The Cartesian Cafe

The Cartesian Cafe is the podcast where an expert guest and Timothy Nguyen map out scientific and mathematical subjects in detail. On the 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. Content also viewable on YouTube: www.youtube.com/timothynguyen and Spotify.

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Episodes

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!   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 If you would like to support this series and future such projects: Paypal: tim@timothynguyen.org Bitcoin: 33thftjoPTHFajj8wJFcCB9sFiyQLFVp8S Ethereum: 0x166a977F411d6f220cF8A56065D16B4FF08a246D

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. 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

Sunday Aug 21, 2022

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. 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"

Saturday Aug 20, 2022

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.   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 : Conversation 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-seenVice 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

Thursday Aug 18, 2022

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 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  

Tuesday Aug 16, 2022

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.

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