Yeah, I really like the the metaphorical aspect you brought here. And if I can reveal how naive I am about Heisenberg's Uncertainty Principle, I didn't know that it was applicable in these different things other than just position and momentum. Maybe I'm the only one. But that's really interesting that there, so are there a lot of other places where this is also the case?

CF: Well, mathematically, it's just a function defined, for example, on Rn that has regularity and decay properties, and so that function's L2 norm--so the statement of the theorem is under those decay and regularity assumptions, that function's L2 norm has to remain less than or equal to 2 over n multiplied by the product of the L2 norm of the first moment of the function times the L2 norm of the gradient of the function. And so in some sense, we can view that as talking about momentum and position, and so that has applications to various physical systems, both momentum and position. But in some sense, whenever you have a gradient of a function, it can also relate to some system's energy. And so mathematically, I think we have the position to view this in a the more abstract way, whereas physically, you tend to only read about the the momentum and position version and less about the energy and time version. So that's why it took me a long time to think about did the writers of Rick and Morty get it right, because they're basically relying on, it seems they're giving the impression that uncertainty is leading to chaos. Because every time someone feels uncertainty, the timeline gets split and multiple, simultaneous versions of reality are going on at the same time, introducing more chaos into the system. And I kept thinking about it: "But mathematically, that's not what I learned, like what's happening?" And so I really think it's at the end where where Rick merges all the timelines together and basically reduces the chaos in the system. I really think that's the moment where we're seeing Heisenberg's Uncertainty Principle at play. We're seeing that in the moment where Rick was the least certain about himself and his abilities to fix this is the moment where the timeline were fixed. I really think someone had to be knowing about the energy and time version of Heisenberg's Uncertainty Principle.

KK: I need to go back and watch this. I've seen all the first two seasons, but I don't remember this one in particular. My son should be here. He could tell you all about it. We could be having this, oh yeah this yeah. It's a bit raw of a show, though, so listener warning, if you don't like obscenities and--

EL: Delicate ears beware.

KK: Really not politically correct humor very often. It's, you know, it's

CF: I agree.

KK: it's a little raw.

CF: Yeah.

KK: It's entertaining. But it's

CF: Yeah, it's definitely dark humor and lots of references to sci-fi horror. And, you know, some references are done well, some are just a little, I don't know.

KK: Yeah.

CF: But I definitely learned about this show from undergraduate students during a conference where we were, you know, stuck in a long commute. And students found themselves talking to me about all sorts of things. And they mentioned this episode where something was proven mathematically, and I'm a huge fan of Back to the Future, I hadn't watched--I only recently, you know, watched the show, even though it's been around for some time, apparently. And I'm a huge fan of Back to the Future. And they're telling me that there's mathematical proofs. And so I'm course I'm like, "Well, I'm gonna have to check out the mathematical proofs." Any mathematician that watches the show could see that the mathematical proof, I'm not sure that it's much of a mathematical proof.

So it got it got me to watch the episode. And once I was watching the episode, what really drew my attention was that I realized they're talking about chaos and uncertainty.

EL: So going back to the theorem itself, where did you encounter that the first time?

CF: First year grad school at UC Santa Barbara. And it's actually--I never told the professor who was teaching a course turned out to eventually go on and become my PhD advisor. And that first year that I was a graduate student at UC Santa Barbara, I was much more interested in differential geometry and topology than I was in analysis. And this was in one of our homework assignments sort of buried in there. And I don't remember exactly who, if it was the professor himself, or maybe one of his current graduate students, or a TA for the course, that explained that inequality and its physical relationship to chaos and uncertainty. And I'm pretty sure that the conversation with whoever it was, it was about chaos and uncertainty. And it wasn't about momentum and position, which I think would have turned me off at the time. But we were talking about this, relating it to uncertainty and measurements and chaos present in the system. And for me, since that moment, I think I've lived by this sort of mantra that if I plan things out, more things are going to go wrong the more planning that I do. But, you know, I kind of have to keep that in mind that I can only plan so much without introducing some chaos into the system. And so it made a huge impression on me. And I asked this professor who assigned this homework assignment, I'm sure it was the first semester, I mean, the first quarter, of graduate real analysis, if he had more reading for me to do. And he became my advisor, and I went into this area, mathematical physics, PDEs, and harmonic analysis. So it made a huge influence on me. And that's why I wanted to include it as my favorite theorem.

EL: Yeah, that's such a great story. It's like your superhero origin story, is this theorem.

KK: Yeah. So surely this L2 norm business, though, came after the fact. Like Heisenberg just sort of figured it out in the physics sense, and then some mathematician must have come up with the L2 norm business.

CF: Right. I actually think Heisenberg came up with it in the physical sense. There was someone who wrote down something mathematically, and I actually haven't gone--I should--I haven't gone through the literature to find out which mathematician wrote down the L2 norm statement of the inequality.

But in the book Introduction to Nonlinear Dispersive Equations by Felipe Linares and Gustavo Ponce-- Gustavo's my advisor--on page 60, it's Exercise 3.14. It proves Heisenberg's inequality and states it the way I've stated it here in this podcast, and it's a really neat analysis exercise. You know, you have to use the density of the Schwartz class functions, you use integration by parts. It's a really neat exercise and really helps you use those tools that PDE people use. And yeah, my advisor doesn't know how much that exercise influenced my decision to study the mathematical physics, PDEs, and harmonic analysis.

KK: Good, then now our listeners have an exercise, too. So,

EL: Yeah.

CF: That's right. Yeah. So my recommendations are watch Rick and Morty, try exercise 3.14 from Introduction to Nonlinear Dispersive Equations, and have a deep philosophical conversation about uncertainty and chaos with your good friends as you Netflix and chill it out.

EL: Nice. Yeah, wise words, definitely. Thanks a lot for joining us. I really need to brush up on some of my physics, I think, and think about this stuff.

CF: I'm happy to talk about it anytime you like. Thank you so much for the invitation. I've really enjoyed talking to you all.

KK: Thanks, Cynthia.

Episode 36 - Nikita Nikolaev & Beatriz Navarro Lameda

14 februari 2019 | 30 min

Kevin Knudson: Welcome to My Favorite Theorem, a special Valentine’s Day edition this year.

Evelyn Lamb: Yes.

KK: I’m one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida. This is your other host.

EL: Hi. I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah.

KK: How’s it going, Evelyn.

EL: It’s going okay. We got, we probably had 15 inches of snow in the past day and a half or so.

KK: Oh. It’s sunny and 80 in Florida, so I’m not going to rub it in. This is a Valentine’s Day edition. Are you and your spouse doing anything special?

EL: We’re not big Valentine’s Day people.

KK: So here’s the nerdy thing I did. So Ellen, my wife, is an artist, and she loves pens and pencils. There’s this great website called CW Pencil Enterprise, and they have a little kit where you can make a bouquet of pencils for your significant other, so this is what I did. Because we’ve been married for almost 27 years. I mean, we don’t have to have the big show anymore. So that’s our Valentine’s Day.

EL: Yeah, we’re not big Valentine’s Day people, but I got very excited about doing a Valentine’s Day episode of My Favorite Theorem because of the guests that we have. Will you introduce them?

KK: I know! We’re pleased to have Nikita Nikolaev and Beatriz Navarro, and they had some popular press. So why don’t you guys introduce yourselves and tell everybody about you?

Nikita Nikolaev: Hi. My name is Nikita. I’m a postdoctoral fellow at the University of Geneva in Switzerland. I study algebraic geometry of singular differential equations.

Beatriz Navarro Lameda: And I’m Beatriz. I’m currently a Ph.D. student at the University of Toronto, but I’m doing an exchange program at the University of Geneva, so that’s why we’re here together. And I’m studying probability, in particular directed polymers in random environments.

EL: Okay, cool! So that is actually applicable in some way.

BNL: Yes, it is.

EL: Oh, great!

KK: So why don’t we talk about this whole thing with the wedding? So we had this conversation before we started recording, but I’m sure our listeners would love to hear this. So what exactly happened at your wedding?

NN: Both of us being mathematicians, of course, we had almost everybody was either a mathematician or somehow mathematically related, most of our guests. So we decided to have a little bit of fun at the wedding, sprinkle a little bit of maths here and there. And one of the ideas was to, when the guests arrive at the dinner, in order for them to find which table they’re sitting at, they would have to solve a small mathematical problem. They would arrive at the venue there, and they would open their name card, and the name card would contain a first coordinate.

BNL: And a question.

NN: And a question. And the questions were very bespoke. It really depended on what we know their mathematical background to be. We had many people in my former research group, so I pulled questions from their papers or some of the talks they’ve given.

EL: This is so great! Yeah.

NN: And there were some people who are, maybe they’re not mathematicians, they’re engineers or chemists or something, and we would have questions which are more mathematically flavored rather than actual mathematical questions just to make everyone feel like they’re at a wedding of mathematicians.

EL: Right.

NN: So right. They had to find out two coordinates. All the tables were named after regular polyhedra, and they had to find out what their polyhedron of the night was.

EL: Okay.

NN: In order to do that, there was a matrix of polyhedra. Each one had two coordinates, and once you find out what the two coordinates are, you look at that matrix, and it gives you what polyhedron you’ve got. So as a guest, you would open the name card, and it would contain your first coordinate and a question and a multiple choice answer. And the answers were,

BNL: Usually it was one right answer and two crazy answers that had nothing to do with the question. Most of them were 2019 because that’s the year we got married, and then some other options. And then once you choose your answer, you would be directed to some other card that had a name of some mathematical term or some theorem, and that one would give you the second coordinate.

NN: So we made this cool what we called maths tree. We had several of these manzanita trees, and we put little cards on them with names, with these mathematical terms, with the answers, with people’s questions, and we just had this tree with cards hanging down, more than 100 cards hanging down. What I liked, in mathematicians it induced this kind of hunting instinct. You somehow look at this tree, and there are all these terms that you recognize, and you’ve seen before in your mathematical career, and you’re searching for that one that you know is the correct one.

BNL: And of course we wanted to make sure everyone found their table, so if they for any reason chose the wrong answer, they would also be directed to some card with a mathematical term. And when they opened it, it would say, “Oops, try again.” So that way you knew, okay, I just have to go and try again and find what the correct coordinate would be.

KK: This is amazing.

NN: And then to foolproof the whole thing, during the cocktail hour, they would do this kind of hunting for mathematical terminology, but then the doors would open into the dinner room, and just like most textbooks, when you open the back of the textbook, there’s the answer key, in the dinner room we had the answer key, which was a poster with everyone’s names

BNL: And their polyhedra

NN: And their polyhedron of the night.

EL: Yeah.

NN: So it was foolproof. I think some commentators on the internet were very concerned that some guests wouldn’t find their seat and starve to death.

EL: Leave hungry.

NN: No, it was all thought through completely.

KK: Some of the internet comments, people were just incredulous. Like, “I can’t believe these people forced their guests to do this!” They don’t understand, we would think this was incredible. This is amazing!

EL: Yeah! So delightfully nerdy and thoughtful. So, yeah, we’ve mentioned, this did end up on the internet in some way, which is how I heard about it, because I sadly was not invited to your wedding. (Since this is the first time I have looked at you at all.) So yeah, how did it end up making the rounds on some weird corner of the internet?

NN: So basically a couple of weeks before the wedding, I made a post on Facebook. It was a private Facebook post, just to my Facebook friends. You know, a Facebook friend is a very general notion.

EL: Right.

NN: I kind of briefly explained that all our guests are mathematicians, so we’re going to do this cool thing, we’re going to come up with mathematical questions, and one of my Facebook “friends,” a Facebook acquaintance, I later found out who it was, they didn’t like it so much, and they did a screengrab, and then they posted, with our names redacted and everything redacted, they made a post on Reddit which was like, “Maths shaming, look, these people are forcing their guests to solve a mathematical question to find their seat, maths shaming.”

BNL: It was in the “bridezilla” thread. “This crazy bride is forcing their guests to solve mathematical problems,” and how evil she is. Which, funny because Nikita was the one who wrote that Facebook post.

NN: I actually was the one.

BNL: So it was not a bridezilla, it’s what I like to call a Groom Kong.

NN: That’s right. So then this Reddit thread kind of got very popular, and later some newspaper in Australia picked it up, and then it just snowballed from there. Fox News, Daily Mail, yeah.

KK: Well, this is great. This is good, now you’ve had your 15 minutes of fame, and now life can get back to normal.

EL: Yeah.

KK: This is a great story. Okay, but this is a podcast about theorems, so what is your favorite theorem?

NN: Right, yeah. So we kind of actually thought long and hard about what theorem to choose. Like, what is our favorite theorem is such a difficult question, actually.

KK: Sure.

NN: It's kind of like, you know, what is your favorite music piece? And it's, I mean, it's so many variables. Depends on the day, right?

EL: Yeah.

NN: But we ended up deciding that we were going to choose the intermediate value theorem.

KK: Oh, nice.

NN: As our favorite theorem.

KK: Good.

BNL: So, yes, the intermediate value theorem is probably one of the first theorems that you learn when you go to university, right? Like calculus you start learning basic calculus, and it's one of the first theorems that you see. Well, what it says is that well, suppose you start with a continuous function f, and you look at some interval (a, b), so the function f sends a to f(a), b to f(b), and then you pick any value y that is between f(a) and f(b). And then you know that you will find a point c that is between a and b, such that f(c) equals y.

So it looks like an incredibly simple statement. Obvious.

KK: Sure.

BNL: Right, but it is a quite powerful statement. Most students believe it without proof. They don't need it. It's, yes, absolutely obvious. But, well, we have lots of things that we like about the theorem.

NN: Yeah, I mean, it feels incredibly simple and completely obvious. You look at it, and, you know, it's the only thing that could possibly be true. And the cool thing about it, of course, is that it represents kind of the essence of what we mean by continuous function.

KK: Sure.

EL: Yeah.

NN: In fact, actually, if you look at the history, before our modern formal definition of continuity, that was part of the property that was a required property that people used to use as part of the definition of continuity. In fact, actually, if you look at the history, before we formalize the definition of continuity, people were very confused about what a continuous function actually should mean. And many thought, erroneously, that this intermediate value property was equivalent to continuity. In some sense, it's what you would want to believe because it really is the property that more or less formalizes what we normally tell our students, that heuristically, a continuous function is one that if you want to draw it, you can draw it without taking off your pencil off of a piece of paper, and that's what the Intermediate Value Theorem, this intermediate value property, it represents that really.

But for all its simplicity and triviality, if you actually look at it properly, if you look at our modern definition of continuity using epsilon-delta, then it becomes not obvious at all.

BNL: Yes. So if you see epsilon, you look at the definition of continuity, and have epsilons, deltas, how is it possible that from this thing, you get such an obvious statement, like the intermediate value theorem? So what the intermediate value theorem is telling us is that, well, continuous functions do exactly what we want them to do. They are what we intuitively think of a continuous function. So in a sense, what the intermediate value theorem is doing for us is serving as a bridge between this formal definition that we encounter in university. So we start first year calculus, and then our professor gives us this epsilon-delta definition of continuity. And it's like, oh, but in high school, I learned that a continuous function is one that I can draw without lifting my pencil. Well, the intermediate value theorem is precisely that. It's connecting the two ideas, just in a very powerful way.

NN: Yeah. And also, you know, it cannot be overstated how useful it is. I mean, we use it all the time. As a geometer, of course, you know, you use some generalization of it, that continuous functions send connected sets to connected, and we use it all the time, absolutely, without thinking, we take it absolutely for granted.

BNL: So even if you do analysis, you are using it all the time, because you can see that the intermediate value theorem is also equivalent to the least upper bound property, so the completeness axiom of the real numbers. Which is quite incredible, to see that just having the intermediate value theorem could be equivalent to such a fundamental axiom for the real numbers, right? So it appears everywhere. It's surprising. We know, it's very easy, when you see the proof of the intermediate value theorem, you see that it is a consequence of this least upper bound property, but the converse is also true. So in a sense, we have that very powerful notion there.

KK: I don't think I knew the converse. So I'm a topologist, right? So to me, this is just a statement that the continuous image of a connected set is connected. But then, of course, the hard part is showing that the connected subsets of the real line are precisely the intervals, which I guess is where the least upper bound property comes in.

BNL: Yes, indeed, yes. Exactly yes.

KK: Okay. I haven't thought about analysis in a while. As it turns out, we're hiring several people this year. And, for some of them, we've asked them to do a teaching demonstration. And we want them all to do the same thing. And as it so happens, it's a calculus one demonstration about continuity and the intermediate value theorem.

BNL: Oh.

EL: Nice.

KK: So in the last month, I've seen, you know, 10 presentations about the intermediate value theorem. And I've come to really appreciate it as a result. My favorite application is, though, that you can use it to prove that you can always make any table level, or not level, but all four legs touch the ground at the same time.

BNL: Yes.

KK: Yeah, that's, that's great fun. The table won't be level necessarily, but all four feet will be on the ground, so it won't wobble.

BNL: Yes.

EL: Right.

NN: If only it were actually applied in classrooms, right?

KK: Right.

EL: Yeah.

NN: The first thing you always do when you come to, you sit at a desk somewhere, is to pull out a piece of paper to actually level it.

EL: Yeah. So was this a theorem that you immediately really appreciated? Or do you feel like your appreciation has grown as you have matured mathematically?

BNL: In my case, I definitely learned to appreciate it more and more as I matured mathematically. The first time I saw the theorem, it's like, "Okay, yes, interesting, very cool theorem." But I didn't realize at the moment how powerful that theorem was. Then as my mathematical learning continued, then I realized, "Oh, this is happening because of the intermediate value theorem. And this is also a consequence of it." So there's so many important things that are a consequence, of the intermediate value theorem. That really makes me appreciate it.

NN: Well, there's also somehow, I think this also comes with maturity, when you realize that some very, what appear to be very hard theorems, if you strip away all the complexity, you realize that they may be really just some clever application of the intermediate value theorem.

BNL: Like Sharkovskii's theorem, for example, is a theorem about periodic points of continuous functions. And it just introduces some new ordering in the natural numbers. And it tells you that if you have a periodic point of some period m, then you will have periodic points of any period that comes after m in that ordering. You can also look at the famous "period three implies chaos."

KK: Right.

BNL: A big component of it is period three implies all other periods. And the proof of it is really just a clever use of the intermediate value theorem. It's so interesting, that such an important and famous theorem is just a very kind of immediate--though, you know, it takes some work to get it--but you can definitely do it with just the Intermediate Value Theorem. And I actually like to present that theorem to students in high school because they can believe the Intermediate Value Theorem.

EL: Yeah.

BNL: That's something that if you tell someone, "This is true," no one is going to question it is definitely true.

KK: Sure.

BNL: And then you tell them, "Oh, using this thing that is obvious, we can also prove these other things." And I've actually done with high school students to, you know, prove Sharkovskii's theorem just starting from the fact that they believe the intermediate value theorem. So they can get to higher-level theorems just from something very simple. I think that's beautiful.

NN: Yeah, that's kind of a very astonishing thing, that from something so simple, and what looks obvious, you can get statements which really are not obvious at all, like what she just explained, Sharkovskii's theorem, that's kind of a mind blowing thing.

EL: Yeah, you're making a pretty good case, I must say.

KK: That's right.

EL: So when we started this podcast, our very first episode was Kevin and I just talking about our own favorite theorems. And I have already since re-, you know, one of our other guests has taken my loyalty elsewhere. And I think you're kind of dragging me. So I think, I think my theorem love is quite fickle, it turns out. I can be persuaded.

KK: You know, in the beginning of our conversation, you pointed out, you know, how does one choose a favorite theorem, right? And, and it's sort of like, your favorite theorem du jour. It has to be.

BNL: Exactly, yes.

EL: Yeah.

KK: All right, so what does one pair with the intermediate value theorem?

BNL: So we thought about it. And to continue with the Valentine's Day theme, we want to pair the intermediate value theorem with love in a relationship.

KK: Ah, okay, good.

BNL: The reason why we want to pair it with love is because when you love someone, it's completely obvious to you. You just know it's true, you know you love someone.

KK: That's true.

BNL: You just feel like there's no proof required. It's just, you know it, you love this person.

NN: It's the only thing that can possibly be true, there's no reason to prove it.

BNL: But also, just like any good theorem, you can also prove, you can provide a proof of love, right? You can show someone that you love them.

NN: Any good mathematical theorem can always be supplied with a very rigorous, detailed to whatever required level proof. And if you truly, really truly love someone, you can prove it. And if someone questions, any part of that proof, you can always supply more details and a more detailed explanation for why why you love that person. And that's why there's a similarity between the intermediate value theorem and love in a relationship.

EL: Yeah, well, I'm thinking of the poem now, "How do I love thee? Let me count the ways," This is a slightly mathematically-flavored poem.

KK: But I think there must be at least, you know, the, the continuity of the continuum ways, right? Or the cardinality of the continuum ways.

NN: Absolutely.

KK: That's an excellent pairing.

EL: Yeah.

BNL: We also thought that love is something that we feel, we take it as an obvious statement, and then from love, we can build so many other things, right? Like in the intermediate value theorem case, we start from a theorem that looks obvious, and using it, we can prove so many other theorems. So it's the same, right, in a relationship. You start from love, and then you can build so many other great things.

EL: Yeah, a marriage for example.

BNL and NN: For example, yes.

EL: Yeah. And a ridiculously amazing wedding game as part of that.

NN: There were some other mathematical tidbits in the wedding. So one of them I'll mention is our rings. Our wedding bands are actually Möbius bands.

KK: Oh, I see.

EL: Okay, very nice.

NN: We had to work with a jeweler. And there's a bit of a trick, because if you just take a wedding band, and you do the twist to make it a Möbius band, than the place where it twists would stick out too much.

EL: Yeah.

NN: So the idea is to try to squish it. And that, of course, is a bit challenging if you want to make a good-looking ring, so that was part of the problem to be solved.

EL: Yeah. Well, my wedding ring is also--it's not a Möbius band. But it's one that I helped design with a particular somewhat math-ish design.

KK: My wife and I are on our second set of wedding bands. The first ones, because we were, I was a graduate student and poor, we got silver ones. And silver doesn't last as long, so we're on our second ones. But the first ones were handmade, and they were, they had sort of like a similar to Evelyn's sort of little crossing thing. So they were a little bit mathy, too. I guess that's a thing that we do, right?

EL: Yeah.

NN: It's inevitable.

KK: Yeah, excellent.

KK: So we like to give our guests a chance to plug anything. Do you have any websites, books, wedding registries that you want to plug?

NN: Actually, in terms of the wedding registry, lots of our guests, of course, were asking. We didn't have a wedding registry because given the career of a postdoc, where you travel from place to place every few years, a wedding registry isn't the most practical thing. Yeah, difficult.

BNL: Yes. So we said, well, you can just give us anything you like, we'll have a box where you can leave envelopes. And some of our guests were very creative. They gave us, some of them decided to give us money. But the amounts they chose were very interesting, because they were, like, some integer times e or times π, or some combination. They wrote the number and then they explained how they came up with that number. And that was very interesting and sweet.

NN: Some of them didn't explain it. But we kind of understood. We cracked the code, essentially, except one. So one of our friends wrote us a check with a very strange number. And to this day,

BNL: We still don't know what the number is.

NN: We kept trying to guess what it could be. But no, I don't know. Maybe eventually I'll just have to ask. I'd like to know.

KK: Maybe it was just random.

NN: Maybe it was just random.

BNL: Yeah, I think one of the best gifts people gave us was their reaction right after seeing their card. In particular, there is a very nice story of a guest who really, really loved the way we set up everything and maybe you can tell us about that.

NN: Yeah, so we, at the dinner we would approach tables to say hi to some guests, and so this particular, he's actually Bea's teaching mentor.

BNL: So I'm very much into teaching. And he's the one who taught me most of the things I know.

NN: So we approached him, and, and he looked at us, and he pulled out the name card out of his breast pocket like, "This. This is the most beautiful thing I've ever seen. This is incredible. It's from my last paper, isn't it?" Yes. Yeah, that's right. He's like, "I have to send it to my collaborator. He's going to love it." And just seeing that reaction, him telling us how much he loved the card, just made all those hours that I and Bea spent reading through papers and trying to come up with some kind of, you know, short sounding question to be put into multiple choice, made all of that worthwhile.

EL: Yeah, I'm just imagining it. Like, usually you don't have to, like, cram for your wedding. But yeah, you've got all these papers you've got to read.

BNL: Yeah, we spent days going through everyone's papers and trying to find questions that were short enough to put in a small card and also easy to answer as a multiple choice question.

NN: Yeah, some were easy. So for example, my former PhD advisor came to our wedding, and I basically gave him a question from my thesis, you know, just to make sure he'd read it.

EL: Yeah.

NN: So when we approached him at the dinner and I said, "Oh, did you like the question?" and he just looked at me like, "Yeah, well I gave you that question two years ago!"

EL: Yeah

NN: So, yes, some questions were easy to come up with. Some questions were a bit more difficult. So we had a number of people from set theory, and neither of us are in set theory. I'd never, ever before opened a paper in set theory. It was all very, very new to me.

EL: Nice.

KK: Well, this has been great fun. Thanks for being such good sports on short notice.

EL: Yeah.

KK: Thank you for joining us.

BNL: Yeah.

EL: Yeah, really fun to talk to you about this. It's so much better than even the Reddit post and weird news stories led me to believe.

KK: Well, congratulations. it's fun meeting you guys. And let me tell you, it's fun being married for 27 years, too.

NN: We're looking forward to that.

KK: All right, take care.

NN: Thank you. Bye bye.

BNL: Bye.

Episode 35 - Nira Chamberlain

24 januari 2019 | 22 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics, favorite theorems, and other random stuff that we never know what it’ll be. I’m one of your hosts, Kevin Knudson. I’m a professor of mathematics at the University of Florida. This is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah, where I forgot to turn on the heat when I first woke up this morning. I've got separate systems. So it is very cold in the basement here where I am recording.

KK: Well, yeah, it's cold in Florida this morning. It was, you know, in the mid-60s. It's very pleasant. I'm still in short sleeves. Our listeners can't see this, but I'm in short sleeves. Evelyn’s in a sweater. And our guest is in a jacket in his attic.

EL: Yes.

KK: So today we are happy to welcome Nira Chamberlain all the way from the UK. Can you tell everyone about yourself a little bit?

Nira Chamberlain: Yes, hello. My name is doctor Nira Chamberlain. I’m a professional mathematical modeler. I'm also the president-designate of the Institute of Mathematics and its Applications.

KK: Fantastic. So tell us about the IMA a little bit. So we have one of those here, but it's a different thing. So what is it?

NC: Right. I mean, the Institute of Mathematics and its Applications is a professional body of mathematicians, of professional mathematicians, and it's a learned society. It's been around since 1964. And it is actually to make sure that UK has a strong mathematical culture and look after the interest of mathematicians by industry, government and academia.

KK: Oh, that's great. Maybe we should have one of them here. So the IMA here is something else. It’s a mathematics institute. But maybe the US should have one of these. We have the AMS, right, the American Mathematical Society.

EL: Or SIAM might be more similar because it does applications, applied math.

KK: Yeah, maybe.

EL: Yeah, we’ve kind of got some.

NC: So we asked you on for lots of reasons. One is, you know, you're just sort of an interesting guy. Two, because you’re an applied mathematician, and we like to have applied mathematicians on as much as we can, Three you actually won something called the Great Internet Math-Off this summer, of which Evelyn was a participant.

EL: Yes. So he has been ruled—he’s not just an interesting guy, he has been officially ruled—the most interesting mathematician in the world…among people who were in the competition. The person who ran it always put this very long disclaimer asterisk, but I think Nira definitely has some claim on the title here. So, yeah. Do you want to talk a little bit about the big, Great Internet Math-Off?

NC: Yes, we have let's say an organization, a group of mathematicians that do a blog, Aperiodical, and they decided to start this competition called the big internet math-off. And it’s a as a knockout tournament, 16 mathematicians, and they put up something interesting about mathematics. It was put up on the internet, it was there for 48 hours, the general public would vote for what they found was their most favorite or most interesting, and the winner would progress to the next round, and it was four rounds all together. And if you reach to the final end, and you win it, you get the title “World's Most Interesting Mathematician.” And when I was invited, I thought, “Oh, isn't this really for those mathematicians that are pure mathematicians and those public communicators and those into puzzles? I mean, I'm a mathematical modeler, I’m in applied mathematics, so what am I really going to talk about?” And then when I saw that when I was actually introduced as the applied mathematician and everybody else was, let's say the public communicator, and here's the applied mathematician. It was almost like: then here's the villain—boo!

I thought, “Okay, there you go.” I’m thinking, “All right, what we're going to do is I'm actually going to stick to being an applied mathematician.” So three out of the four topics I actually introduced were about applied mathematics, and yes, the fourth topic was actually about the history of mathematics. And I was fortunate enough to get through each of the rounds and win the overall competition. It was very interesting and very good.

EL: Yeah, and I do wish — I think you look very interesting right now—I wish our listeners could see that you've got headphones on that make you look a little bit like a pilot, and behind you are these V-shaped beams, I guess in the attic, where I can totally imagine you, like, piloting some ship here, so you're really looking the part this morning, or this afternoon for you.

NC: Thank you very much indeed. I mean that’s what I call my mathematics attack room, which is the attic, and I have 200 math books behind me. And I’ve got three whiteboards in front of me, quite a number of computer screens. And I’ve got all my mathematical resources all in one place.

KK: Okay, so I just took a screenshot. So maybe with your permission, we’ll put this up somewhere.

So this is a podcast about theorems. So, Nira what is your favorite theorem?

NC: Okay, my favorite theorem is actually to do with the Lorenz equation, the Lorenz attractor. Now it was done in the 1960s by a meteorologist called Edward Lorenz. And what he wanted to do was to take a a partial differential equation, see if he could make some simplifications to it, and he came up with three nonlinear ordinary differential equations to actually look at, let's say, the convection and the movement, to see where we can actually use that to do some meteorological predictions. And then he got this set of equations, went to work solving it numerically, and then he decided, “Actually, I’d better restart my computer game because I've done something wrong.” So he went back, he restarted the computer, but he actually changed the initial conditions by a little bit. And then when he came back, he actually saw that the trajectory of the solution was different from what he had started with. When he went back and started checking, he actually saw that the initial conditions only changed by a little bit, and what was this? It was probably one of the first examples of the “butterfly effect.” The butterfly effect is saying that if, let's say, a butterfly flaps its wings, then that will prevent a hurricane going into Florida — topical.

KK: Yeah, it’s been a rough month.

NC: Yeah, or, if, let's say, another butterfly flaps its wings, then maybe another hurricane may go into Salt Lake City, for example. And this is, like I said, an example of chaotic behavior once you choose certain parameters now. The reason why I like this theorem so much is I was actually introduced to this topic when I was in my final year of my mathematics degree. And it probably was one of the introductions to the field of mathematical modeling, recognizing that when you actually model reality, mathematics is powerful, but also has its limitations. And you’re just trying to find that boundary between what can be done and what can't be done. Mathematical modeling has a part to play in that.

KK: Right. What's so interesting about meteorological modeling is that I've noticed that forecasts are really good for about two days.

NC: Yeah.

KK: So with modern computing power, I mean, of course, as you pointed out, everything is so sensitive to initial conditions, that if you have good initial data, you can get a good forecast for a couple of days, but I never believe them beyond that. It's not because the models are bad. It's because the computation is so precise now that the errors can propagate, and you sort of get these problems. Do you have any sense of how we might extend those models out better, or is it just a lost cause, is it hopeless?

NC: It's probably a lost cause. I agree with you to a certain extent. But it's a case of when we're dealing with, let’s say, meteorological equations, if they have chaotic behavior, if you put down initial conditions, and it's changed, you know, it's going out and it's changing, it just shows that, yeah, we may have good predictions to begin with, but as we go on into the future, those rounding errors will come, those differences will come. And it's almost like, let's use an analogy, let's say you go to whatever computer algebra software you have, and you get π, and let’s say you square root it 10 times, and then you raise it to a power 10 times, and then if you square root it 100 times and then you raise it to a power 100 times, and if you keep on repeating that, then actually, when you come back to the figure, you're thinking, “Is this actually π?” No, it's not. And also different calculator and different computer algebra softwares, you’ll see that they will have actually their difference. It’s that point where in terms of when we're doing, predicting a weather system, because of the chaotic behavior of the actual nonlinear differential equations, coupled with those rounding errors, it is very difficult to do that long-term weather forecast. So nobody can really say to me, “By the way, in five years’ time, on the 17th of June, the weather will be this.” That’s very much a nonsense.

KK: Sure, sure. Well, I guess orbital mechanics are that way too, right? I mean, the planetary orbits. I mean, we understand them, but we also can't predict anything in some sense.

NC: Yeah.

KK: Right, right. Living in Florida, I pay a lot of attention to hurricane models. And it's actually really fascinating to go to these various sites. So windy.com is a good one of these. They show the wind field over the whole planet if you want. And they'll also, when there are hurricanes, they have the separate models. So the European model actually turns out to be better than the American one a lot, which is sort of interesting because hurricanes affect us a lot more than— I mean the remnants get to the UK and all of that. But so you’re right, it's sort of interesting, the different implementations—the same equations, essentially, right, that underlie everything get built into different models. And then different computing systems have the different rounding error. And the models, they’re sort of, they're usually pretty close, but they do diverge. It's really very fascinating.

NC: Yeah, I mean, over in the United Kingdom, we had an interesting case in 1987 where the French meteorology office says, “By the way, people in the north of France, they should be aware that there's going to be a hurricane approaching.” While the British meteorologic office was saying, “Oh, there's no way that there's going to be a hurricane. There's no hurricane. Our model says there’s going to be no hurricane.” So the French are saying there’s going to be a hurricane. The British say there’s not going to be a hurricane. And guess what? The French were right and a hurricane hit the United Kingdom.

And because of that what they did is that now the Met Office, which is the main weather place in Britain, what they've done is they put quite a number of boats out in the Atlantic to measure, to come up with a much more accurate measure of the weather system so that they can actually feed their models, and they also use more powerful models because he equation itself remains the same, it’s the information that actually goes into it which is which is the difference, yeah? So in terms of what you said in the American models, it's all dependent on who you get the measurements from because you may not get exactly the measurement from the same boat. You may get it from a different boat, from different boats in a different location, different people. This is where you come to that human factor. Some people will say, “Oh, round it to this significant figure,” while someone else will say, “Round it to that significant figure,” and guess what? All of that actually affects your final results.

EL: Yeah, that matters.

NC: Yes.

KK: So do you do this kind of modeling yourself, or are you in other applications?

NC: Oh, I'm very much in other applications. I mean, I'm still very much a a mathematical modeler. I mean, my research now is to do with—to minimize the probability of artificial intelligence takeover. That’s what my current research I'm doing at Loughborough University.

EL: Well that, you know, the robots will have you first in the line or something in the robot uprising.

NC: Well, we talk about robots, but this is quite interesting. When we're talking about, let's say, artificial intelligence takeover, everybody thinks about the Hollywood Terminator matrix, I Robot, you know, robots marching down the street. But there are different types of AI takeovers, and some of them are much more subtle than that. For instance, one scenario is, let's say for instance, you have a company, and they decide to really upgrade their artificial intelligence, their machine learning, to a certain sense it's more advanced than their competition is. And by doing so, they actually put all their competitors out of business. And so what you have is you have this one company almost running the world economy. Now the question is, would that company make decisions (based on its AI), would they make decisions that are conducive with social cohesion? And you can't put your hand on your heart and say, “Absolutely, yes, because a machine, it’s largely, like, 1-0, it doesn't really care about the consequences of social cohesion. So henceforth, we can actually to a model of that, saying could we ever get to a situation where one company actually dominates all different industrial sectors and ends up, let’s say, running the world economy? And if that's the case, what can what strategies can we actually implement to try and minimize that risk?

EL: It sounds not entirely hypothetical.

KK: No, no. Well, you know, of course the conspiracy theorists types in the US would have you believe that this already exists, right? The Deep State and the Illuminati run everything, right?

EL: But getting back to the Lorenz system and everything, you were saying that this is one of the earliest examples of mathematical modeling you saw. Was it one of the things that inspired you to go that direction when you got your PhD?

NC: Yes, so I was doing that as part of my final year mathematics degree, and I thought, well, this whole idea that, you know, here’s applied mathematics, using mathematics in the real world, saying that there are problems that some people say it's impossible, you can't use mathematics. And you're just trying to push the boundaries of mathematics and say, “This is how we actually model reality.” It was one of the things that actually did inspire me, so Edward Lorenz actually inspire me, just saying, wait a minute, applied mathematics is not necessarily about: here’s a problem, here’s an equation, put the numbers in the right places, and here's a solution. It's about gaining that insight into the real world, learning more about the world around you learning more about the universe around you through through mathematics. And that's what inspired me.

KK: And it's very imprecise, but that's sort of what makes it intriguing, right? I mean, you have to come up with simplifying assumptions to even build a model, and then how much information can we extract from that?

NC: That’s one of the key things about mathematical modeling. I mean, you're looking at the world. The world is complex, full of uncertainty, and it’s messy. And you are making some simplifying assumptions, but the key thing is: do you make simplifying assumptions to an extent that it actually corrupts and compromises your solution, or do you make simplifying assumptions that say, “Actually, this gives me insight into how the world actually works.”? And recognizing which factors do you include, which factors do you exclude, and bring a model that is what I call useful.

KK: Right. That’s the art, right?

NC: Yeah, that's the art. That's the art of mathematical modeling.

KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. So what pairs well with the Lorenz equation?

NC: I chose to pair it with the Jamaican dish called ackee yam and saltfish[16:48] . Now the reason why is with ackee yam and saltfish, if you cook it right, it is delicious, but if you cook it wrong, the Ackee turns out to be poisonous and that’s a bit like the Lorenz equation.

KK: What is ackee? I don't think I know what this is.

NC: Okay. Ackee is actually a vegetable, but if you actually were to look at it, it looks like scrambled egg, but it's actually a vegetable. It's like a yellow vegetable.

EL: Huh.

KK: Interesting.

NC: And yam, it’s like an overgrown, very hard potato. It’s looks like a very overgrown, hard potato.

KK: Sure, yeah.

NC: And saltfish is just a Jamican saying for cod. Even though you could really say ackee yam and cod, they don't call it cod, they call it saltfish.

KK: Okay. All right. So I've never heard ackee.

EL: Yeah, me neither.

KK: I mean, I knew that yams, so in the United States, most people will call sweet potatoes yams, they’ll use those two words interchangeably. But of course, yams are distinct, and I think they can be poisonous if you don't cook them right, right? Or some varieties can. But so ackee is something separate from the yam.

NC: Yeah.

EL: Also poisonous if you don't cook it right.

NC: Absolutely.

KK: So can you actually access this in England, or do you have to go to Jamaica to get this?

NC: Yes, we can access this this in England because in England we have a large West Indian diaspora community.

KK: Sure, right.

NC: And also we do get lots of variety of foods from different countries around the world. So we can, it's relatively easy to to access ackee yam. And also we’ve got quite a number of Caribbean restaurants, so definitely there they are going to cook it right.

KK: So it's interesting, we have a Caribbean restaurant here in town in Gainesville, which of course we're not as far away as you are, but they don't try to poison us. The food is delicious.  EL: That you know of.

KK: Well that's right. I love eating there. The food is really spectacular. But this is interesting.

EL: And is this a family recipe? Do you have roots in in the West Indies, or…

NC: Yes, my parents were from from Jamaica. I still have relatives in Jamaica, and my wife’s descent is Jamaican. Now and again we do have that Caribbean meal. I thought, “Well, what shall I say as a food? I thought, “Well, should I go for the British fish and chips?” I thought, “No, let's go for ackee yam and saltfish.”

KK: Sure, well and actually I think your jacket looks like a Jamaican-influenced thing, right? With the black, green, and yellow, right?

NC: Yes, absolutely. And that's because it's quite cold in the in the attic. This is the same style of jacket as the Jamaican bobsled team, so I decided to wear it, as it’s quite cold up here.

EL: Yeah, Cool Runnings, the movie about that, was an integral part of my childhood. My brother and sister and I watched that movie a lot. So I’m curious about this ackee vegetable, like how sensitive are we talking for the dependence on initial conditions, the dependence on cooking this correctly to be safe? Is it pretty good, or do you have to be pretty careful?

NC: You have to be pretty good, you have to be pretty careful. As long as you follow the instructions you’re okay, but in this case, if you don't cook it long enough, you don't cook it at a high enough temperature, whatever you do, please do not eat it cold, do not eat it raw.

EL: Okay.

KK: Like actually it might kill you, or it just makes you really sick?

NC: It will make you really sick. I haven't heard— well let’s put it this way, I do not wish to carry out the experiments to see what would happen.

KK: Understood.

EL: Yes.

KK: Well this has been great fun. I've learned a lot.

EL: Yeah

KK: Thanks for joining us, Nira.

NC: Thank you very much indeed for inviting me.

Episode 34 - Skip Garibaldi

10 januari 2019 | 24 min

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics, theorems, and, I don't know, just about anything else under the sun, apparently. I'm Kevin Knudson. I'm one of your hosts. I'm a professor of mathematics at the University of Florida. This is your other host.

Evelyn Lamb: Hi, I'm Evelyn lamb. I'm a freelance math and science writer based in Salt Lake City. So how are things going?

KK: It's homecoming weekend. We're recording this on a Friday, and for people who might not be familiar with Southeastern Conference football, it is an enormous thing here. And so today is is a university holiday. Campus is closed. In fact, the local schools are closed. There's a big parade that starts in about 20 minutes. My son marched in it for four years. So I've seen it. I don't need to go again.

EL: Yeah.

KK: I had brunch at the president's house this morning, you know. It's a big festive time. I hope it doesn't get rained out, though. It's looking kind of gross outside. How are things for you?

EL: All right. Yeah, thankfully, no parades going on near me. Far too much of a misanthrope to enjoy that. Things are fight here. My alarm clock-- we're also recording in the the week in between the last Sunday of October and the first Sunday of November.

KK: Right.

EL: In 2007, the US switched when it went away from Daylight Saving back to Standard Time to the first Sunday of November. But my alarm clock, which automatically adjusts, was manufactured before 2007.

KK: I have one of those too.

EL: Yeah, so it's constantly teasing me this week. Like, "Oh, wouldn't it be nice if it were only 7am now?" So yeah.

KK: All right. Well, yeah, first world problems, right?

EL: Yes. Very, very much.

KK: All right. So today, we are thrilled to have Skip Garibaldi join us. Skip, why don't you introduce yourself?

Skip Garibaldi: My name is Skip Garibaldi. I'm the director at the Center for Communications Research in La Jolla.

KK: You're from San Diego, aren't you?

SG: Well, I got my PhD there.

KK: Ish?

SG: Yeah, ish.

KK: Okay.

SG: So I actually grew up in Northern California. But once I went to San Diego to get my degree, I decided that that was really the place to be.

KK: Well, who can blame you, really?

EL: Yeah, a lot to love there.

KK: It's hard to argue with San Diego. Yeah. So you've been all over. For a while you're at the Institute for Pure and Applied Math at UCLA.

SG: Yeah, that was my job before I came to the Center for Communications Research. I was associate director there. That was an amazing experience. So their job is to host conferences and workshops which bring together mathematicians in areas where there's application, or maybe mathematicians with different kinds of mathematicians where the two groups don't really talk to each other. And so the fact that they have this vision of how to do that in an effective way is pretty amazing. So that was a great experience for me.

KK: Yeah, and you even got in the news for a while. Didn't you and a reporter, like, uncover some crime syndicate? What am I remembering?

SG: That's right. Somehow, I became known for writing things about the lottery. And so a reporter who was doing an investigative piece on lottery crime in Florida contacted me, and I worked closely with him and some other mathematicians, and some people got arrested. The FBI got involved and it was a big adventure.

KK: So Florida man got arrested. Never heard of that. That's so weird.

SG: There's a story about someone in Gainesville in the newspaper article. You could take a look.

KK: It wasn't me. It wasn't me, I promise.

EL: Whoever said math wasn't an exciting field?

KK: That's right.

Alright, so, you must have a favorite theorem, Skip, what is it?

SG: I do. So you know, I listened to some of your other podcasts. And I have to confess, my favorite theorem is a little bit different from what your other guests picked.

EL: Good. We like the the great range of things that we get on here.

SG: So my favorite theorem for this podcast answers a question that I had when I was young. It's not something that is part of my research today. It's never helped me prove another theorem. But it answers some question I had from being junior high. And so the way it goes, I'm going to call it the unknowability of irrational numbers.

So let me explain. When you're a kid, and you're in school, you probably had a number line on the wall in your classroom. And so it's just a line going left to right on the wall. And it's got some markings on it for your integers, your 0,1,2,3, your -1,-2,-3, maybe it has some rational numbers, like 1/2 and 3/4 marked, but there's all these other points on that number line. And we know some of them, like the square root of two or e. Those are irrational, they're decimals that when you write them down as a number-- like π is 3.14, we know that you can't really write it down that way because the decimal keeps on going, it never repeats. So wherever you stop writing, you still haven't quite captured π.

So what I wondered about was like, "Can we name all those points on the number line?

EL: Yeah.

SG: Are π and e and the square root of two special? Or can we get all of them? And it comes up because your teacher assigns you these math problems. And it's like "x^2+3x+5=0. Tell me what x is." And then you name the answer. And it's something involving a square root and division and addition, and you use the quadratic formula, and you get the answer.

So that's the question. How many of those irrational can you actually name? And the answer is, well, it's hard.

EL: Yeah.

SG: Right?

KK: Like weirdly, like a lot of them, but not many.

SG: Exactly!

EL: Yeah.

SG: So if we just think about it, what would it mean to name one of those numbers? It would mean that, well, you'd have to write down some symbols into a coherent math problem, or a sentence or something, like π is the circumference of a circle over a diameter. And when you think about that, well, there's only finitely many choices for that first letter and finitely many choices for that second letter. So it doesn't matter how many teachers there are, and students, or planets with people on them, or alternate universes with extra students. There's only so many of those numbers you can name. And in fact, there's countably many.

EL: Right.

KK: Right. Yeah. So are we talking about just the class of algebraic numbers? Or are we even thinking a little more expansively?

SG: Absolutely more expansive than that. So for your audience members with more sophisticated tastes, you know, maybe you want to talk about periods where you can talk about the value of any integral over some kind of geometric object.

KK: Oh, right. Okay.

SG: You still have to describe the object, and you have to describe the function that you're integrating. And you have to take the integral. So it's still a finite list of symbols. And once you end up in that realm, numbers that we can describe explicitly with our language, or with an alien language, you're stuck with only a countable number of things you can name precisely.

EL: Yeah.

KK: Well, yeah, that makes sense, I suppose.

SG: Yeah. And so, Kevin, you asked about algebraic numbers. There are other classes of numbers you can think about, which, the ones I'm talking about include all of those. You can talk about something called closed form numbers, which means, like, you can take roots of polynomials and take exp and log.

KK: Right.

SG: That doesn't change the setup. That doesn't give you anything more than what I'm talking about.

EL: Yeah. And just to back up a sec, algebraic numbers, basically, it's like roots of polynomials, and then doing, like, multiplication and division with them. That kind of thing. So, like, closed form, then you're expanding that a little bit, but still in a sort of countable way.

SG: Yes. Like, what kinds of numbers could you express precisely if you had a calculator with sort of infinite precision, right? You're going to start with an integer. You can take it square root, maybe you can take its sine, you know. You can think about those kinds of numbers. That's another notion, and you still end up with a countable list of numbers.

KK: Right. So this sounds like a logic problem.

SG: Yes, it does feel that way.

KK: Yeah.

SG: So, Kevin and Evelyn, I can already imagine what you're thinking. But let me say it for the benefit of the people for whom the word "countable" is maybe a new thing thing. It means that you can imagine there's a way to order these in a list so that it makes sense to talk about the next one. And if you march down that list, you'll eventually reach all of them. That's what it means. But the interesting thing is, if you think about the numbers on the number line, we know going back to Cantor in the 1800s that those are not countable. You use the so-called diagonalization argument, if you happen to have seen that.

KK: Right.

EL: Yeah. Which is just a beautiful, beautiful thing. Just, I have to put a plug in for diagonalization.

KK: Oh, it's wonderful.

SG: I've been thinking about it a lot in preparation for this podcast. I agree.

KK: Sure.

SG: So what that means is that that's the statement, these irrational numbers, you can't name all of them, because there are uncountably many of them, but only countably many numbers you can name.

It sort of has a hideous consequence that I want to mention. And it's why this is my favorite theorem. Because it says, it's not just that you can't name all of them. It's just much worse than that. So the reason I love this theorem is not just that it answers a question from my childhood. But it tells you something kind of shocking about the universe. So when you--if you could somehow magically pick a specific point on the number line, which you can't, because you know, there's--

KK: Right.

SG: You have finite resolution when you pick points in the real world. But pretend you could, then the statement is the chance that the number you picked was a number you could name precisely is very low. Exactly. It's essentially zero.

KK: Yeah.

SG: So the technical way to say this is that the countable subset of real numbers has Lebesgue measure zero.

KK: Right.

SG: So I was feeling a little awkward about using this as my theorem for your podcast, because, you know, the proof is not much. If you know about countable and uncountable, I just told you the whole proof. And you might ask, "What else can I prove using this fact?" And the answer is, I don't know. But we've just learned something about irrational numbers that I think some of your listeners haven't known before. And I think it's a little shocking.

EL: Yeah, yeah. Well, it sounds like I was maybe more of a late bloomer on thinking about this than you, because I remember being in grad school, and just feeling really frustrated one day. I was like, you know, transcendental numbers, the non-algebraic numbers are, you know, 100% of the number line, Lebesgue measure one, and I know like, three of them, essentially. I know, like, e, π, and natural log two. And, you know, really, two of them are already kind of, in a relationship with each other. They're both related to e or the natural log idea. It's just like, okay, 2π. Oh, that's kind of a cheap transcendental number.

Like there's, there's really not that much difference. I mean, I guess then, in a sense, I only know, like, one irrational number, which is square root of 2, like, any other roots of things are non-transcendental, and then I know the rationals, but yeah, it's just like, there are all these numbers, and I know so few of them.

SG: Yeah.

KK: Right. And these other these other things, of course, when you start dealing with infinite series, and you know, you realize that, say, the Sierpinski carpet has area zero, right? But it's uncountable, and you're like, wait a minute, this can't be right. I mean, this is, I think why Cantor was so ridiculed in his time, because it does just seem ridiculous. So you were sitting around in middle school just thinking about this, and your teacher led you down this path? Or was it much later that you figured this out?

SG: Well, I figured out the answer much later. But I worried about it a lot as a child. I used to worry about a lot of things like, your classic question is--if you really want to talk about things I worried about as a child--back in seventh grade, I was really troubled about .99999 with all the nines and whether or not that was one.

EL: Oh yeah.

SG: And I have a terrible story about my eighth grade education regarding that. But in the end, I discovered that they are they are actually equal.

KK: Well, if you make some assumptions, right? I mean, there are number systems, where they're not equal.

SG: Ah, yeah, I'd be happy--I'm not prepared to get into a detailed discussion of the hyperreals.

KK: Neither am I. But what's nice about that idea is that, of course, a lot depends on our assumptions. We we set up rules, and then with the rules that we're used to, .999 repeating is equal to one. But you know, mathematicians like sandboxes, right? Okay, let's go into this sandbox and throw out this rule and see what happens. And then you get non Euclidean geometry, right, or whatever.

SG: Right.

KK: Really beautiful stuff.

SG: I have an analogy for this statement about real numbers that I don't know if your listeners will find compelling or not, but I do, so I'm going to say it unless you stop me.

KK: Okay.

EL: Go for it.

SG: Exactly. So one of the things I find totally amazing about geology is that, you know, we can see rocks that are on the surface of the earth and inspect them, and we can drill down in mines, and we can look at some rocks down there. But fundamentally, most of the geology of the earth, we can't see directly. We've never seen the mantle, we're never going to see the core. And that's most Earth. So nonetheless, there's a lot of great science you can do indirectly by analyzing as an aggregate, by studying the way, earthquake waves propagate and so on. But we're not able to look at things directly. And I think that has an analogy here with the number line, where the rocks can see on the surface are the integers and rationals. You drill down, and you can find some gems or something, and there's your irrational numbers you can name, and then all the ones you'll never be able to name, no matter how hard you try, how much time there is, how many alternate universes filled with people there are, you'll never be able to name, somehow that's like the core because you can't ever actually get directly at them.

EL: Yeah. I like this analogy a lot, because I was just reading about Inge Lehmann who is the Danish seismologist (who I think of as an applied mathematician) who was one of the people who found these different seismic waves that showed that the inner core had the liquid part--or I guess the core had the liquid part and then the solid inner core. She determined that it couldn't all be uniform, basically by doing inverse problems where, like, "Oh, these waves would not have come from this." So that's very relevant to something I just read. Christiane Rousseau actually wrote a really cool article about Inge Lehmann.

SG: Yes, that's a great article.

EL: So yeah, people should look that up.

KK: I'll have to find this.

EL: great analogy. Yeah.

KK: So, we know now that this, this is a long time there for you. So that's another question we've already answered. So, okay, what does one pair with this unknowability?

SG: Ah, so I I think I'm going to have to pair it with one of my favorite TV shows, which is Twin Peaks.

EL: Okay.

SG: So I watch the show, I really enjoy it. But there's a lot of stuff in there that just is impossible to understand.

And you can go read the stuff the people wrote about it on the side, and you can understand a little bit of it. But you know, most of it's clearly never meant to be understood. You're supposed to enjoy it as an aggregate.

KK: That's true. So you and I are the same age, roughly. We were in college when Twin Peaks was a thing. Did you did you watch it then?

SG: No, I just remember that personal ads in the school paper saying, "Anyone who has a video recording of Twin Peaks last week, please tell me. I'll bring doughnuts."

EL: You grew up in a dark time.

SG: Before DVRs, yeah.

KK: That's right. Well, yeah. Before Facebook or anything like that. You had to put an ad in the paper for stuff like this, yeah.

EL: Yeah, I'm really, really understanding the angst of your generation now.

KK: You know what, I kind of preferred it. I kind of like not being reached. Cell phones are kind of a nuisance that way. Although I don't miss paying for phone calls. Remember that, staying up till 11 to not have to pay long distance?

SG: Yeah.

KK: Alright, so Twin Peaks. So you like pie.

SG: Yeah, clearly. And coffee.

KK: And coffee.

SG: And Snoqualmie.

KK: Very good.

SG: I don't know if you--

KK: Sure. I only sort of vaguely remember-- what I remember most about that show is just being frustrated by it, right? Sometimes you'd watch it and a lot would happen. It's like, "Wow, this is bizarre and weird, and David Lynch is a genius." And then there'd be other shows where nothing would happen.

SG: Yes.

KK: I mean, nothing! And, you know, also see Book II of Game of Thrones, for example, where nothing happens, right? Yeah. And David Lynch, of course, was sort of at his peak at that time.

SG: Right.

KK: All right. So Twin Peaks. That's a good pairing because you're right, you'll never figure that out. I think a lot of it was meant to be unknowable.

SG: Yes. Yeah. Have you seen season three of Twin Peaks? The one that was out recently?

KK: No, I don't have cable anymore.

SG: About halfway through that season, there's an episode that is intensely hard to watch because so little happened on it. And if you look at the list the viewership ratings for each episode, there's a steep drop-off in the series at that episode. So this is like the most unknowable part of the number line if you if you follow the analogy.

KK: Okay. All right. That's interesting. So I assume that these these knowable numbers are probably fairly evenly distributed. I guess the rationals are pretty evenly distributed. So yeah.

So So our listeners might wonder if there's some sort of weird distribution to these things, like the ones that you can't name, do they live in certain parts? And the answer is no, they live everywhere.

SG: Yes. That's absolutely right.

EL: I wonder, though, if you can kind of--I'm thinking of continued fraction representations, where there is an explicit definition of number that's well-approximable versus badly-approximable numbers. I guess those are approximable by rationales, not by finite operations or closed form. So maybe that's a bad analogy.

KK: Mm hmm.

SG: Well, if you or your listeners are interested in, thinking about this question some more, then you can Google closed-form number. There's a Wikipedia entry to get people started. And there are a couple of references in there to some really well-written articles on the subject, one by my friend Tim Chow, that was an American Mathematical Monthly, and another one by Borwein and Crandall that was in the Notices of the AMS that's for free on the internet.

EL: Oh, great.

KK: Okay, great. We'll link to those.

EL: And actually, here's, this question, I'm not sure, so I'll just say, is this the same as computable or is closed form a different thing from computable numbers?

SG: Yeah, that's a good question. So there's not a widely-agreed upon definition of the term closed form number. So that's already a question. And then I'm not sure what your definition of computable is.

EL: Me neither.

SG: Okay.

EL: No, I've just heard of the term computable. But yeah, I guess the nice thing is no matter how you define it, your theorem will still be true.

SG: That's right. Exactly.

EL: There's still only countable.

KK: And now we've found something else unknowable: are these the same thing?

SG: Those are really hard questions in general. Yeah. That's the main question plumbed in those articles and referred to is: if you define them in these different ways, how different are they?

EL: Oh, cool.

SG: If you take a particular number, does it sit in which set? Those kinds of questions. Yeah, those are really hard usually, much like you said, what are the transcendental numbers that are--are certain numbers transcendental or not can be a hard question to answer.

EL: Yeah, yeah, even if you think, "Oh yeah, this certainly has to be transcendental, it takes a while to actually prove it."

SG: Yes.

KK: Or maybe you can't. I wonder if some of those statements are even actually undecidable, but again, we don't know. All right, we're going down weird rabbit holes here. Maybe David Lynch could just do a show.

SG: That would be great.

KK: Yeah, there would just be a lot of mathematicians, and nothing would happen

SG: And maybe owls.

KK: And maybe owls. Well, this has been great fun. Thanks for joining us before you head off to work, Skip. Our listeners don't know that it's you know, well, it's now nine in the morning where you are. So thanks for joining us, and I hope your traffic isn't so bad in La Jolla today.

SG: Every day's a great day here. Thank you so much for having me.

KK: Yeah. Thanks, Skip.

21 juli 2017 | 14 min

KK: Welcome to My Favorite Theorem. I’m Kevin Knudson, and I’m joined by my cohost. EL: I’m Evelyn Lamb. KK: This is Episode 0, in which we’ll lay out our ground rules for what we’re going to do. The idea is every week we’ll have a guest, and that guest will tell us what his or her favorite theorem is, and they’ll tell us some fun things about themselves, and Evelyn had good ideas here. What else are we going to do? EL: Yeah, well, with any great thing in life, pairings are important. So we’ll find the perfect wine, or ice cream, or work of 19th century German romanticism to include with the theorem. We’ll ask our guests to help us with that. KK: Since this is episode 0, we thought we should probably set the tone and let you know what our favorite theorems are. I’m going to defer. I’m going to let Evelyn go first here. What’s your favorite theorem? EL: OK, so we’re recording this on March 23rd, which is Emmy Noether’s birthday, her 135th, to be precise. I feel like I should say Noether’s theorem. It’s a theorem in physics that relates, that says basically conserved quantities in physics come from symmetries in nature. So time translation symmetry yields conservation of energy and things like that. But I’m not going to say that one. I’m sorry, physics, I just like math more. So I’m going to pick the uniformization theorem as my favorite theorem. KK: I don’t think I know that theorem. Which one is this? EL: It’s a great theorem. When I was doing math research, I was working in Teichmüller theory, which is related to hyperbolic geometry. This is a theorem about two-dimensional surfaces. The upshot of this theorem is that every two-dimensional surface can be given geometry that is either spherical, flat — so, Euclidean, like the flat plane — or hyperbolic. The uniformization itself is related to simply connected Riemann surfaces, the ones with no holes, but using this theorem you can show that 2-d surfaces with any number of holes have one of these kinds of 2-d geometry. This is a great theorem. I just love that part of topology where you’re classifying surfaces and everything. I think it’s nice A little of the history is that it was conjectured by Poincaré in 1882 and Klein in 1883. I think the first proof was by Poincaré in the early 1900s. There are a lot of proofs of it that come from different approaches. KK: Now that you tell me what the theorem is, of course I know what it is. Being a topologist, I know how to classify surfaces, I think. That is a great theorem. There’s so much going on there. You can think about Riemann surfaces as quotients of hyperbolic space, and you have all this fun geometry going. I love that theorem. In fact, I’m teaching our graduate topology course this year, and I didn’t do this. I’m sorry. I had to get through homology and cohomology. So yeah, surfaces are classified. We know surfaces. So what are you going to pair this with? EL: So my pairing is Neapolitan ice cream. I’m going a bit literal with this. Neapolitan ice cream is the ice cream that has part of it vanilla, part of it chocolate, and part of it strawberry. So this theorem says that surfaces come in three flavors. KK: Nice. EL: When I was a little kid, when we had our birthday parties at home, my mom always let us pick what ice cream we wanted to have, and I always picked Neapolitan so that if my friends liked one of the flavors but not the others, they could have whichever flavor they wanted. KK: You’re too kind. EL: Really, I’m just such a good-hearted person. KK: Clearly. EL: Yeah, Neapolitan. Three flavors of surfaces, three flavors of ice cream. KK: Nice. Although nobody ever eats the strawberry, right? EL: Yeah, I love strawberry ice cream now, but yeah, when I was a little kid chocolate and vanilla were a little more my thing. KK: I remember my mother would sometimes buy the Neapolitan, and I remember the strawberry would just sit there, uneaten, until it got freezer burn, and we just threw it away at that point. EL: I guess the question is, which of the kinds of geometry is strawberry? KK: Well, vanilla is clearly flat, right? EL: Yeah, that’s good. I guess that means strawberry must be spherical. KK: That seems right. It’s pretty unique, right? Spherical geometry is kind of dull, right? There’s just the sphere. There’s a lot more variation in hyperbolic geometry, right? EL: Yeah, I guess so. I feel like there are more different kinds of chocolate-flavored ice cream, and hyperbolic, there are so many different hyperbolic surfaces. KK: Right. Here in Gainesville, we have a really wonderful local ice cream place, and twice a year they have chocolate night, and they have 32 different varieties of chocolate. EL: Oh my gosh. KK: So you can go and you can get a ginormous bowl of all 36 flavors if you want, but we usually get a little sample of eight different flavors and try them out. It’s really wonderful. I think that’s the right classification. EL: OK. So Kevin? KK: Yes? EL: What is your favorite theorem? KK: Well, yeah, I thought about this for a long time, and what I came up with was that my favorite theorem is the ham sandwich theorem. I think it’s largely because it’s got a fun name, right? EL: Yeah. KK: And I remember hearing about this theorem as an undergrad for the first time. This was a general topology course, and you don’t prove it in that, I think. You need some algebraic topology to prove this well. I thought, wow, what a cool thing! There’s something called the ham sandwich theorem. So what is the ham sandwich theorem? It says: say you have a ham sandwich, which consists of two pieces of bread and a chunk of ham. And maybe you got a little nuts and you put one piece of bread on top of the fridge, and one on the floor, and your ham is sitting on the counter, and the theorem is that if you have a long enough knife, you can make one cut and cut all of those things in half. Mathematically what that means is that you have three blobs in space, and there is a single plane that cuts each of those blobs in half exactly. I just thought that was a pretty remarkable theorem, and I still think it’s kind of remarkable theorem because it’s kind of hard to picture, right? Your blobs could be anywhere. They could be really far apart, as long as they have positive measure, so as long as they’re not some flat thing, they actually have some 3-d-ness to them, then you can actually find a plane that does this. What’s even more fun, I think, is that this is a consequence of the Borsuk-Ulam theorem, which in this case would say that if you have a continuous function from the 2-sphere to the plane, then two antipodal points have to go to the same place. And that’s always a fun theorem to explain to people who don’t know any mathematics, because you can say, somewhere, right now, there are two opposite points on the surface of the earth where the temperature and the humidity are the same, for example. EL: Yeah. KK: I love that kind of theorem, where there’s a good physical interpretation for it. And of course there are higher-dimensional analogues, but the idea of the ham sandwich theorem is great. Everybody’s had a ham sandwich, probably, or some kind of sandwich. It doesn’t have to be ham. Maybe we should be more politically correct. What’s a good sandwich? EL: A peanut butter sandwich is a great sandwich. KK: A peanut butter sandwich. But the peanut butter is kind of hard to get going, right? You don’t really want that anywhere except in the middle of the sandwich. You don’t want to imagine this blob of peanut butter. The ham you can kind of imagine. EL: It’s really saying that you don’t even have to remove the peanut butter from the jar. You can leave the peanut butter in the jar. KK: There you go. EL: You can cut this sandwich in half. KK: Your knife’s going to have to cut through the whole jar. It’s gotta be a pretty strong knife. EL: Yeah. We’re already asking for an arbitrarily long knife. KK: Yes. EL: You don’t think our arbitrarily long knife can cut through glass? Come on. KK: It probably can, you’re right. How silly of me. If we’re being so silly and hyperbolic, we might as well. EL: We’re mathematicians, after all. KK: You’re right, we are. So I thought about the pairing, too. Basically, I’ve got a croque monsieur, right? EL: Right. KK: You’re in France. You probably eat these all the time. So what does one have with a croque monsieur? It’s not really fancy food. So I think you’ve got to go with a beer for this, and if I’m getting to choose any beer, we have a wonderful local brewery here, First Magnitude brewery, it’s owned by a good friend of mine. They have a really nice pale ale. It’s called 72 Pale Ale. I invite everyone to look up First Magnitude Brewing on the internet there and check them out. It’s a good beer. Not too hoppy. EL: OK. KK: It’s hoppy enough, but it’s not one of those West Coast IPA’s that makes your mouth shrivel up. EL: Yeah, socks you in the face with the hops. KK: Yeah, you don’t need all of that. EL: So actually, if you think of the two pieces of bread as one mass of bread and the ham as its own thing, then you could also bisect the bread, the ham, and the beer with one knife. KK: That’s right, we could do that. EL: Yeah, if you really wanted to make sure to eat your meal in two identical halves. KK: Right. So you have vanilla donuts and balls of chocolate, no, no, the donuts, wait a minute. The hyperbolic spaces were chocolate. This is starting to break down. But the flat geometry is the plane. But there’s a flat torus too, right? So you could have a flat donut, or a flat plane. Very cool. This is fun. I think we’re going to have a good time doing this. EL: I think so too. And I think we’re going to end each episode hungry. KK: It sounds that way, yeah. In the weeks to come, we have a pretty good lineup of interesting people from all areas of mathematics and all parts of the world, hopefully. I’m excited about this project. So thanks, Evelyn, for coming along with me on this. EL: Yeah. Thank you for inviting me. I’m looking forward to this. KK: Until next time, this has been My Favorite Theorem. KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at [email protected]. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.

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