# The Cartesian Cafe

The Cartesian Cafe is the podcast where an expert guest and Timothy Nguyen map out scientific and mathematical subjects in detail. This collaborative journey with other experts will have us 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. Topics covered include mathematics, physics, computer science, machine learning, and artificial intelligence. Content also viewable on YouTube: www.youtube.com/timothynguyen and Spotify. Timothy Nguyen is a mathematician and AI researcher working in industry. Homepage: www.timothynguyen.com, Twitter: @IAmTimNguyen Patreon: www.patreon.com/timothynguyen

## Episodes

Friday May 10, 2024

Friday May 10, 2024

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 IntelligenceM. Hutter. Universal Artificial IntelligenceS. Legg and M. Hutter. Universal Intelligence: A Definition of Machine Intelligence

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Friday Feb 02, 2024

Friday Feb 02, 2024

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

Tuesday Jan 09, 2024

Tuesday Jan 09, 2024

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

Friday Dec 01, 2023

Friday Dec 01, 2023

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

Wednesday Sep 27, 2023

Wednesday Sep 27, 2023

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

Wednesday Aug 02, 2023

Wednesday Aug 02, 2023

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

Wednesday Jun 14, 2023

Wednesday Jun 14, 2023

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

Tuesday May 02, 2023

Tuesday May 02, 2023

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

Tuesday Mar 21, 2023

Tuesday Mar 21, 2023

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 Everythinghttps://youtu.be/DPvD4VnD5Z4

Twitter: @iamtimnguyenWebpage: http://www.timothynguyen.org

Wednesday Feb 15, 2023

Wednesday Feb 15, 2023

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/