# 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, machine learning, artificial intelligence, and computer science. 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

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/

Wednesday Jan 04, 2023

Wednesday Jan 04, 2023

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

Tuesday Nov 22, 2022

Tuesday Nov 22, 2022

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

Thursday Oct 13, 2022

Thursday Oct 13, 2022

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:

L. Goldmakher. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf

B. Katz. https://www.youtube.com/watch?v=RhpVSV6iCko

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Wednesday Sep 07, 2022

Wednesday Sep 07, 2022

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:

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!

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

Sunday Aug 21, 2022

Sunday Aug 21, 2022

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