The Cartesian Cafe

Michael Freedman is a mathematician who was awarded the Fields Medal in 1986 for his solution of the 4-dimensional Poincare conjecture. Mike has also received numerous other awards for his scientific contributions including a MacArthur Fellowship and the National Medal of Science. In 1997, Mike joined Microsoft Research and in 2005 became the director of Station Q, Microsoft’s quantum computing research lab. As of 2023, Mike is a Senior Research Scientist at the Center for Mathematics and Scientific Applications at Harvard University.

Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen

In this wide-ranging conversation, we give a panoramic view of Mike’s extensive body of work over the span of his career. It is divided into three parts: early, middle, and present day, which respectively include his work on the 4-dimensional Poincare conjecture, his transition to topological physics, and finally his recent work in applying ideas from mathematics and philosophy to social economics. Our conversation is a blend of both the nitty-gritty details and the anecdotal story-telling that can only be obtained from a living legend.

I. Introduction

• 00:00 : Preview
• 01:34 : Fields Medalist working in industry
• 03:24 : Academia vs industry
• 04:59 : Mathematics and art
• 06:33 : Technical overview

II. Early Mike: The Poincare Conjecture (PC)

• 08:14 : Introduction, statement, and history
• 14:30 : Three categories for PC (topological, smooth, PL)
• 17:09 : Smale and PC for d at least 5
• 17:59 : Homotopy equivalence vs homeomorphism
• 22:08 : Joke
• 23:24 : Morse flow
• 33:21 : Whitney Disk
• 41:47 : Casson handles
• 50:24 : Manifold factors and the Whitehead continuum
• 1:00:39 : Donaldson’s results in the smooth category
• 1:04:54 : (Not) writing up full details of the proof then and now
• 1:08:56 : Why Perelman succeeded

II. Mid Mike: Topological Quantum Field Theory (TQFT) and Quantum Computing (QC)

• 1:10:54: Introduction
• 1:11:42: Cliff Taubes, Raoul Bott, Ed Witten
• 1:12:40 : Computational complexity, Church-Turing, and Mike’s motivations
• 1:24:01 : Why Mike left academia, Microsoft’s offer, and Station Q
• 1:29:23 : Topological quantum field theory (according to Atiyah)
• 1:34:29 : Anyons and a theorem on Chern-Simons theories
• 1:38:57 : Relation to QC
• 1:46:08 : Universal TQFT
• 1:55:57 : Witten: Donalson theory cannot be a unitary TQFT
• 2:01:22 : Unitarity is possible in dimension 3
• 2:05:12 : Relations to a theory of everything?
• 2:07:21 : Where topological QC is now

III. Present Mike: Social Economics

• 2:11:08 : Introduction
• 2:14:02 : Lionel Penrose and voting schemes
• 2:21:01 : Radical markets (pun intended)
• 2:25:45 : Quadratic finance/funding
• 2:30:51 : Kant’s categorical imperative and a paper of Vitalik Buterin, Zoe Hitzig, Glen Weyl
• 2:36:54 : Gauge equivariance
• 2:38:32 : Bertrand Russell: philosophers and differential equations

IV: Outro

• 2:46:20 : Final thoughts on math, science, philosophy
• 2:51:22 : Career advice

 

Some Further Reading:
Mike’s Harvard lecture on PC4: https://www.youtube.com/watch?v=TSF0i6BO1Ig
Behrens et al. The Disc Embedding Theorem.
M. Freedman. Spinoza, Leibniz, Kant, and Weyl. arxiv:2206.14711

 

Twitter:
@iamtimnguyen

 

Webpage:
http://www.timothynguyen.org

Marcus Hutter is an artificial intelligence researcher who is both a Senior Researcher at Google DeepMind and an Honorary Professor in the Research School of Computer Science at Australian National University. He is responsible for the development of the theory of Universal Artificial Intelligence, for which he has written two books, one back in 2005 and one coming right off the press as we speak. Marcus is also the creator of the Hutter prize, for which you can win a sizable fortune for achieving state of the art lossless compression of Wikipedia text.

Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen

In this technical conversation, we cover material from Marcus’s two books “Universal Artificial Intelligence” (2005) and “Introduction to Universal Artificial Intelligence” (2024). The main goal is to develop a mathematical theory for combining sequential prediction (which seeks to predict the distribution of the next observation) together with action (which seeks to maximize expected reward), since these are among the problems that intelligent agents face when interacting in an unknown environment. Solomonoff induction provides a universal approach to sequence prediction in that it constructs an optimal prior (in a certain sense) over the space of all computable distributions of sequences, thus enabling Bayesian updating to enable convergence to the true predictive distribution (assuming the latter is computable). Combining Solomonoff induction with optimal action leads us to an agent known as AIXI, which in this theoretical setting, can be argued to be a mathematical incarnation of artificial general intelligence (AGI): it is an agent which acts optimally in general, unknown environments. The second half of our discussion concerning agents assumes familiarity with the basic setup of reinforcement learning.

I. Introduction

• 00:38 : Biography
• 01:45 : From Physics to AI
• 03:05 : Hutter Prize
• 06:25 : Overview of Universal Artificial Intelligence
• 11:10 : Technical outline

II. Universal Prediction

• 18:27 : Laplace’s Rule and Bayesian Sequence Prediction
• 40:54 : Different priors: KT estimator
• 44:39 : Sequence prediction for countable hypothesis class
• 53:23 : Generalized Solomonoff Bound (GSB)
• 57:56 : Example of GSB for uniform prior
• 1:04:24 : GSB for continuous hypothesis classes
• 1:08:28 : Context tree weighting
• 1:12:31 : Kolmogorov complexity
• 1:19:36 : Solomonoff Bound & Solomonoff Induction
• 1:21:27 : Optimality of Solomonoff Induction
• 1:24:48 : Solomonoff a priori distribution in terms of random Turing machines
• 1:28:37 : Large Language Models (LLMs)
• 1:37:07 : Using LLMs to emulate Solomonoff induction
• 1:41:41 : Loss functions
• 1:50:59 : Optimality of Solomonoff induction revisited
• 1:51:51 : Marvin Minsky

III. Universal Agents

• 1:52:42 : Recap and intro
• 1:55:59 : Setup
• 2:06:32 : Bayesian mixture environment
• 2:08:02 : AIxi. Bayes optimal policy vs optimal policy
• 2:11:27 : AIXI (AIxi with xi = Solomonoff a priori distribution)
• 2:12:04 : AIXI and AGI. Clarification: ASI (Artificial Super Intelligence) would be a more appropriate term than AGI for the AIXI agent.
• 2:12:41 : Legg-Hutter measure of intelligence
• 2:15:35 : AIXI explicit formula
• 2:23:53 : Other agents (optimistic agent, Thompson sampling, etc)
• 2:33:09 : Multiagent setting
• 2:39:38 : Grain of Truth problem
• 2:44:38 : Positive solution to Grain of Truth guarantees convergence to a Nash equilibria
• 2:45:01 : Computable approximations (simplifying assumptions on model classes): MDP, CTW, LLMs
• 2:56:13 : Outro: Brief philosophical remarks

 

Further Reading:
M. Hutter, D. Quarrel, E. Catt. An Introduction to Universal Artificial Intelligence
M. Hutter. Universal Artificial Intelligence
S. Legg and M. Hutter. Universal Intelligence: A Definition of Machine Intelligence

 

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Richard Borcherds is a mathematician and professor at University of California Berkeley known for his work on lattices, group theory, and infinite-dimensional algebras. His numerous accolades include being awarded the Fields Medal in 1998 and being elected a fellow of the American Mathematical Society and the National Academy of Sciences.

Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen

In this episode, Richard and I give an overview of Richard's most famous result: his proof of the Monstrous Moonshine conjecture relating the monster group on the one hand and modular forms on the other. A remarkable feature of the proof is that it involves vertex algebras inspired from elements of string theory. Some familiarity with group theory and representation theory are assumed in our discussion.

I. Introduction

• 00:25: Biography
• 02:51 : Success in mathematics
• 04:04 : Monstrous Moonshine overview and John Conway
• 09:44 : Technical overview

II. Group Theory

• 11:31 : Classification of finite-simple groups + history of the monster group
• 18:03 : Conway groups + Leech lattice
• 22:13 : Why was the monster conjectured to exist + more history 28:43 : Centralizers and involutions
• 32:37: Griess algebra

III. Modular Forms

• 36:42 : Definitions
• 40:06 : The elliptic modular function
• 48:58 : Subgroups of SL_2(Z)

IV. Monstrous Moonshine Conjecture Statement

• 57:17: Representations of the monster
• 59:22 : Hauptmoduls
• 1:03:50 : Statement of the conjecture
• 1:07:06 : Atkin-Fong-Smith's first proof
• 1:09:34 : Frenkel-Lepowski-Meurman's work + significance of Borcherd's proof

V. Sketch of Proof

• 1:14:47: Vertex algebra and monster Lie algebra
• 1:21:02 : No ghost theorem from string theory
• 1:25:24 : What's special about dimension 26?
• 1:28:33 : Monster Lie algebra details
• 1:32:30 : Dynkin diagrams and Kac-Moody algebras
• 1:43:21 : Simple roots and an obscure identity
• 1:45:13: Weyl denominator formula, Vandermonde identity
• 1:52:14 : Chasing down where modular forms got smuggled in
• 1:55:03 : Final calculations

VI. Epilogue

• 1:57:53 : Your most proud result?
• 2:00:47 : Monstrous moonshine for other sporadic groups?
• 2:02:28 : Connections to other fields. Witten and black holes and mock modular forms.

 

Further reading: V Tatitschef. A short introduction to Monstrous Moonshine. https://arxiv.org/pdf/1902.03118.pdf

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Thought I'd share some exciting news about what's happening at The Cartesian Cafe in 2024 and also a personal message to viewers on how they can support the cafe.

Patreon:

https://www.patreon.com/timothynguyen

Tim Maudlin is a philosopher of science specializing in the foundations of physics, metaphysics, and logic. He is a professor at New York University, a member of the Foundational Questions Institute, and the founder and director of the John Bell Institute for the Foundations of Physics.

Patreon (bonus materials + video chat):
https://www.patreon.com/timothynguyen

In this very in-depth discussion, Tim and I probe the foundations of science through the avenues of locality and determinism as arising from the Einstein-Poldosky-Rosen (EPR) paradox and Bell's Theorem. These issues are so intricate that even the Nobel Prize committee incorrectly described the significance of Bell's work in their press release for the 2022 prize in physics. Viewers motivated enough to think deeply about these ideas will be rewarded with a conceptually proper understanding of the nonlocal nature of physics and its manifestation in quantum theory.

I. Introduction 00:00 :

• 00:25: Biography
• 05:26: Interdisciplinary work
• 11:54 : Physicists working on the wrong things
• 16:47 : Bell's Theorem soft overview
• 24:14: Common misunderstanding of "God does not play dice."
• 25:59: Technical outline

II. EPR Paradox / Argument

• 29:14 : EPR is not a paradox
• 34:57 : Criterion of reality
• 43:57 : Mathematical formulation
• 46:32 : Locality: No spooky action at a distance
• 49:54 : Bertlmann's socks
• 53:17 : EPR syllogism summarized
• 54:52 : Determinism is inferred not assumed
• 1:02:18 : Clarifying analogy: Coin flips
• 1:06:39 : Einstein's objection to determinism revisited

III. Bohm Segue

• 1:11:05 : Introduction
• 1:13:38: Bell and von Neumann's error
• 1:20:14: Bell's motivation: Can I remove Bohm's nonlocality?

IV. Bell's Theorem and Related Examples

• 1:25:13 : Setup
• 1:27:59 : Decoding Bell's words: Locality is the key!
• 1:34:16 : Bell's inequality (overview)
• 1:36:46 : Bell's inequality (math)
• 1:39:15 : Concrete example of violation of Bell's inequality
• 1:49:42: GHZ Example

V. Miscellany

• 2:06:23 : Statistical independence assumption
• 2:13:18: The 2022 Nobel Prize
• 2:17:43: Misconceptions and hidden variables
• 2:22:28: The assumption of local realism? Repeat: Determinism is a conclusion not an assumption.

VI. Interpretations of Quantum Mechanics

• 2:28:44: Interpretation is a misnomer
• 2:29:48: Three requirements. You can only pick two.
• 2:34:52: Copenhagen interpretation?

 

Further Reading:

J. Bell. Speakable and Unspeakable in Quantum Mechanics

T. Maudlin. Quantum Non-Locality and Relativity

Wikipedia: Mermin's device, GHZ experiment

 

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Antonio (Tony) Padilla is a theoretical physicist and cosmologist at the University of Nottingham. He serves as the Associate Director of the Nottingham Centre of Gravity, and in 2016, Tony shared the Buchalter Cosmology Prize for his work on the cosmological constant. Tony is also a star of the Numberphile YouTube channel, where his videos have received millions of views and he is also the author of the book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity.

Patreon: https://www.patreon.com/timothynguyen

This episode combines some of the greatest cosmological questions together with mathematical imagination. Tony and I go through the math behind some oft-quoted numbers in cosmology and calculate the age, size, and number of atoms in the universe. We then stretch our brains and consider how likely it would be to find your Doppelganger in a truly large universe, which takes us on a detour through black hole entropy. We end with a discussion of naturalness and the anthropic principle to round out our discussion of fantastic numbers in physics.

Part I. Introduction

• 00:00 : Introduction
• 01:06 : Math and or versus physics
• 12:09 : Backstory behind Tony's book
• 14:12 : Joke about theoreticians and numbers
• 16:18 : Technical outline

Part II. Size, Age, and Quantity in the Universe

• 21:42 : Size of the observable universe
• 22:32 : Standard candles
• 27:39 : Hubble rate
• 29:02 : Measuring distances and time
• 37:15 : Einstein and Minkowski
• 40:52 : Definition of Hubble parameter
• 42:14 : Friedmann equation
• 47:11 : Calculating the size of the observable universe
• 51:24 : Age of the universe
• 56:14 : Number of atoms in the observable universe
• 1:01:08 : Critical density
• 1:03:16: 10^80 atoms of hydrogen
• 1:03:46 : Universe versus observable universe

Part III. Extreme Physics and Doppelgangers

• 1:07:27 : Long-term fate of the universe
• 1:08:28 : Black holes and a googol years
• 1:09:59 : Poincare recurrence
• 1:13:23 : Doppelgangers in a googolplex meter wide universe
• 1:16:40 : Finitely many states and black hole entropy
• 1:25:00 : Black holes have no hair
• 1:29:30 : Beckenstein, Christodolou, Hawking
• 1:33:12 : Susskind's thought experiment: Maximum entropy of space
• 1:42:58 : Estimating the number of doppelgangers
• 1:54:21 : Poincare recurrence: Tower of four exponents.

Part IV: Naturalness and Anthropics

• 1:54:34 : What is naturalness? Examples.
• 2:04:09 : Cosmological constant problem: 10^120 discrepancy
• 2:07:29 : Interlude: Energy shift clarification. Gravity is key.
• 2:15:34 : Corrections to the cosmological constant
• 2:18:47 : String theory landscape: 10^500 possibilities
• 2:20:41 : Anthropic selection
• 2:25:59 : Is the anthropic principle unscientific? Weinberg and predictions.
• 2:29:17 : Vacuum sequestration

Further reading: Antonio Padilla. Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Boaz Barak is a professor of computer science at Harvard University, having previously been a principal researcher at Microsoft Research and a professor at Princeton University. His research interests span many areas of theoretical computer science including cryptography, computational complexity, and the foundations of machine learning. Boaz serves on the scientific advisory boards for Quanta Magazine and the Simons Institute for the Theory of Computing and he was selected for Foreign Policy magazine’s list of 100 leading global thinkers for 2014.

www.patreon.com/timothynguyen

Cryptography is about maintaining the privacy and security of communication. In this episode, Boaz and I go through the fundamentals of cryptography from a foundational mathematical perspective. We start with some historical examples of attempts at encrypting messages and how they failed. After some guesses as to how one might mathematically define security, we arrive at the one due to Shannon. The resulting definition of perfect secrecy turns out to be too rigid, which leads us to the notion of computational secrecy that forms the foundation of modern cryptographic systems. We then show how the existence of pseudorandom generators (which remains a conjecture) ensures that such computational secrecy is achievable, assuming P does not equal NP. Having covered private key cryptography in detail, we then give a brief overview of public key cryptography. We end with a brief discussion of Bitcoin, machine learning, deepfakes, and potential doomsday scenarios.

I. Introduction

• 00:17 : Biography: Academia vs Industry
• 10:07 : Military service
• 12:53 : Technical overview
• 17:01 : Whiteboard outline

II. Warmup

• 24:42 : Substitution ciphers
• 27:33 : Viginere cipher
• 29:35 : Babbage and Kasiski
• 31:25 : Enigma and WW2
• 33:10 : Alan Turing

III. Private Key Cryptography: Perfect Secrecy

• 34:32 : Valid encryption scheme
• 40:14 : Kerckhoffs's Principle
• 42:41 : Cryptography = steelman your adversary
• 44:40 : Attempt #1 at perfect secrecy
• 49:58 : Attempt #2 at perfect secrecy
• 56:02 : Definition of perfect secrecy (Shannon)
• 1:05:56 : Enigma was not perfectly secure
• 1:08:51 : Analogy with differential privacy
• 1:11:10 : Example: One-time pad (OTP)
• 1:20:07 : Drawbacks of OTP and Soviet KGB misuse
• 1:21:43 : Important: Keys cannot be reused!
• 1:27:48 : Shannon's Impossibility Theorem

IV. Computational Secrecy

• 1:32:52 : Relax perfect secrecy to computational secrecy
• 1:41:04 : What computational secrecy buys (if P is not NP)
• 1:44:35 : Pseudorandom generators (PRGs)
• 1:47:03 : PRG definition
• 1:52:30 : PRGs and P vs NP
• 1:55:47: PRGs enable modifying OTP for computational secrecy

V. Public Key Cryptography

• 2:00:32 : Limitations of private key cryptography
• 2:09:25 : Overview of public key methods
• 2:13:28 : Post quantum cryptography

VI. Applications

• 2:14:39 : Bitcoin
• 2:18:21 : Digital signatures (authentication)
• 2:23:56 : Machine learning and deepfakes
• 2:30:31 : A conceivable doomsday scenario: P = NP

Further reading: Boaz Barak. An Intensive Introduction to Cryptography

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Sean Carroll is a theoretical physicist and philosopher who specializes in quantum mechanics, cosmology, and the philosophy of science. He is the Homewood Professor of Natural Philosophy at Johns Hopkins University and an external professor at the Sante Fe Institute. Sean has contributed prolifically to the public understanding of science through a variety of mediums: as an author of several physics books including Something Deeply Hidden and The Biggest Ideas in the Universe, as a public speaker and debater on a wide variety of scientific and philosophical subjects, and also as a host of his podcast Mindscape which covers topics spanning science, society, philosophy, culture, and the arts.

www.patreon.com/timothynguyen

In this episode, we take a deep dive into The Many Worlds (Everettian) Interpretation of quantum mechanics. While there are many philosophical discussions of the Many Worlds Interpretation available, ours marries philosophy with the technical, mathematical details. As a bonus, the whole gamut of topics from philosophy and physics arise, including the nature of reality, emergence, Bohmian mechanics, Bell's Theorem, and more. We conclude with some analysis of Sean's speculative work on the concept of emergent spacetime, a viewpoint which naturally arises from Many Worlds. This video is most suitable for those with a basic technical understanding of quantum mechanics.

Part I: Introduction

• 00:00:00 : Introduction
• 00:05:42 : Philosophy and science: more interdisciplinary work?
• 00:09:14 : How Sean got interested in Many Worlds (MW)
• 00:13:04 : Technical outline

Part II: Quantum Mechanics in a Nutshell

• 00:14:58 : Textbook QM review
• 00:24:25 : The measurement problem
• 00:25:28 : Einstein: "God does not play dice"
• 00:27:49 : The reality problem

Part III: Many Worlds

• 00:31:53 : How MW comes in
• 00:34:28 : EPR paradox (original formulation)
• 00:40:58 : Simpler to work with spin
• 00:42:03 : Spin entanglement
• 00:44:46 : Decoherence
• 00:49:16 : System, observer, environment clarification for decoherence
• 00:53:54 : Density matrix perspective (sketch)
• 00:56:21 : Deriving the Born rule
• 00:59:09 : Everett: right answer, wrong reason. The easy and hard part of Born's rule.
• 01:03:33 : Self-locating uncertainty: which world am I in?
• 01:04:59 : Two arguments for Born rule credences
• 01:11:28 : Observer-system split: pointer-state problem
• 01:13:11 : Schrodinger's cat and decoherence
• 01:18:21 : Consciousness and perception
• 01:21:12 : Emergence and MW
• 01:28:06 : Sorites Paradox and are there infinitely many worlds
• 01:32:50 : Bad objection to MW: "It's not falsifiable."

Part IV: Additional Topics

• 01:35:13 : Bohmian mechanics
• 01:40:29 : Bell's Theorem. What the Nobel Prize committee got wrong
• 01:41:56 : David Deutsch on Bohmian mechanics
• 01:46:39 : Quantum mereology
• 01:49:09 : Path integral and double slit: virtual and distinct worlds

Part V. Emergent Spacetime

• 01:55:05 : Setup
• 02:02:42 : Algebraic geometry / functional analysis perspective
• 02:04:54 : Relation to MW

Part VI. Conclusion

• 02:07:16 : Distribution of QM beliefs
• 02:08:38 : Locality

 

Further reading:

• Hugh Everett. The Theory of the Universal Wave Function, 1956.
• Sean Carroll. Something Deeply Hidden, 2019.

More Sean Carroll & Timothy Nguyen:

Fragments of the IDW: Joe Rogan, Sam Harris, Eric Weinstein: https://youtu.be/jM2FQrRYyas

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

 

Daniel Schroeder is a particle and accelerator physicist and an editor for The American Journal of Physics. Dan received his PhD from Stanford University, where he spent most of his time at the Stanford Linear Accelerator, and he is currently a professor in the department of physics and astronomy at Weber State University. Dan is also the author of two revered physics textbooks, the first with Michael Peskin called An Introduction to Quantum Field Theory (or simply Peskin & Schroeder within the physics community) and the second An Introduction to Thermal Physics. Dan enjoys teaching physics courses at all levels, from Elementary Astronomy through Quantum Mechanics.

In this episode, I get to connect with one of my teachers, having taken both thermodynamics and quantum field theory courses when I was a university student based on Dan's textbooks. We take a deep dive towards answering two fundamental questions in the subject of thermodynamics: what is temperature and what is entropy? We provide both a qualitative and quantitative analysis, discussing good and bad definitions of temperature, microstates and macrostates, the second law of thermodynamics, and the relationship between temperature and entropy. Our discussion was also a great chance to shed light on some of the philosophical assumptions and conundrums in thermodynamics that do not typically come up in a physics course: the fundamental assumption of statistical mechanics, Laplace's demon, and the arrow of time problem (Loschmidt's paradox) arising from the second law of thermodynamics (i.e. why is entropy increasing in the future when mechanics has time-reversal symmetry).

Patreon: https://www.patreon.com/timothynguyen

Outline:

• 00:00:00 : Introduction
• 00:01:54 : Writing Books
• 00:06:51 : Academic Track: Research vs Teaching
• 00:11:01 : Charming Book Snippets
• 00:14:54 : Discussion Plan: Two Basic Questions
• 00:17:19 : Temperature is What You Measure with a Thermometer
• 00:22:50 : Bad definition of Temperature: Measure of Average Kinetic Energy
• 00:25:17 : Equipartition Theorem
• 00:26:10 : Relaxation Time
• 00:27:55 : Entropy from Statistical Mechanics
• 00:30:12 : Einstein solid
• 00:32:43 : Microstates + Example Computation
• 00:38:33: Fundamental Assumption of Statistical Mechanics (FASM)
• 00:46:29 : Multiplicity is highly concentrated about its peak
• 00:49:50 : Entropy is Log(Multiplicity)
• 00:52:02 : The Second Law of Thermodynamics
• 00:56:13 : FASM based on our ignorance?
• 00:57:37 : Quantum Mechanics and Discretization
• 00:58:30 : More general mathematical notions of entropy
• 01:02:52 : Unscrambling an Egg and The Second Law of Thermodynamics
• 01:06:49 : Principle of Detailed Balance
• 01:09:52 : How important is FASM?
• 01:12:03 : Laplace's Demon
• 01:13:35 : The Arrow of Time (Loschmidt's Paradox)
• 01:15:20 : Comments on Resolution of Arrow of Time Problem
• 01:16:07 : Temperature revisited: The actual definition in terms of entropy
• 01:25:24 : Historical comments: Clausius, Boltzmann, Carnot
• 01:29:07 : Final Thoughts: Learning Thermodynamics

 

Further Reading:

• Daniel Schroeder. An Introduction to Thermal Physics
• L. Landau & E. Lifschitz. Statistical Physics.

 

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Ethan Siegel is a theoretical astrophysicist and science communicator. He received his PhD from the University of Florida and held academic positions at the University of Arizona, University of Oregon, and Lewis & Clark College before moving on to become a full-time science writer. Ethan is the author of the book Beyond The Galaxy, which is the story of “How Humanity Looked Beyond Our Milky Way And Discovered The Entire Universe” and he has contributed numerous articles to ScienceBlogs, Forbes, and BigThink. Today, Ethan is the face and personality behind Starts With A Bang, both a website and podcast by the same name that is dedicated to explaining and exploring the deepest mysteries of the cosmos.

In this episode, Ethan and I discuss the mysterious nature of dark matter: the evidence for it and the proposals for what it might be.

Patreon: https://www.patreon.com/timothynguyen

Part I. Introduction

• 00:00:00 : Biography and path to science writing
• 00:07:26 : Keeping up with the field outside academia
• 00:11:42 : If you have a bone to pick with Ethan...
• 00:12:50 : On looking like a scientist and words of wisdom
• 00:18:24 : Understanding dark matter = one of the most important open problems
• 00:21:07 : Technical outline

Part II. Ordinary Matter

• 23:28 : Matter and radiation scaling relations
• 29:36 : Hubble constant
• 31:00 : Components of rho in Friedmann's equations
• 34:14 : Constituents of the universe
• 41:21 : Big Bang nucleosynthesis (BBN)
• 45:32 : eta: baryon to photon ratio and deuterium formation
• 53:15 : Mass ratios vs eta

Part III. Dark Matter

• 1:01:02 : rho = radiation + ordinary matter + dark matter + dark energy
• 1:05:25 : nature of peaks and valleys in cosmic microwave background (CMB): need dark matter
• 1:07:39: Fritz Zwicky and mass mismatch among galaxies of a cluster
• 1:10:40 : Kent Ford and Vera Rubin and and mass mismatch within a galaxy
• 1:11:56 : Recap: BBN tells us that only about 5% of matter is ordinary
• 1:15:55 : Concordance model (Lambda-CDM)
• 1:21:04 : Summary of how dark matter provides a common solution to many problems
• 1:23:29 : Brief remarks on modified gravity
• 1:24:39 : Bullet cluster as evidence for dark matter
• 1:31:40 : Candidates for dark matter (neutrinos, WIMPs, axions)
• 1:38:37 : Experiment vs theory. Giving up vs forging on
• 1:48:34 : Conclusion

Image Credits: http://timothynguyen.org/image-credits/

Further learning:

• E. Siegel. Beyond the Galaxy
• Ethan Siegel's webpage: www.startswithabang.com

 

More Ethan Siegel & Timothy Nguyen videos:

• Brian Keating’s Losing the Nobel Prize Makes a Good Point but …
  https://youtu.be/iJ-vraVtCzw
• Testing Eric Weinstein's and Stephen Wolfram's Theories of Everything
  https://youtu.be/DPvD4VnD5Z4

 

Twitter: @iamtimnguyen
Webpage: http://www.timothynguyen.org

Alex Kontorovich is a Professor of Mathematics at Rutgers University and served as the Distinguished Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics in 2020–2021. Alex has received numerous awards for his illustrious mathematical career, including the Levi L. Conant Prize in 2013 for mathematical exposition, a Simons Foundation Fellowship, an NSF career award, and being elected Fellow of the American Mathematical Society in 2017. He currently serves on the Scientific Advisory Board of Quanta Magazine and as Editor-in-Chief of the Journal of Experimental Mathematics.

In this episode, Alex takes us from the ancient beginnings to the present day on the subject of circle packings. We start with the Problem of Apollonius on finding tangent circles using straight-edge and compass and continue forward in basic Euclidean geometry up until the time of Leibniz whereupon we encounter the first complete notion of a circle packing. From here, the plot thickens with observations on surprising number theoretic coincidences, which only received full appreciation through the craftsmanship of chemistry Nobel laureate Frederick Soddy. We continue on with more advanced mathematics arising from the confluence of geometry, group theory, and number theory, including fractals and their dimension, hyperbolic dynamics, Coxeter groups, and the local to global principle of advanced number theory. We conclude with a brief discussion on extensions to sphere packings.

Patreon: http://www.patreon.com/timothynguyen

I. Introduction

• 00:00: Biography
• 11:08: Lean and Formal Theorem Proving
• 13:05: Competitiveness and academia
• 15:02: Erdos and The Book
• 19:36: I am richer than Elon Musk
• 21:43: Overview

II. Setup

• 24:23: Triangles and tangent circles
• 27:10: The Problem of Apollonius
• 28:27: Circle inversion (Viette’s solution)
• 36:06: Hartshorne’s Euclidean geometry book: Minimal straight-edge & compass constructions

III. Circle Packings

• 41:49: Iterating tangent circles: Apollonian circle packing
• 43:22: History: Notebooks of Leibniz
• 45:05: Orientations (inside and outside of packing)
• 45:47: Asymptotics of circle packings
• 48:50: Fractals
• 50:54: Metacomment: Mathematical intuition
• 51:42: Naive dimension (of Cantor set and Sierpinski Triangle)
• 1:00:59: Rigorous definition of Hausdorff measure & dimension

IV. Simple Geometry and Number Theory

• 1:04:51: Descartes’s Theorem
• 1:05:58: Definition: bend = 1/radius
• 1:11:31: Computing the two bends in the Apollonian problem
• 1:15:00: Why integral bends?
• 1:15:40: Frederick Soddy: Nobel laureate in chemistry
• 1:17:12: Soddy’s observation: integral packings

V. Group Theory, Hyperbolic Dynamics, and Advanced Number Theory

• 1:22:02: Generating circle packings through repeated inversions (through dual circles)
• 1:29:09: Coxeter groups: Example
• 1:30:45: Coxeter groups: Definition
• 1:37:20: Poincare: Dynamics on hyperbolic space
• 1:39:18: Video demo: flows in hyperbolic space and circle packings
• 1:42:30: Integral representation of the Coxeter group
• 1:46:22: Indefinite quadratic forms and integer points of orthogonal groups
• 1:50:55: Admissible residue classes of bends
• 1:56:11: Why these residues? Answer: Strong approximation + Hasse principle
• 2:04:02: Major conjecture
• 2:06:02: The conjecture restores the "Local to Global" principle (for thin groups instead of orthogonal groups)
• 2:09:19: Confession: What a rich subject
• 2:10:00: Conjecture is asymptotically true
• 2:12:02: M. C. Escher

VI. Dimension Three: Sphere Packings

• 2:13:03: Setup + what Soddy built
• 2:15:57: Local to Global theorem holds

VII. Conclusion

• 2:18:20: Wrap up
• 2:19:02: Russian school vs Bourbaki

Image Credits: http://timothynguyen.org/image-credits/

Greg Yang is a mathematician and AI researcher at Microsoft Research who for the past several years has done incredibly original theoretical work in the understanding of large artificial neural networks. Greg received his bachelors in mathematics from Harvard University in 2018 and while there won the Hoopes prize for best undergraduate thesis. He also received an Honorable Mention for the Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student in 2018 and was an invited speaker at the International Congress of Chinese Mathematicians in 2019.

In this episode, we get a sample of Greg's work, which goes under the name "Tensor Programs" and currently spans five highly technical papers. The route chosen to compress Tensor Programs into the scope of a conversational video is to place its main concepts under the umbrella of one larger, central, and time-tested idea: that of taking a large N limit. This occurs most famously in the Law of Large Numbers and the Central Limit Theorem, which then play a fundamental role in the branch of mathematics known as Random Matrix Theory (RMT). We review this foundational material and then show how Tensor Programs (TP) generalizes this classical work, offering new proofs of RMT. We conclude with the applications of Tensor Programs to a (rare!) rigorous theory of neural networks.

Patreon: https://www.patreon.com/timothynguyen

Part I. Introduction

• 00:00:00 : Biography
• 00:02:45 : Harvard hiatus 1: Becoming a DJ
• 00:07:40 : I really want to make AGI happen (back in 2012)
• 00:09:09 : Impressions of Harvard math
• 00:17:33 : Harvard hiatus 2: Math autodidact
• 00:22:05 : Friendship with Shing-Tung Yau
• 00:24:06 : Landing a job at Microsoft Research: Two Fields Medalists are all you need
• 00:26:13 : Technical intro: The Big Picture
• 00:28:12 : Whiteboard outline

Part II. Classical Probability Theory

• 00:37:03 : Law of Large Numbers
• 00:45:23 : Tensor Programs Preview
• 00:47:26 : Central Limit Theorem
• 00:56:55 : Proof of CLT: Moment method
• 1:00:20 : Moment method explicit computations

Part III. Random Matrix Theory

• 1:12:46 : Setup
• 1:16:55 : Moment method for RMT
• 1:21:21 : Wigner semicircle law

Part IV. Tensor Programs

• 1:31:03 : Segue using RMT
• 1:44:22 : TP punchline for RMT
• 1:46:22 : The Master Theorem (the key result of TP)
• 1:55:04 : Corollary: Reproof of RMT results
• 1:56:52 : General definition of a tensor program

Part V. Neural Networks and Machine Learning

• 2:09:05 : Feed forward neural network (3 layers) example
• 2:19:16 : Neural network Gaussian Process
• 2:23:59 : Many distinct large N limits for neural networks
• 2:27:24 : abc parametrizations (Note: "a" is absorbed into "c" here): variance and learning rate scalings
• 2:36:54 : Geometry of space of abc parametrizations
• 2:39:41: Kernel regime
• 2:41:32 : Neural tangent kernel
• 2:43:35: (No) feature learning
• 2:48:42 : Maximal feature learning
• 2:52:33 : Current problems with deep learning
• 2:55:02 : Hyperparameter transfer (muP)
• 3:00:31 : Wrap up

Further Reading:

Tensor Programs I, II, III, IV, V by Greg Yang and coauthors.

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Scott Aaronson is a professor of computer science at University of Texas at Austin and director of its Quantum Information Center. Previously he received his PhD at UC Berkeley and was a faculty member at MIT in Electrical Engineering and Computer Science from 2007-2016. Scott has won numerous prizes for his research on quantum computing and complexity theory, including the Alan T Waterman award in 2012 and the ACM Prize in Computing in 2020. In addition to being a world class scientist, Scott is famous for his highly informative and entertaining blog Schtetl Optimized, which has kept the scientific community up to date on quantum hype for nearly the past two decades.

In this episode, Scott Aaronson gives a crash course on quantum computing, diving deep into the details, offering insights, and clarifying misconceptions surrounding quantum hype.

Patreon: https://www.patreon.com/timothynguyen

Correction: 59:03: The matrix denoted as "Hadamard gate" is actually a 45 degree rotation matrix. The Hadamard gate differs from this matrix by a sign flip in the last column. See 1:11:00 for the Hadamard gate.

Part I. Introduction (Personal)

• 00:00: Biography
• 01:02: Shtetl Optimized and the ways of blogging
• 09:56: sabattical at OpenAI, AI safety, machine learning
• 10:54: "I study what we can't do with computers we don't have"

Part II. Introduction (Technical)

• 22:57: Overview
• 24:13: SMBC Cartoon: "The Talk". Summary of misconceptions of the field
• 33:09: How all quantum algorithms work: choreograph pattern of interference
• 34:38: Outline

Part III. Setup

• 36:10: Review of classical bits
• 40:46: Tensor product and computational basis
• 42:07: Entanglement
• 44:25: What is not spooky action at a distance
• 46:15: Definition of qubit
• 48:10: bra and ket notation
• 50:48: Superposition example
• 52:41: Measurement, Copenhagen interpretation

Part IV. Working with qubits

• 57:02: Unitary operators, quantum gates
• 1:03:34: Philosophical aside: How to "store" 2^1000 bits of information.
• 1:08:34: CNOT operation
• 1:09:45: quantum circuits
• 1:11:00: Hadamard gate
• 1:12:43: circuit notation, XOR notation
• 1:14:55: Subtlety on preparing quantum states
• 1:16:32: Building and decomposing general quantum circuits: Universality
• 1:21:30: Complexity of circuits vs algorithms
• 1:28:45: How quantum algorithms are physically implemented
• 1:31:55: Equivalence to quantum Turing Machine

Part V. Quantum Speedup

• 1:35:48: Query complexity (black box / oracle model)
• 1:39:03: Objection: how is quantum querying not cheating?
• 1:42:51: Defining a quantum black box
• 1:45:30: Efficient classical f yields efficient U_f
• 1:47:26: Toffoli gate
• 1:50:07: Garbage and quantum uncomputing
• 1:54:45: Implementing (-1)^f(x))
• 1:57:54: Deutsch-Jozsa algorithm: Where quantum beats classical
• 2:07:08: The point: constructive and destructive interference

Part VI. Complexity Classes

• 2:08:41: Recap. History of Simon's and Shor's Algorithm
• 2:14:42: BQP
• 2:18:18: EQP
• 2:20:50: P
• 2:22:28: NP
• 2:26:10: P vs NP and NP-completeness
• 2:33:48: P vs BQP
• 2:40:48: NP vs BQP
• 2:41:23: Where quantum computing explanations go off the rails

Part VII. Quantum Supremacy

• 2:43:46: Scalable quantum computing
• 2:47:43: Quantum supremacy
• 2:51:37: Boson sampling
• 2:52:03: What Google did and the difficulties with evaluating supremacy
• 3:04:22: Huge open question

Twitter: @IAmTimNguyen

Homepage: www.timothynguyen.org

Grant Sanderson is a mathematician who is the author of the YouTube channel “3Blue1Brown”, viewed by millions for its beautiful blend of visual animation and mathematical pedagogy. His channel covers a wide range of mathematical topics, which to name a few include calculus, quaternions, epidemic modeling, and artificial neural networks. Grant received his bachelor's degree in mathematics from Stanford University and has worked with a variety of mathematics educators and outlets, including Khan Academy, The Art of Problem Solving, MIT OpenCourseWare, Numberphile, and Quanta Magazine.

In this episode, we discuss the famous unsolvability of quintic polynomials: there exists no formula, consisting only of finitely many arithmetic operations and radicals, for expressing the roots of a general fifth degree polynomial in terms of the polynomial's coefficients. The standard proof that is taught in abstract algebra courses uses the machinery of Galois theory. Instead of following that route, Grant and I proceed in barebones style along (somewhat) historical lines by first solving quadratics, cubics, and quartics. Along the way, we present the insights obtained by Lagrange that motivate a very natural combinatorial question, which contains the germs of modern group theory and Galois theory and whose answer suggests that the quintic is unsolvable (later confirmed through the work of Abel and Galois). We end with some informal discussions about Abel's proof and the topological proof due to Vladimir Arnold.

Patreon: https://www.patreon.com/timothynguyen

Part I. Introduction

• 00:00:Introduction
• 00:52: How did you get interested in math?
• 06:30: Future of math pedagogy and AI 
• 12:03: Overview. How Grant got interested in unsolvability of the quintic
• 15:26: Problem formulation
• 17:42: History of solving polynomial equations
• 19:50: Po-Shen Loh 

Part II. Working Up to the Quintic

• 28:06: Quadratics
• 34:38 : Cubics
• 37:20: Viete’s formulas
• 48:51: Math duels over solving cubics: del Ferro, Fiorre, Tartaglia, Cardano, Ferrari
• 53:24: Prose poetry of solving cubics
• 54:30: Cardano’s Formula derivation
• 1:03:22: Resolvent 
• 1:04:10: Why exactly 3 roots from Cardano’s formula?

Part III. Thinking More Systematically

• 1:12:25: Takeaways and Lagrange’s insight into why quintic might be unsolvable
• 1:17:20: Origins of group theory?
• 1:23:29: History’s First Whiff of Galois Theory
• 1:25:24: Fundamental Theorem of Symmetric Polynomials
• 1:30:18: Solving the quartic from the resolvent
• 1:40:08: Recap of overall logic

Part IV. Unsolvability of the Quintic

• 1:52:30: S_5 and A_5 group actions
• 2:01:18: Lagrange’s approach fails!
• 2:04:01: Abel’s proof
• 2:06:16: Arnold’s Topological Proof
• 2:18:22: Closing Remarks

Further Reading on Arnold's Topological Proof of Unsolvability of the Quintic:

1. L. Goldmakher. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf
2. B. Katz. https://www.youtube.com/watch?v=RhpVSV6iCko

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

John Baez is a mathematical physicist, professor of mathematics at UC Riverside, a researcher at the Centre for Quantum Technologies in Singapore, and a researcher at the Topos Institute in Berkeley, CA. John has worked on an impressively wide range of topics, pure and applied, ranging from loop quantum gravity, applications of higher categories to physics, applied category theory, environmental issues and math related to engineering and biology, and most recently on applying network theory to scientific software.Additionally, John is a prolific writer and blogger. This first began with John’s column This Week's Finds in Mathematical Physics, which ran 300 issues between 1993 and 2010, which then continued in the form of his ongoing blog Azimuth. Last but not least, John is also a host and contributor of the popular blog The n-category Cafe.

In this episode, we dive into John Baez and John Huerta’s paper “The Algebra of Grand Unified Theories” which was awarded the Levi Conant Prize in 2013. The paper gives a crash course in the representation theory underlying the Standard Model of particle physics and its three most well known Grand Unified Theories (GUTs): the SU(5), SO(10) (aka Spin(10)), and Pati-Salam theories. The main result of Baez-Huerta is that the particle representations underlying the three GUTs can in fact be unified via a commutative diagram. We dive deep into the numerology of the standard model to see how the SU(5) theory naturally arises. We then make brief remarks about SO(10) and Pati-Salam theories in order to state the Baez-Huerta theorem about their organization into a commutative square: a unification among grand unifications!

Patreon: https://www.patreon.com/timothynguyen

Correction:

• 1:29:01: The formula for hypercharge in the bottom right note should be Y = 2(Q-I_3) instead of Y = (Q-I_3)/2.

Notes:

1. While we do provide a crash course on SU(2) and spin, some representation theory jargon is used at times in our discussion. Those unfamiliar should just forge ahead!
2. We work in Euclidean signature instead of Lorentzian signature. Other than keeping track of minus signs, no essential details are changed. 

Part I. Introduction

• 00:00: Introduction
• 05:50: Climate change
• 09:40: Crackpot index
• 14:50: Eric Weinstein, Brian Keating, Geometric Unity
• 18:13: Overview of “The Algebra of Grand Unified Theories” paper
• 25:40: Overview of Standard Model and GUTs
• 34:25: SU(2), spin, isospin of nucleons 40:22: SO(4), Spin(4), double cover
• 44:24: three kinds of spin

Part II. Zoology of Standard Model

• 49:35: electron and neutrino
• 58:40: quarks
• 1:04:51: the three generations of the Standard Model
• 1:08:25: isospin quantum numbers
• 1:17:11: U(1) representations (“charge”)
• 1:29:01: hypercharge
• 1:34:00: strong force and color
• 1:36:50: SU(3)
• 1:40:45: antiparticles

Part III. SU(5) numerology

• 1:41:16: 32 = 2^5 particles
• 1:45:05: Mapping SU(3) x SU(2) x U(1) to SU(5) and hypercharge matching
• 2:05:17: Exterior algebra of C^5 and more hypercharge matching
• 2:37:32: SU(5) rep extends Standard Model rep

Part IV. How the GUTs fit together

• 2:41:42: SO(10) rep: brief remarks
• 2:46:28: Pati-Salam rep: brief remarks
• 2:47:17: Commutative diagram: main result
• 2:49:12: What about the physics? Spontaneous symmetry breaking and the Higgs mechanism

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Tai-Danae Bradley is a mathematician who received her Ph.D. in mathematics from the CUNY Graduate Center. She was formerly at Alphabet and is now at Sandbox AQ, a startup focused on combining machine learning and quantum physics. Tai-Danae is a visiting research professor of mathematics at The Master’s University and the executive director of the Math3ma Institute, where she hosts her popular blog on category theory. She is also a co-author of the textbook Topology: A Categorical Approach that presents basic topology from the modern perspective of category theory.

In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions.

Patreon: https://www.patreon.com/timothynguyen

Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc

Timestamps:

• 00:00:00 : Introduction
• 00:03:07 : How did you get into category theory?
• 00:06:29 : Outline of podcast
• 00:09:21 : Motivating category theory
• 00:11:35 : Analogy: Object Oriented Programming
• 00:12:32 : Definition of category
• 00:18:50 : Example: Category of sets
• 00:20:17 : Example: Matrix category
• 00:25:45 : Example: Preordered set (poset) is a category
• 00:33:43 : Example: Category of finite-dimensional vector spaces
• 00:37:46 : Forgetful functor
• 00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity!
• 00:40:06 : Definition of functor
• 00:42:01 : Example: API change between programming languages is a functor
• 00:44:23 : Example: Groups, group homomorphisms are categories and functors
• 00:47:33 : Resume definition of functor
• 00:49:14 : Example: Functor between poset categories = order-preserving function
• 00:52:28 : Hom Functors. Things are getting meta (no not the tech company)
• 00:57:27 : Category theory is beautiful because of its rigidity
• 01:00:54 : Contravariant functor
• 01:03:23 : Definition: Presheaf
• 01:04:04 : Why are things meta? Arrows, arrows between arrows, categories of categories, ad infinitum.
• 01:07:38 : Probing a space with maps (prelude to Yoneda Lemma)
• 01:12:10 : Algebraic topology motivated category theory
• 01:15:44 : Definition: Natural transformation
• 01:19:21 : Example: Indexing category
• 01:21:54 : Example: Change of currency as natural transformation
• 01:25:35 : Isomorphism and natural isomorphism
• 01:27:34 : Notion of isomorphism in different categories
• 01:30:00 : Yoneda Lemma
• 01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation
• 01:42:33 : Analogy between Yoneda Lemma and linear algebra
• 01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors
• 01:50:40 : Yoneda embedding is fully faithful. Reasoning about this.
• 01:55:15 : Language Category
• 02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics"

Further Reading:

• Tai-Danae's Blog: https://www.math3ma.com/categories
• Tai-Danae Bradley. "What is applied category theory?" https://arxiv.org/pdf/1809.05923.pdf
• Tai-Danae Bradley, John Terilla, Yiannis Vlassopoulos. "An enriched category theory of language: from syntax to semantics." https://arxiv.org/pdf/2106.07890.pdf

John Urschel received his bachelors and masters in mathematics from Penn State and then went on to become a professional football player for the Baltimore Ravens in 2014. During his second season, Urschel began his graduate studies in mathematics at MIT alongside his professional football career. Urschel eventually decided to retire from pro football to pursue his real passion, the study of mathematics, and he completed his doctorate in 2021. Urschel is currently a scholar at the Institute for Advanced Study where he is actively engaged in research on graph theory, numerical analysis, and machine learning. In addition, Urschel is the author of Mind and Matter, a New York Times bestseller about his life as an athlete and mathematician, and has been named as one of Forbes 30 under 30 for being an outstanding young scientist.

In this episode, John and I discuss a hodgepodge of topics in spectral graph theory. We start off light and discuss the famous Braess's Paradox in which traffic congestion can be increased by opening a road. We then discuss the graph Laplacian to enable us to present Cheeger's Theorem, a beautiful result relating graph bottlenecks to graph eigenvalues. We then discuss various graph embedding and clustering results, and end with a discussion of the PageRank algorithm that powers Google search.

Patreon: https://www.patreon.com/timothynguyen

Originally published on June 9, 2022 on YouTube: https://youtu.be/O6k0JRpA2mg

Timestamps:

I. Introduction

• 00:00: Introduction
• 04:30: Being a professional mathematician and academia vs industry
• 09:41: John's taste in mathematics
• 13:00: Outline
• 17:23: Braess's Paradox: "Opening a highway can increase traffic congestion."
• 25:34: Prisoner's Dilemma. We need social forcing mechanisms to avoid undesirable outcomes (traffic jams).

II. Spectral Graph Theory Basics

• 31:20: What is a graph
• 36:33: Graph bottlenecks. Practical situations: Task assignment, the economy, organizational management.
• 42:44: Quantifying bottlenecks: Cheeger's constant
• 46:43: Cheeger's constant sample computations
• 52:07: NP Hardness
• 55:48: Graph Laplacian
• 1:00:27: Graph Laplacian: 1-dimensional example

III. Cheeger's Inequality and Harmonic Oscillators

• 1:07:35: Cheeger's Inequality: Statement
• 1:09:37: Cheeger's Inequality: A great example of beautiful mathematics
• 1:10:46: Cheeger's Inequality: Computationally tractable approximation of Cheeger's constant
• 1:19:16: Harmonic oscillators: Springs heuristic for lambda_2 and Cheeger's inequality
• 1:22:20: Interlude: Tutte's Spring Embedding Theorem (planar embeddings in terms of springs)
• 1:29:45: Interlude: Graph drawing using eigenfunction

IV. Graph bisection and clustering

• 1:38:26: Summary thus far and graph bisection
• 1:42:44: Graph bisection: Large eigenvalues for PCA vs low eigenvalues for spectral bisection
• 1:43:40: Graph bisection: 1-dimensional intuition
• 1:47:43: Spectral graph clustering (complementary to graph bisection)

V. Markov chains and PageRank

• 1:52:10: PageRank: Google's algorithm for ranking search results
• 1:53:44: PageRank: Markov chain (Markov matrix)
• 1:57:32: PageRank: Stationary distribution
• 2:00:20: Perron-Frobenius Theorem
• 2:06:10: Spectral gap: Analogy between bottlenecks for graphs and bottlenecks for Markov chain mixing
• 2:07:56: Conclusion: State of the field, Urschel's recent results
• 2:10:28: Joke: Two kinds of mathematicians

Further Reading:

• A. Ng, M. Jordan, Y. Weiss. "On Spectral Clustering: Analysis and an algorithm"
• D. Spielman. "Spectral and Algebraic Graph Theory"

Richard Easther is a scientist, teacher, and communicator. He has been a Professor of Physics at the University of Auckland for over the last 10 years and was previously a professor of physics at Yale University. As a scientist, Richard covers ground that crosses particle physics, cosmology, astrophysics and astronomy, and in particular, focuses on the physics of the very early universe and the ways in which the universe changes between the Big Bang and the present day.

In this episode, Richard and I discuss the details of cosmology at large, both technically and historically. We dive into Einstein's equations from general relativity and see what implications they have for an expanding universe alongside a discussion of the cast of characters involved in 20th century cosmology (Einstein, Hubble, Friedmann, Lemaitre, and others). We also discuss inflation, gravitational waves, the story behind Brian Keating's book Losing the Nobel Prize, and the current state of experiments and cosmology as a field.

Patreon: https://www.patreon.com/timothynguyen

Originally published on May 3, 2022 on YouTube: https://youtu.be/DiXyZgukRmE

Timestamps:

• 00:00:00 : Introduction
• 00:02:42 : Astronomy must have been one of the earliest sciences
• 00:03:57 : Eric Weinstein and Geometric Unity
• 00:13:47 : Outline of podcast
• 00:15:10 : Brian Keating, Losing the Nobel Prize, Geometric Unity
• 00:16:38 : Big Bang and General Relativity
• 00:21:07 : Einstein's equations
• 00:26:27 : Einstein and Hilbert
• 00:27:47 : Schwarzschild solution (typo in video)
• 00:33:07 : Hubble
• 00:35:54 : One galaxy versus infinitely many
• 00:36:16 : Olbers' paradox
• 00:39:55 : Friedmann and FRLW metric
• 00:41:53 : Friedmann metric was audacious?
• 00:46:05 : Friedmann equation
• 00:48:36 : How to start a fight in physics: West coast vs East coast metric and sign conventions.
• 00:50:05 : Flat vs spherical vs hyperbolic space
• 00:51:40 : Stress energy tensor terms
• 00:54:15 : Conservation laws and stress energy tensor
• 00:58:28 : Acceleration of the universe
• 01:05:12 : Derivation of a(t) ~ t^2/3 from preceding computations
• 01:05:37 : a = 0 is the Big Bang. How seriously can we take this?
• 01:07:09 : Lemaitre
• 01:11:51 : Was Hubble's observation of an expanding universe in 1929 a fresh observation?
• 01:13:45 : Without Einstein, no General Relativity?
• 01:14:45 : Two questions: General Relativity vs Quantum Mechanics and how to understand time and universe's expansion velocity (which can exceed the speed of light!)
• 01:17:58 : How much of the universe is observable
• 01:24:54 : Planck length
• 01:26:33 : Physics down to the Big Bang singularity
• 01:28:07 : Density of photons vs matter
• 01:33:41 : Inflation and Alan Guth
• 01:36:49 : No magnetic monopoles?
• 01:38:30 : Constant density requires negative pressure
• 01:42:42 : Is negative pressure contrived?
• 01:49:29 : Marrying General Relativity and Quantum Mechanics
• 01:51:58 : Symmetry breaking
• 01:53:50 : How to corroborate inflation?
• 01:56:21 : Sabine Hossenfelder's criticisms
• 02:00:19 : Gravitational waves
• 02:01:31 : LIGO
• 02:04:13 : CMB (Cosmic Microwave Background)
• 02:11:27 : Relationship between detecting gravitational waves and inflation
• 02:16:37 : BICEP2
• 02:19:06 : Brian Keating's Losing the Nobel Prize and the problem of dust
• 02:24:40 : BICEP3
• 02:26:26 : Wrap up: current state of cosmology

Notes:

• Easther's blogpost on Eric Weinstein: http://excursionset.com/blog/2013/5/25/trainwrecks-i-have-seen
• Vice article on Eric Weinstein and Geometric Unity:
  https://www.vice.com/en/article/z3xbz4/eric-weinstein-says-he-solved-the-universes-mysteries-scientists-disagree

Further learning:

• Matts Roos. "Introduction to Cosmology"
• Barbara Ryden. "Introduction to Cosmology"
• Our Cosmic Mistake About Gravitational Waves: https://www.youtube.com/watch?v=O0D-COVodzY

Po-Shen Loh is a professor at Carnegie Mellon University and a coach for the US Math Olympiad. He is also a social entrepreneur where he has his used his passion and expertise in mathematics in the service of education (expii.com) and epidemiology (novid.org).

In this episode, we discuss the mathematics behind Loh's novel approach to contact tracing in the fight against COVID, which involves a beautiful blend of graph theory and computer science.

Originally published on March 3, 2022 on Youtube: https://youtu.be/8CLxLBMGxLE

Patreon: https://www.patreon.com/timothynguyen

Timestamps:

• 00:00:00 : Introduction
• 00:01:11 : About Po-Shen Loh
• 00:03:49 : NOVID app
• 00:04:47 : Graph theory and quarantining
• 00:08:39 : Graph adjacency definition for contact tracing
• 00:16:01 : Six degrees of separation away from anyone?
• 00:21:13 : Getting the game theory and incentives right
• 00:30:40 : Conventional approach to contact tracing
• 00:34:47 : Comparison with big tech
• 00:39:19 : Neighbor search complexity
• 00:45:15 : Watts-Strogatz small networks phenomenon
• 00:48:37 : Storing neighborhood information
• 00:57:00 : Random hashing to reduce computational burden
• 01:05:24 : Logarithmic probing of sparsity
• 01:09:56 : Two math PhDs struggle to do division
• 01:11:17 : Bitwise-or for union of bounded sets
• 01:16:21 : Step back and recap
• 01:26:15 : Tradeoff between number of hash bins and sparsity
• 01:29:12 : Conclusion

Further reading:

Po-Shen Loh. "Flipping the Perspective in Contact Tracing" https://arxiv.org/abs/2010.03806

Hello everyone, this is Tim Nguyen and welcome to The Cartesian Cafe. On this podcast we embark on a collaborative journey with other experts, to discuss mathematical and scientific topics in faithful detail, which means writing down formulas, drawing pictures, and reasoning about them together on a whiteboard. If you’ve been longing for a deeper dive into the intricacies of scientific subjects, then this is the podcast for you. Welcome to The Cartesian Cafe.

Patreon: https://www.patreon.com/timothynguyen