The Cartesian Cafe

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

More Sean Carroll & Timothy Nguyen:

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

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). #physics #thermodynamics #philosophy #mathematics 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

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 group
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

Twitter: @iamtimnguyen

Webpage: timothynguyen.org

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

Further Reading:
Scott Aaronson's Lecture Notes: https://www.scottaaronson.com/qclec.pdf
Scott Aaronson's Blog: https://scottaaronson.blog
Nielsen & Chuang. Quantum Computation and Quantum Information

Twitter: @iamtimnguyen

Webpage: http://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.

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

If you would like to support this series and future such projects:

Paypal: tim@timothynguyen.org

Bitcoin: 33thftjoPTHFajj8wJFcCB9sFiyQLFVp8S

Ethereum: 0x166a977F411d6f220cF8A56065D16B4FF08a246D

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

• 00:00:00: Introduction
• 00:05:50: Climate change
• 00:09:40: 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

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.

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

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

Corrections:
01:14:24 : The inequalities are reversed here. It is corrected at 01:16:16.

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"

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