Richard Borcherds is a mathematician and professor at University of California Berkeley known for his work on lattices, group theory, and infinite-dimensional algebras. His numerous accolades include being awarded the Fields Medal in 1998 and being elected a fellow of the American Mathematical Society and the National Academy of Sciences. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen In this episode, Richard and I give an overview of Richard's most famous result: his proof of the Monstrous Moonshine conjecture relating the monster group on the one hand and modular forms on the other. A remarkable feature of the proof is that it involves vertex algebras inspired from elements of string theory. Some familiarity with group theory and representation theory are assumed in our discussion. I. Introduction 00:25: Biography 02:51 : Success in mathematics 04:04 : Monstrous Moonshine overview and John Conway 09:44 : Technical overview II. Group Theory 11:31 : Classification of finite-simple groups + history of the monster group 18:03 : Conway groups + Leech lattice 22:13 : Why was the monster conjectured to exist + more history 28:43 : Centralizers and involutions 32:37: Griess algebra III. Modular Forms 36:42 : Definitions 40:06 : The elliptic modular function 48:58 : Subgroups of SL_2(Z) IV. Monstrous Moonshine Conjecture Statement 57:17: Representations of the monster 59:22 : Hauptmoduls 1:03:50 : Statement of the conjecture 1:07:06 : Atkin-Fong-Smith's first proof 1:09:34 : Frenkel-Lepowski-Meurman's work + significance of Borcherd's proof V. Sketch of Proof 1:14:47: Vertex algebra and monster Lie algebra 1:21:02 : No ghost theorem from string theory 1:25:24 : What's special about dimension 26? 1:28:33 : Monster Lie algebra details 1:32:30 : Dynkin diagrams and Kac-Moody algebras 1:43:21 : Simple roots and an obscure identity 1:45:13: Weyl denominator formula, Vandermonde identity 1:52:14 : Chasing down where modular forms got smuggled in 1:55:03 : Final calculations VI. Epilogue 1:57:53 : Your most proud result? 2:00:47 : Monstrous moonshine for other sporadic groups? 2:02:28 : Connections to other fields. Witten and black holes and mock modular forms. Further reading: V Tatitschef. A short introduction to Monstrous Moonshine. https://arxiv.org/pdf/1902.03118.pdf Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

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

Tim Maudlin is a philosopher of science specializing in the foundations of physics, metaphysics, and logic. He is a professor at New York University, a member of the Foundational Questions Institute, and the founder and director of the John Bell Institute for the Foundations of Physics. Patreon (bonus materials + video chat):https://www.patreon.com/timothynguyen In this very in-depth discussion, Tim and I probe the foundations of science through the avenues of locality and determinism as arising from the Einstein-Poldosky-Rosen (EPR) paradox and Bell's Theorem. These issues are so intricate that even the Nobel Prize committee incorrectly described the significance of Bell's work in their press release for the 2022 prize in physics. Viewers motivated enough to think deeply about these ideas will be rewarded with a conceptually proper understanding of the nonlocal nature of physics and its manifestation in quantum theory. I. Introduction 00:00 : 00:25: Biography 05:26: Interdisciplinary work 11:54 : Physicists working on the wrong things 16:47 : Bell's Theorem soft overview 24:14: Common misunderstanding of "God does not play dice." 25:59: Technical outline II. EPR Paradox / Argument 29:14 : EPR is not a paradox 34:57 : Criterion of reality 43:57 : Mathematical formulation 46:32 : Locality: No spooky action at a distance 49:54 : Bertlmann's socks 53:17 : EPR syllogism summarized 54:52 : Determinism is inferred not assumed 1:02:18 : Clarifying analogy: Coin flips 1:06:39 : Einstein's objection to determinism revisited III. Bohm Segue 1:11:05 : Introduction 1:13:38: Bell and von Neumann's error 1:20:14: Bell's motivation: Can I remove Bohm's nonlocality? IV. Bell's Theorem and Related Examples 1:25:13 : Setup 1:27:59 : Decoding Bell's words: Locality is the key! 1:34:16 : Bell's inequality (overview) 1:36:46 : Bell's inequality (math) 1:39:15 : Concrete example of violation of Bell's inequality 1:49:42: GHZ Example V. Miscellany 2:06:23 : Statistical independence assumption 2:13:18: The 2022 Nobel Prize 2:17:43: Misconceptions and hidden variables 2:22:28: The assumption of local realism? Repeat: Determinism is a conclusion not an assumption. VI. Interpretations of Quantum Mechanics 2:28:44: Interpretation is a misnomer 2:29:48: Three requirements. You can only pick two. 2:34:52: Copenhagen interpretation? Further Reading: J. Bell. Speakable and Unspeakable in Quantum Mechanics T. Maudlin. Quantum Non-Locality and Relativity Wikipedia: Mermin's device, GHZ experiment Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Antonio (Tony) Padilla is a theoretical physicist and cosmologist at the University of Nottingham. He serves as the Associate Director of the Nottingham Centre of Gravity, and in 2016, Tony shared the Buchalter Cosmology Prize for his work on the cosmological constant. Tony is also a star of the Numberphile YouTube channel, where his videos have received millions of views and he is also the author of the book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity. Patreon: https://www.patreon.com/timothynguyen This episode combines some of the greatest cosmological questions together with mathematical imagination. Tony and I go through the math behind some oft-quoted numbers in cosmology and calculate the age, size, and number of atoms in the universe. We then stretch our brains and consider how likely it would be to find your Doppelganger in a truly large universe, which takes us on a detour through black hole entropy. We end with a discussion of naturalness and the anthropic principle to round out our discussion of fantastic numbers in physics. Part I. Introduction 00:00 : Introduction 01:06 : Math and or versus physics 12:09 : Backstory behind Tony's book 14:12 : Joke about theoreticians and numbers 16:18 : Technical outline Part II. Size, Age, and Quantity in the Universe 21:42 : Size of the observable universe 22:32 : Standard candles 27:39 : Hubble rate 29:02 : Measuring distances and time 37:15 : Einstein and Minkowski 40:52 : Definition of Hubble parameter 42:14 : Friedmann equation 47:11 : Calculating the size of the observable universe 51:24 : Age of the universe 56:14 : Number of atoms in the observable universe 1:01:08 : Critical density 1:03:16: 10^80 atoms of hydrogen 1:03:46 : Universe versus observable universe Part III. Extreme Physics and Doppelgangers 1:07:27 : Long-term fate of the universe 1:08:28 : Black holes and a googol years 1:09:59 : Poincare recurrence 1:13:23 : Doppelgangers in a googolplex meter wide universe 1:16:40 : Finitely many states and black hole entropy 1:25:00 : Black holes have no hair 1:29:30 : Beckenstein, Christodolou, Hawking 1:33:12 : Susskind's thought experiment: Maximum entropy of space 1:42:58 : Estimating the number of doppelgangers 1:54:21 : Poincare recurrence: Tower of four exponents. Part IV: Naturalness and Anthropics 1:54:34 : What is naturalness? Examples. 2:04:09 : Cosmological constant problem: 10^120 discrepancy 2:07:29 : Interlude: Energy shift clarification. Gravity is key. 2:15:34 : Corrections to the cosmological constant 2:18:47 : String theory landscape: 10^500 possibilities 2:20:41 : Anthropic selection 2:25:59 : Is the anthropic principle unscientific? Weinberg and predictions. 2:29:17 : Vacuum sequestration Further reading: Antonio Padilla. Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Boaz Barak is a professor of computer science at Harvard University, having previously been a principal researcher at Microsoft Research and a professor at Princeton University. His research interests span many areas of theoretical computer science including cryptography, computational complexity, and the foundations of machine learning. Boaz serves on the scientific advisory boards for Quanta Magazine and the Simons Institute for the Theory of Computing and he was selected for Foreign Policy magazine’s list of 100 leading global thinkers for 2014. www.patreon.com/timothynguyen Cryptography is about maintaining the privacy and security of communication. In this episode, Boaz and I go through the fundamentals of cryptography from a foundational mathematical perspective. We start with some historical examples of attempts at encrypting messages and how they failed. After some guesses as to how one might mathematically define security, we arrive at the one due to Shannon. The resulting definition of perfect secrecy turns out to be too rigid, which leads us to the notion of computational secrecy that forms the foundation of modern cryptographic systems. We then show how the existence of pseudorandom generators (which remains a conjecture) ensures that such computational secrecy is achievable, assuming P does not equal NP. Having covered private key cryptography in detail, we then give a brief overview of public key cryptography. We end with a brief discussion of Bitcoin, machine learning, deepfakes, and potential doomsday scenarios. I. Introduction 00:17 : Biography: Academia vs Industry 10:07 : Military service 12:53 : Technical overview 17:01 : Whiteboard outline II. Warmup 24:42 : Substitution ciphers 27:33 : Viginere cipher 29:35 : Babbage and Kasiski 31:25 : Enigma and WW2 33:10 : Alan Turing III. Private Key Cryptography: Perfect Secrecy 34:32 : Valid encryption scheme 40:14 : Kerckhoffs's Principle 42:41 : Cryptography = steelman your adversary 44:40 : Attempt #1 at perfect secrecy 49:58 : Attempt #2 at perfect secrecy 56:02 : Definition of perfect secrecy (Shannon) 1:05:56 : Enigma was not perfectly secure 1:08:51 : Analogy with differential privacy 1:11:10 : Example: One-time pad (OTP) 1:20:07 : Drawbacks of OTP and Soviet KGB misuse 1:21:43 : Important: Keys cannot be reused! 1:27:48 : Shannon's Impossibility Theorem IV. Computational Secrecy 1:32:52 : Relax perfect secrecy to computational secrecy 1:41:04 : What computational secrecy buys (if P is not NP) 1:44:35 : Pseudorandom generators (PRGs) 1:47:03 : PRG definition 1:52:30 : PRGs and P vs NP 1:55:47: PRGs enable modifying OTP for computational secrecy V. Public Key Cryptography 2:00:32 : Limitations of private key cryptography 2:09:25 : Overview of public key methods 2:13:28 : Post quantum cryptography VI. Applications 2:14:39 : Bitcoin 2:18:21 : Digital signatures (authentication) 2:23:56 : Machine learning and deepfakes 2:30:31 : A conceivable doomsday scenario: P = NP Further reading: Boaz Barak. An Intensive Introduction to Cryptography Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Sean Carroll is a theoretical physicist and philosopher who specializes in quantum mechanics, cosmology, and the philosophy of science. He is the Homewood Professor of Natural Philosophy at Johns Hopkins University and an external professor at the Sante Fe Institute. Sean has contributed prolifically to the public understanding of science through a variety of mediums: as an author of several physics books including Something Deeply Hidden and The Biggest Ideas in the Universe, as a public speaker and debater on a wide variety of scientific and philosophical subjects, and also as a host of his podcast Mindscape which covers topics spanning science, society, philosophy, culture, and the arts. www.patreon.com/timothynguyen In this episode, we take a deep dive into The Many Worlds (Everettian) Interpretation of quantum mechanics. While there are many philosophical discussions of the Many Worlds Interpretation available, ours marries philosophy with the technical, mathematical details. As a bonus, the whole gamut of topics from philosophy and physics arise, including the nature of reality, emergence, Bohmian mechanics, Bell's Theorem, and more. We conclude with some analysis of Sean's speculative work on the concept of emergent spacetime, a viewpoint which naturally arises from Many Worlds. This video is most suitable for those with a basic technical understanding of quantum mechanics. Part I: Introduction 00:00:00 : Introduction 00:05:42 : Philosophy and science: more interdisciplinary work? 00:09:14 : How Sean got interested in Many Worlds (MW) 00:13:04 : Technical outline Part II: Quantum Mechanics in a Nutshell 00:14:58 : Textbook QM review 00:24:25 : The measurement problem 00:25:28 : Einstein: "God does not play dice" 00:27:49 : The reality problem Part III: Many Worlds 00:31:53 : How MW comes in 00:34:28 : EPR paradox (original formulation) 00:40:58 : Simpler to work with spin 00:42:03 : Spin entanglement 00:44:46 : Decoherence 00:49:16 : System, observer, environment clarification for decoherence 00:53:54 : Density matrix perspective (sketch) 00:56:21 : Deriving the Born rule 00:59:09 : Everett: right answer, wrong reason. The easy and hard part of Born's rule. 01:03:33 : Self-locating uncertainty: which world am I in? 01:04:59 : Two arguments for Born rule credences 01:11:28 : Observer-system split: pointer-state problem 01:13:11 : Schrodinger's cat and decoherence 01:18:21 : Consciousness and perception 01:21:12 : Emergence and MW 01:28:06 : Sorites Paradox and are there infinitely many worlds 01:32:50 : Bad objection to MW: "It's not falsifiable." Part IV: Additional Topics 01:35:13 : Bohmian mechanics 01:40:29 : Bell's Theorem. What the Nobel Prize committee got wrong 01:41:56 : David Deutsch on Bohmian mechanics 01:46:39 : Quantum mereology 01:49:09 : Path integral and double slit: virtual and distinct worlds Part V. Emergent Spacetime 01:55:05 : Setup 02:02:42 : Algebraic geometry / functional analysis perspective 02:04:54 : Relation to MW Part VI. Conclusion 02:07:16 : Distribution of QM beliefs 02:08:38 : Locality Further reading: Hugh Everett. The Theory of the Universal Wave Function, 1956. Sean Carroll. Something Deeply Hidden, 2019. More Sean Carroll & Timothy Nguyen: Fragments of the IDW: Joe Rogan, Sam Harris, Eric Weinstein: https://youtu.be/jM2FQrRYyas Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Daniel Schroeder is a particle and accelerator physicist and an editor for The American Journal of Physics. Dan received his PhD from Stanford University, where he spent most of his time at the Stanford Linear Accelerator, and he is currently a professor in the department of physics and astronomy at Weber State University. Dan is also the author of two revered physics textbooks, the first with Michael Peskin called An Introduction to Quantum Field Theory (or simply Peskin & Schroeder within the physics community) and the second An Introduction to Thermal Physics. Dan enjoys teaching physics courses at all levels, from Elementary Astronomy through Quantum Mechanics. In this episode, I get to connect with one of my teachers, having taken both thermodynamics and quantum field theory courses when I was a university student based on Dan's textbooks. We take a deep dive towards answering two fundamental questions in the subject of thermodynamics: what is temperature and what is entropy? We provide both a qualitative and quantitative analysis, discussing good and bad definitions of temperature, microstates and macrostates, the second law of thermodynamics, and the relationship between temperature and entropy. Our discussion was also a great chance to shed light on some of the philosophical assumptions and conundrums in thermodynamics that do not typically come up in a physics course: the fundamental assumption of statistical mechanics, Laplace's demon, and the arrow of time problem (Loschmidt's paradox) arising from the second law of thermodynamics (i.e. why is entropy increasing in the future when mechanics has time-reversal symmetry). Patreon: https://www.patreon.com/timothynguyen Outline: 00:00:00 : Introduction 00:01:54 : Writing Books 00:06:51 : Academic Track: Research vs Teaching 00:11:01 : Charming Book Snippets 00:14:54 : Discussion Plan: Two Basic Questions 00:17:19 : Temperature is What You Measure with a Thermometer 00:22:50 : Bad definition of Temperature: Measure of Average Kinetic Energy 00:25:17 : Equipartition Theorem 00:26:10 : Relaxation Time 00:27:55 : Entropy from Statistical Mechanics 00:30:12 : Einstein solid 00:32:43 : Microstates + Example Computation 00:38:33: Fundamental Assumption of Statistical Mechanics (FASM) 00:46:29 : Multiplicity is highly concentrated about its peak 00:49:50 : Entropy is Log(Multiplicity) 00:52:02 : The Second Law of Thermodynamics 00:56:13 : FASM based on our ignorance? 00:57:37 : Quantum Mechanics and Discretization 00:58:30 : More general mathematical notions of entropy 01:02:52 : Unscrambling an Egg and The Second Law of Thermodynamics 01:06:49 : Principle of Detailed Balance 01:09:52 : How important is FASM? 01:12:03 : Laplace's Demon 01:13:35 : The Arrow of Time (Loschmidt's Paradox) 01:15:20 : Comments on Resolution of Arrow of Time Problem 01:16:07 : Temperature revisited: The actual definition in terms of entropy 01:25:24 : Historical comments: Clausius, Boltzmann, Carnot 01:29:07 : Final Thoughts: Learning Thermodynamics Further Reading: Daniel Schroeder. An Introduction to Thermal Physics L. Landau & E. Lifschitz. Statistical Physics. Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Ethan Siegel is a theoretical astrophysicist and science communicator. He received his PhD from the University of Florida and held academic positions at the University of Arizona, University of Oregon, and Lewis & Clark College before moving on to become a full-time science writer. Ethan is the author of the book Beyond The Galaxy, which is the story of “How Humanity Looked Beyond Our Milky Way And Discovered The Entire Universe” and he has contributed numerous articles to ScienceBlogs, Forbes, and BigThink. Today, Ethan is the face and personality behind Starts With A Bang, both a website and podcast by the same name that is dedicated to explaining and exploring the deepest mysteries of the cosmos. In this episode, Ethan and I discuss the mysterious nature of dark matter: the evidence for it and the proposals for what it might be. Patreon: https://www.patreon.com/timothynguyen Part I. Introduction 00:00:00 : Biography and path to science writing 00:07:26 : Keeping up with the field outside academia 00:11:42 : If you have a bone to pick with Ethan... 00:12:50 : On looking like a scientist and words of wisdom 00:18:24 : Understanding dark matter = one of the most important open problems 00:21:07 : Technical outline Part II. Ordinary Matter 23:28 : Matter and radiation scaling relations 29:36 : Hubble constant 31:00 : Components of rho in Friedmann's equations 34:14 : Constituents of the universe 41:21 : Big Bang nucleosynthesis (BBN) 45:32 : eta: baryon to photon ratio and deuterium formation 53:15 : Mass ratios vs eta Part III. Dark Matter 1:01:02 : rho = radiation + ordinary matter + dark matter + dark energy 1:05:25 : nature of peaks and valleys in cosmic microwave background (CMB): need dark matter 1:07:39: Fritz Zwicky and mass mismatch among galaxies of a cluster 1:10:40 : Kent Ford and Vera Rubin and and mass mismatch within a galaxy 1:11:56 : Recap: BBN tells us that only about 5% of matter is ordinary 1:15:55 : Concordance model (Lambda-CDM) 1:21:04 : Summary of how dark matter provides a common solution to many problems 1:23:29 : Brief remarks on modified gravity 1:24:39 : Bullet cluster as evidence for dark matter 1:31:40 : Candidates for dark matter (neutrinos, WIMPs, axions) 1:38:37 : Experiment vs theory. Giving up vs forging on 1:48:34 : Conclusion Image Credits: http://timothynguyen.org/image-credits/ Further learning: E. Siegel. Beyond the Galaxy Ethan Siegel's webpage: www.startswithabang.com More Ethan Siegel & Timothy Nguyen videos: Brian Keating’s Losing the Nobel Prize Makes a Good Point but …https://youtu.be/iJ-vraVtCzw Testing Eric Weinstein's and Stephen Wolfram's Theories of Everythinghttps://youtu.be/DPvD4VnD5Z4 Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Alex Kontorovich is a Professor of Mathematics at Rutgers University and served as the Distinguished Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics in 2020–2021. Alex has received numerous awards for his illustrious mathematical career, including the Levi L. Conant Prize in 2013 for mathematical exposition, a Simons Foundation Fellowship, an NSF career award, and being elected Fellow of the American Mathematical Society in 2017. He currently serves on the Scientific Advisory Board of Quanta Magazine and as Editor-in-Chief of the Journal of Experimental Mathematics. In this episode, Alex takes us from the ancient beginnings to the present day on the subject of circle packings. We start with the Problem of Apollonius on finding tangent circles using straight-edge and compass and continue forward in basic Euclidean geometry up until the time of Leibniz whereupon we encounter the first complete notion of a circle packing. From here, the plot thickens with observations on surprising number theoretic coincidences, which only received full appreciation through the craftsmanship of chemistry Nobel laureate Frederick Soddy. We continue on with more advanced mathematics arising from the confluence of geometry, group theory, and number theory, including fractals and their dimension, hyperbolic dynamics, Coxeter groups, and the local to global principle of advanced number theory. We conclude with a brief discussion on extensions to sphere packings. Patreon: http://www.patreon.com/timothynguyen I. Introduction 00:00: Biography 11:08: Lean and Formal Theorem Proving 13:05: Competitiveness and academia 15:02: Erdos and The Book 19:36: I am richer than Elon Musk 21:43: Overview II. Setup 24:23: Triangles and tangent circles 27:10: The Problem of Apollonius 28:27: Circle inversion (Viette’s solution) 36:06: Hartshorne’s Euclidean geometry book: Minimal straight-edge & compass constructions III. Circle Packings 41:49: Iterating tangent circles: Apollonian circle packing 43:22: History: Notebooks of Leibniz 45:05: Orientations (inside and outside of packing) 45:47: Asymptotics of circle packings 48:50: Fractals 50:54: Metacomment: Mathematical intuition 51:42: Naive dimension (of Cantor set and Sierpinski Triangle) 1:00:59: Rigorous definition of Hausdorff measure & dimension IV. Simple Geometry and Number Theory 1:04:51: Descartes’s Theorem 1:05:58: Definition: bend = 1/radius 1:11:31: Computing the two bends in the Apollonian problem 1:15:00: Why integral bends? 1:15:40: Frederick Soddy: Nobel laureate in chemistry 1:17:12: Soddy’s observation: integral packings V. Group Theory, Hyperbolic Dynamics, and Advanced Number Theory 1:22:02: Generating circle packings through repeated inversions (through dual circles) 1:29:09: Coxeter groups: Example 1:30:45: Coxeter groups: Definition 1:37:20: Poincare: Dynamics on hyperbolic space 1:39:18: Video demo: flows in hyperbolic space and circle packings 1:42:30: Integral representation of the Coxeter group 1:46:22: Indefinite quadratic forms and integer points of orthogonal groups 1:50:55: Admissible residue classes of bends 1:56:11: Why these residues? Answer: Strong approximation + Hasse principle 2:04:02: Major conjecture 2:06:02: The conjecture restores the "Local to Global" principle (for thin groups instead of orthogonal groups) 2:09:19: Confession: What a rich subject 2:10:00: Conjecture is asymptotically true 2:12:02: M. C. Escher VI. Dimension Three: Sphere Packings 2:13:03: Setup + what Soddy built 2:15:57: Local to Global theorem holds VII. Conclusion 2:18:20: Wrap up 2:19:02: Russian school vs Bourbaki Image Credits: http://timothynguyen.org/image-credits/

Greg Yang is a mathematician and AI researcher at Microsoft Research who for the past several years has done incredibly original theoretical work in the understanding of large artificial neural networks. Greg received his bachelors in mathematics from Harvard University in 2018 and while there won the Hoopes prize for best undergraduate thesis. He also received an Honorable Mention for the Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student in 2018 and was an invited speaker at the International Congress of Chinese Mathematicians in 2019. In this episode, we get a sample of Greg's work, which goes under the name "Tensor Programs" and currently spans five highly technical papers. The route chosen to compress Tensor Programs into the scope of a conversational video is to place its main concepts under the umbrella of one larger, central, and time-tested idea: that of taking a large N limit. This occurs most famously in the Law of Large Numbers and the Central Limit Theorem, which then play a fundamental role in the branch of mathematics known as Random Matrix Theory (RMT). We review this foundational material and then show how Tensor Programs (TP) generalizes this classical work, offering new proofs of RMT. We conclude with the applications of Tensor Programs to a (rare!) rigorous theory of neural networks. Patreon: https://www.patreon.com/timothynguyen Part I. Introduction 00:00:00 : Biography 00:02:45 : Harvard hiatus 1: Becoming a DJ 00:07:40 : I really want to make AGI happen (back in 2012) 00:09:09 : Impressions of Harvard math 00:17:33 : Harvard hiatus 2: Math autodidact 00:22:05 : Friendship with Shing-Tung Yau 00:24:06 : Landing a job at Microsoft Research: Two Fields Medalists are all you need 00:26:13 : Technical intro: The Big Picture 00:28:12 : Whiteboard outline Part II. Classical Probability Theory 00:37:03 : Law of Large Numbers 00:45:23 : Tensor Programs Preview 00:47:26 : Central Limit Theorem 00:56:55 : Proof of CLT: Moment method 1:00:20 : Moment method explicit computations Part III. Random Matrix Theory 1:12:46 : Setup 1:16:55 : Moment method for RMT 1:21:21 : Wigner semicircle law Part IV. Tensor Programs 1:31:03 : Segue using RMT 1:44:22 : TP punchline for RMT 1:46:22 : The Master Theorem (the key result of TP) 1:55:04 : Corollary: Reproof of RMT results 1:56:52 : General definition of a tensor program Part V. Neural Networks and Machine Learning 2:09:05 : Feed forward neural network (3 layers) example 2:19:16 : Neural network Gaussian Process 2:23:59 : Many distinct large N limits for neural networks 2:27:24 : abc parametrizations (Note: "a" is absorbed into "c" here): variance and learning rate scalings 2:36:54 : Geometry of space of abc parametrizations 2:39:41: Kernel regime 2:41:32 : Neural tangent kernel 2:43:35: (No) feature learning 2:48:42 : Maximal feature learning 2:52:33 : Current problems with deep learning 2:55:02 : Hyperparameter transfer (muP) 3:00:31 : Wrap up Further Reading: Tensor Programs I, II, III, IV, V by Greg Yang and coauthors. Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org

Scott Aaronson is a professor of computer science at University of Texas at Austin and director of its Quantum Information Center. Previously he received his PhD at UC Berkeley and was a faculty member at MIT in Electrical Engineering and Computer Science from 2007-2016. Scott has won numerous prizes for his research on quantum computing and complexity theory, including the Alan T Waterman award in 2012 and the ACM Prize in Computing in 2020. In addition to being a world class scientist, Scott is famous for his highly informative and entertaining blog Schtetl Optimized, which has kept the scientific community up to date on quantum hype for nearly the past two decades. In this episode, Scott Aaronson gives a crash course on quantum computing, diving deep into the details, offering insights, and clarifying misconceptions surrounding quantum hype. Patreon: https://www.patreon.com/timothynguyen Correction: 59:03: The matrix denoted as "Hadamard gate" is actually a 45 degree rotation matrix. The Hadamard gate differs from this matrix by a sign flip in the last column. See 1:11:00 for the Hadamard gate. Part I. Introduction (Personal) 00:00: Biography 01:02: Shtetl Optimized and the ways of blogging 09:56: sabattical at OpenAI, AI safety, machine learning 10:54: "I study what we can't do with computers we don't have" Part II. Introduction (Technical) 22:57: Overview 24:13: SMBC Cartoon: "The Talk". Summary of misconceptions of the field 33:09: How all quantum algorithms work: choreograph pattern of interference 34:38: Outline Part III. Setup 36:10: Review of classical bits 40:46: Tensor product and computational basis 42:07: Entanglement 44:25: What is not spooky action at a distance 46:15: Definition of qubit 48:10: bra and ket notation 50:48: Superposition example 52:41: Measurement, Copenhagen interpretation Part IV. Working with qubits 57:02: Unitary operators, quantum gates 1:03:34: Philosophical aside: How to "store" 2^1000 bits of information. 1:08:34: CNOT operation 1:09:45: quantum circuits 1:11:00: Hadamard gate 1:12:43: circuit notation, XOR notation 1:14:55: Subtlety on preparing quantum states 1:16:32: Building and decomposing general quantum circuits: Universality 1:21:30: Complexity of circuits vs algorithms 1:28:45: How quantum algorithms are physically implemented 1:31:55: Equivalence to quantum Turing Machine Part V. Quantum Speedup 1:35:48: Query complexity (black box / oracle model) 1:39:03: Objection: how is quantum querying not cheating? 1:42:51: Defining a quantum black box 1:45:30: Efficient classical f yields efficient U_f 1:47:26: Toffoli gate 1:50:07: Garbage and quantum uncomputing 1:54:45: Implementing (-1)^f(x)) 1:57:54: Deutsch-Jozsa algorithm: Where quantum beats classical 2:07:08: The point: constructive and destructive interference Part VI. Complexity Classes 2:08:41: Recap. History of Simon's and Shor's Algorithm 2:14:42: BQP 2:18:18: EQP 2:20:50: P 2:22:28: NP 2:26:10: P vs NP and NP-completeness 2:33:48: P vs BQP 2:40:48: NP vs BQP 2:41:23: Where quantum computing explanations go off the rails Part VII. Quantum Supremacy 2:43:46: Scalable quantum computing 2:47:43: Quantum supremacy 2:51:37: Boson sampling 2:52:03: What Google did and the difficulties with evaluating supremacy 3:04:22: Huge open question Twitter: @IAmTimNguyen Homepage: www.timothynguyen.org

Grant Sanderson is a mathematician who is the author of the YouTube channel “3Blue1Brown”, viewed by millions for its beautiful blend of visual animation and mathematical pedagogy. His channel covers a wide range of mathematical topics, which to name a few include calculus, quaternions, epidemic modeling, and artificial neural networks. Grant received his bachelor's degree in mathematics from Stanford University and has worked with a variety of mathematics educators and outlets, including Khan Academy, The Art of Problem Solving, MIT OpenCourseWare, Numberphile, and Quanta Magazine. In this episode, we discuss the famous unsolvability of quintic polynomials: there exists no formula, consisting only of finitely many arithmetic operations and radicals, for expressing the roots of a general fifth degree polynomial in terms of the polynomial's coefficients. The standard proof that is taught in abstract algebra courses uses the machinery of Galois theory. Instead of following that route, Grant and I proceed in barebones style along (somewhat) historical lines by first solving quadratics, cubics, and quartics. Along the way, we present the insights obtained by Lagrange that motivate a very natural combinatorial question, which contains the germs of modern group theory and Galois theory and whose answer suggests that the quintic is unsolvable (later confirmed through the work of Abel and Galois). We end with some informal discussions about Abel's proof and the topological proof due to Vladimir Arnold. Patreon: https://www.patreon.com/timothynguyen Part I. Introduction 00:00:Introduction 00:52: How did you get interested in math? 06:30: Future of math pedagogy and AI 12:03: Overview. How Grant got interested in unsolvability of the quintic 15:26: Problem formulation 17:42: History of solving polynomial equations 19:50: Po-Shen Loh Part II. Working Up to the Quintic 28:06: Quadratics 34:38 : Cubics 37:20: Viete’s formulas 48:51: Math duels over solving cubics: del Ferro, Fiorre, Tartaglia, Cardano, Ferrari 53:24: Prose poetry of solving cubics 54:30: Cardano’s Formula derivation 1:03:22: Resolvent 1:04:10: Why exactly 3 roots from Cardano’s formula? Part III. Thinking More Systematically 1:12:25: Takeaways and Lagrange’s insight into why quintic might be unsolvable 1:17:20: Origins of group theory? 1:23:29: History’s First Whiff of Galois Theory 1:25:24: Fundamental Theorem of Symmetric Polynomials 1:30:18: Solving the quartic from the resolvent 1:40:08: Recap of overall logic Part IV. Unsolvability of the Quintic 1:52:30: S_5 and A_5 group actions 2:01:18: Lagrange’s approach fails! 2:04:01: Abel’s proof 2:06:16: Arnold’s Topological Proof 2:18:22: Closing Remarks Further Reading on Arnold's Topological Proof of Unsolvability of the Quintic: 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

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

Tai-Danae Bradley is a mathematician who received her Ph.D. in mathematics from the CUNY Graduate Center. She was formerly at Alphabet and is now at Sandbox AQ, a startup focused on combining machine learning and quantum physics. Tai-Danae is a visiting research professor of mathematics at The Master’s University and the executive director of the Math3ma Institute, where she hosts her popular blog on category theory. She is also a co-author of the textbook Topology: A Categorical Approach that presents basic topology from the modern perspective of category theory. In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions. Patreon: https://www.patreon.com/timothynguyen Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc Timestamps: 00:00:00 : Introduction 00:03:07 : How did you get into category theory? 00:06:29 : Outline of podcast 00:09:21 : Motivating category theory 00:11:35 : Analogy: Object Oriented Programming 00:12:32 : Definition of category 00:18:50 : Example: Category of sets 00:20:17 : Example: Matrix category 00:25:45 : Example: Preordered set (poset) is a category 00:33:43 : Example: Category of finite-dimensional vector spaces 00:37:46 : Forgetful functor 00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity! 00:40:06 : Definition of functor 00:42:01 : Example: API change between programming languages is a functor 00:44:23 : Example: Groups, group homomorphisms are categories and functors 00:47:33 : Resume definition of functor 00:49:14 : Example: Functor between poset categories = order-preserving function 00:52:28 : Hom Functors. Things are getting meta (no not the tech company) 00:57:27 : Category theory is beautiful because of its rigidity 01:00:54 : Contravariant functor 01:03:23 : Definition: Presheaf 01:04:04 : Why are things meta? Arrows, arrows between arrows, categories of categories, ad infinitum. 01:07:38 : Probing a space with maps (prelude to Yoneda Lemma) 01:12:10 : Algebraic topology motivated category theory 01:15:44 : Definition: Natural transformation 01:19:21 : Example: Indexing category 01:21:54 : Example: Change of currency as natural transformation 01:25:35 : Isomorphism and natural isomorphism 01:27:34 : Notion of isomorphism in different categories 01:30:00 : Yoneda Lemma 01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation 01:42:33 : Analogy between Yoneda Lemma and linear algebra 01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors 01:50:40 : Yoneda embedding is fully faithful. Reasoning about this. 01:55:15 : Language Category 02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics" Further Reading: Tai-Danae's Blog: https://www.math3ma.com/categories Tai-Danae Bradley. "What is applied category theory?" https://arxiv.org/pdf/1809.05923.pdf Tai-Danae Bradley, John Terilla, Yiannis Vlassopoulos. "An enriched category theory of language: from syntax to semantics." https://arxiv.org/pdf/2106.07890.pdf

John Urschel received his bachelors and masters in mathematics from Penn State and then went on to become a professional football player for the Baltimore Ravens in 2014. During his second season, Urschel began his graduate studies in mathematics at MIT alongside his professional football career. Urschel eventually decided to retire from pro football to pursue his real passion, the study of mathematics, and he completed his doctorate in 2021. Urschel is currently a scholar at the Institute for Advanced Study where he is actively engaged in research on graph theory, numerical analysis, and machine learning. In addition, Urschel is the author of Mind and Matter, a New York Times bestseller about his life as an athlete and mathematician, and has been named as one of Forbes 30 under 30 for being an outstanding young scientist. In this episode, John and I discuss a hodgepodge of topics in spectral graph theory. We start off light and discuss the famous Braess's Paradox in which traffic congestion can be increased by opening a road. We then discuss the graph Laplacian to enable us to present Cheeger's Theorem, a beautiful result relating graph bottlenecks to graph eigenvalues. We then discuss various graph embedding and clustering results, and end with a discussion of the PageRank algorithm that powers Google search. Patreon: https://www.patreon.com/timothynguyen Originally published on June 9, 2022 on YouTube: https://youtu.be/O6k0JRpA2mg Timestamps: I. Introduction 00:00: Introduction 04:30: Being a professional mathematician and academia vs industry 09:41: John's taste in mathematics 13:00: Outline 17:23: Braess's Paradox: "Opening a highway can increase traffic congestion." 25:34: Prisoner's Dilemma. We need social forcing mechanisms to avoid undesirable outcomes (traffic jams). II. Spectral Graph Theory Basics 31:20: What is a graph 36:33: Graph bottlenecks. Practical situations: Task assignment, the economy, organizational management. 42:44: Quantifying bottlenecks: Cheeger's constant 46:43: Cheeger's constant sample computations 52:07: NP Hardness 55:48: Graph Laplacian 1:00:27: Graph Laplacian: 1-dimensional example III. Cheeger's Inequality and Harmonic Oscillators 1:07:35: Cheeger's Inequality: Statement 1:09:37: Cheeger's Inequality: A great example of beautiful mathematics 1:10:46: Cheeger's Inequality: Computationally tractable approximation of Cheeger's constant 1:19:16: Harmonic oscillators: Springs heuristic for lambda_2 and Cheeger's inequality 1:22:20: Interlude: Tutte's Spring Embedding Theorem (planar embeddings in terms of springs) 1:29:45: Interlude: Graph drawing using eigenfunction IV. Graph bisection and clustering 1:38:26: Summary thus far and graph bisection 1:42:44: Graph bisection: Large eigenvalues for PCA vs low eigenvalues for spectral bisection 1:43:40: Graph bisection: 1-dimensional intuition 1:47:43: Spectral graph clustering (complementary to graph bisection) V. Markov chains and PageRank 1:52:10: PageRank: Google's algorithm for ranking search results 1:53:44: PageRank: Markov chain (Markov matrix) 1:57:32: PageRank: Stationary distribution 2:00:20: Perron-Frobenius Theorem 2:06:10: Spectral gap: Analogy between bottlenecks for graphs and bottlenecks for Markov chain mixing 2:07:56: Conclusion: State of the field, Urschel's recent results 2:10:28: Joke: Two kinds of mathematicians Further Reading: A. Ng, M. Jordan, Y. Weiss. "On Spectral Clustering: Analysis and an algorithm" D. Spielman. "Spectral and Algebraic Graph Theory"

Richard Easther is a scientist, teacher, and communicator. He has been a Professor of Physics at the University of Auckland for over the last 10 years and was previously a professor of physics at Yale University. As a scientist, Richard covers ground that crosses particle physics, cosmology, astrophysics and astronomy, and in particular, focuses on the physics of the very early universe and the ways in which the universe changes between the Big Bang and the present day. In this episode, Richard and I discuss the details of cosmology at large, both technically and historically. We dive into Einstein's equations from general relativity and see what implications they have for an expanding universe alongside a discussion of the cast of characters involved in 20th century cosmology (Einstein, Hubble, Friedmann, Lemaitre, and others). We also discuss inflation, gravitational waves, the story behind Brian Keating's book Losing the Nobel Prize, and the current state of experiments and cosmology as a field. Patreon: https://www.patreon.com/timothynguyen Originally published on May 3, 2022 on YouTube: https://youtu.be/DiXyZgukRmE Timestamps: 00:00:00 : Introduction 00:02:42 : Astronomy must have been one of the earliest sciences 00:03:57 : Eric Weinstein and Geometric Unity 00:13:47 : Outline of podcast 00:15:10 : Brian Keating, Losing the Nobel Prize, Geometric Unity 00:16:38 : Big Bang and General Relativity 00:21:07 : Einstein's equations 00:26:27 : Einstein and Hilbert 00:27:47 : Schwarzschild solution (typo in video) 00:33:07 : Hubble 00:35:54 : One galaxy versus infinitely many 00:36:16 : Olbers' paradox 00:39:55 : Friedmann and FRLW metric 00:41:53 : Friedmann metric was audacious? 00:46:05 : Friedmann equation 00:48:36 : How to start a fight in physics: West coast vs East coast metric and sign conventions. 00:50:05 : Flat vs spherical vs hyperbolic space 00:51:40 : Stress energy tensor terms 00:54:15 : Conservation laws and stress energy tensor 00:58:28 : Acceleration of the universe 01:05:12 : Derivation of a(t) ~ t^2/3 from preceding computations 01:05:37 : a = 0 is the Big Bang. How seriously can we take this? 01:07:09 : Lemaitre 01:11:51 : Was Hubble's observation of an expanding universe in 1929 a fresh observation? 01:13:45 : Without Einstein, no General Relativity? 01:14:45 : Two questions: General Relativity vs Quantum Mechanics and how to understand time and universe's expansion velocity (which can exceed the speed of light!) 01:17:58 : How much of the universe is observable 01:24:54 : Planck length 01:26:33 : Physics down to the Big Bang singularity 01:28:07 : Density of photons vs matter 01:33:41 : Inflation and Alan Guth 01:36:49 : No magnetic monopoles? 01:38:30 : Constant density requires negative pressure 01:42:42 : Is negative pressure contrived? 01:49:29 : Marrying General Relativity and Quantum Mechanics 01:51:58 : Symmetry breaking 01:53:50 : How to corroborate inflation? 01:56:21 : Sabine Hossenfelder's criticisms 02:00:19 : Gravitational waves 02:01:31 : LIGO 02:04:13 : CMB (Cosmic Microwave Background) 02:11:27 : Relationship between detecting gravitational waves and inflation 02:16:37 : BICEP2 02:19:06 : Brian Keating's Losing the Nobel Prize and the problem of dust 02:24:40 : BICEP3 02:26:26 : Wrap up: current state of cosmology Notes: Easther's blogpost on Eric Weinstein: http://excursionset.com/blog/2013/5/25/trainwrecks-i-have-seen Vice article on Eric Weinstein and Geometric Unity:https://www.vice.com/en/article/z3xbz4/eric-weinstein-says-he-solved-the-universes-mysteries-scientists-disagree Further learning: Matts Roos. "Introduction to Cosmology" Barbara Ryden. "Introduction to Cosmology" Our Cosmic Mistake About Gravitational Waves: https://www.youtube.com/watch?v=O0D-COVodzY

Po-Shen Loh is a professor at Carnegie Mellon University and a coach for the US Math Olympiad. He is also a social entrepreneur where he has his used his passion and expertise in mathematics in the service of education (expii.com) and epidemiology (novid.org). In this episode, we discuss the mathematics behind Loh's novel approach to contact tracing in the fight against COVID, which involves a beautiful blend of graph theory and computer science. Originally published on March 3, 2022 on Youtube: https://youtu.be/8CLxLBMGxLE Patreon: https://www.patreon.com/timothynguyen Timestamps: 00:00:00 : Introduction 00:01:11 : About Po-Shen Loh 00:03:49 : NOVID app 00:04:47 : Graph theory and quarantining 00:08:39 : Graph adjacency definition for contact tracing 00:16:01 : Six degrees of separation away from anyone? 00:21:13 : Getting the game theory and incentives right 00:30:40 : Conventional approach to contact tracing 00:34:47 : Comparison with big tech 00:39:19 : Neighbor search complexity 00:45:15 : Watts-Strogatz small networks phenomenon 00:48:37 : Storing neighborhood information 00:57:00 : Random hashing to reduce computational burden 01:05:24 : Logarithmic probing of sparsity 01:09:56 : Two math PhDs struggle to do division 01:11:17 : Bitwise-or for union of bounded sets 01:16:21 : Step back and recap 01:26:15 : Tradeoff between number of hash bins and sparsity 01:29:12 : Conclusion Further reading: Po-Shen Loh. "Flipping the Perspective in Contact Tracing" https://arxiv.org/abs/2010.03806

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