Episodi

  • Justin Clarke-Doane | Mathematics, Reality, and Morality

    Justin Clarke-Doane | Mathematics, Reality, and Morality

    · The Cartesian Cafe

    Justin Clarke-Doane is a professor of philosophy at Columbia University, whose interests span metaethics, epistemology, and the philosophy of logic & mathematics.

    In this thought provoking-discussion, Justin and I go deep into topics that are typically neglected by most mathematicians and scientists, namely the philosophy of mathematics and morality. Justin has contributed to both these areas via his book Morality and Mathematics, which takes the view that the standard position of being both a mathematical realist and moral antirealist is incoherent. Perhaps the most novel aspect of Justin's work is the treatment of the philosophy of mathematics and morality side-by-side, showing how these two topics, which are usually thought of as being unrelated, in fact have strong analogies. Along the way, we discuss many other foundational topics in epistemology and ethics, with elements of set theory, metaphysics, and logic sprinkled in.

    Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen

    Part I. Introduction

    00:00 : Preview01:56 : Naturalism & Mathematical vs Moral Realism05:34 : Outline of the Discussion

    Part II. Philosophy of Mathematics

    13:25 : Mathematical Realism18:36 : The Reality of Numbers27:58 : Anti-Realist Positions in Mathematics41:49 : Fictionalism in Mathematics44:06 : Distinguishing Metaphysics from Epistemology45:39 : The Role of Naturalism and Fictionalism

    Part III. Philosophy of Morality (vs Mathematics)

    50:24 : Moral Realism and Anti-Realism58:31 : Analogies Between Mathematical and Moral Realism01:05:30 : Kant's Constructivism and Ethical Contextualism01:10:40 : Error Theory in Ethics01:16:02 : Mathematical Realism and Moral Anti-Realism01:17:22 : Contextualism and Moral Realism

    Part IV. Select Topics from Justin's Book

    01:19:11 : Justification and Self-Evidence01:21:24 : The Practice of Axiomatization: Mathematics vs Ethics01:24:51 : Pushback: Is there really controversy in math?01:30:24 : Justification and Belief: Quinean Empiricism and Harman's Thesis01:41:44 : Observations, Explanations, and Moral Facts01:48:41 : Supervenience and High-Level Descriptions02:00:43 : Justification vs Truth: Reliability Challenge in Mathematics and Morality02:03:53 : 2+2 not equaling 4: Accidental Truth vs Truth per se02:13:10 : Pluralism in Mathematics and Ethics02:31:27 : Concluding Thoughts02:32:49 : Correction: "relativism" should be "realism"

    Further reading:Justin Clarke-Doane. Morality and Mathematics.

    X: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

  • Jay McClelland | Neural Networks: Artificial and Biological

    Jay McClelland | Neural Networks: Artificial and Biological

    · The Cartesian Cafe

    Jay McClelland is a pioneer in the field of artificial intelligence and is a cognitive psychologist and professor at Stanford University in the psychology, linguistics, and computer science departments. Together with David Rumelhart, Jay published the two volume work Parallel Distributed Processing, which has led to the flourishing of the connectionist approach to understanding cognition.

    In this conversation, Jay gives us a crash course in how neurons and biological brains work. This sets the stage for how psychologists such as Jay, David Rumelhart, and Geoffrey Hinton historically approached the development of models of cognition and ultimately artificial intelligence. We also discuss alternative approaches to neural computation such as symbolic and neuroscientific ones.

    Patreon (bonus materials + video chat):https://www.patreon.com/timothynguyen

    Part I. Introduction

    00:00 : Preview01:10 : Cognitive psychology07:14 : Interdisciplinary work and Jay's academic journey12:39 : Context affects perception13:05 : Chomsky and psycholinguists8:03 : Technical outline

    Part II. The Brain

    00:20:20 : Structure of neurons00:25:26 : Action potentials00:27:00 : Synaptic processes and neuron firing00:29:18 : Inhibitory neurons00:33:10 : Feedforward neural networks00:34:57 : Visual system00:39:46 : Various parts of the visual cortex00:45:31 : Columnar organization in the cortex00:47:04 : Colocation in artificial vs biological networks00:53:03 : Sensory systems and brain maps

    Part III. Approaches to AI, PDP, and Learning Rules

    01:12:35 : Chomsky, symbolic rules, universal grammar01:28:28 : Neuroscience, Francis Crick, vision vs language01:32:36 : Neuroscience = bottom up01:37:20 : Jay’s path to AI01:43:51 : James Anderson01:44:51 : Geoff Hinton01:54:25 : Parallel Distributed Processing (PDP)02:03:40 : McClelland & Rumelhart’s reading model02:31:25 : Theories of learning02:35:52 : Hebbian learning02:43:23 : Rumelhart’s Delta rule02:44:45 : Gradient descent02:47:04 : Backpropagation02:54:52 : Outro: Retrospective and looking ahead

    Image credits:http://timothynguyen.org/image-credits/

    Further reading:

    Rumelhart, McClelland. Parallel Distributed Processing.

    McClelland, J. L. (2013). Integrating probabilistic models of perception and interactive neural networks: A historical and tutorial review

    Twitter: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

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  • Michael Freedman | A Fields Medalist Panorama

    Michael Freedman | A Fields Medalist Panorama

    · The Cartesian Cafe

    Michael Freedman is a mathematician who was awarded the Fields Medal in 1986 for his solution of the 4-dimensional Poincare conjecture. Mike has also received numerous other awards for his scientific contributions including a MacArthur Fellowship and the National Medal of Science. In 1997, Mike joined Microsoft Research and in 2005 became the director of Station Q, Microsoft’s quantum computing research lab. As of 2023, Mike is a Senior Research Scientist at the Center for Mathematics and Scientific Applications at Harvard University.

    Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen

    In this wide-ranging conversation, we give a panoramic view of Mike’s extensive body of work over the span of his career. It is divided into three parts: early, middle, and present day, which respectively include his work on the 4-dimensional Poincare conjecture, his transition to topological physics, and finally his recent work in applying ideas from mathematics and philosophy to social economics. Our conversation is a blend of both the nitty-gritty details and the anecdotal story-telling that can only be obtained from a living legend.

    I. Introduction

    00:00 : Preview01:34 : Fields Medalist working in industry03:24 : Academia vs industry04:59 : Mathematics and art06:33 : Technical overview

    II. Early Mike: The Poincare Conjecture (PC)

    08:14 : Introduction, statement, and history14:30 : Three categories for PC (topological, smooth, PL)17:09 : Smale and PC for d at least 517:59 : Homotopy equivalence vs homeomorphism22:08 : Joke23:24 : Morse flow33:21 : Whitney Disk41:47 : Casson handles50:24 : Manifold factors and the Whitehead continuum1:00:39 : Donaldson’s results in the smooth category1:04:54 : (Not) writing up full details of the proof then and now1:08:56 : Why Perelman succeeded

    II. Mid Mike: Topological Quantum Field Theory (TQFT) and Quantum Computing (QC)

    1:10:54: Introduction1:11:42: Cliff Taubes, Raoul Bott, Ed Witten1:12:40 : Computational complexity, Church-Turing, and Mike’s motivations1:24:01 : Why Mike left academia, Microsoft’s offer, and Station Q1:29:23 : Topological quantum field theory (according to Atiyah)1:34:29 : Anyons and a theorem on Chern-Simons theories1:38:57 : Relation to QC1:46:08 : Universal TQFT1:55:57 : Witten: Donalson theory cannot be a unitary TQFT2:01:22 : Unitarity is possible in dimension 32:05:12 : Relations to a theory of everything?2:07:21 : Where topological QC is now

    III. Present Mike: Social Economics

    2:11:08 : Introduction2:14:02 : Lionel Penrose and voting schemes2:21:01 : Radical markets (pun intended)2:25:45 : Quadratic finance/funding2:30:51 : Kant’s categorical imperative and a paper of Vitalik Buterin, Zoe Hitzig, Glen Weyl2:36:54 : Gauge equivariance2:38:32 : Bertrand Russell: philosophers and differential equations

    IV: Outro

    2:46:20 : Final thoughts on math, science, philosophy2:51:22 : Career advice

    Some Further Reading:Mike’s Harvard lecture on PC4: https://www.youtube.com/watch?v=TSF0i6BO1IgBehrens et al. The Disc Embedding Theorem.M. Freedman. Spinoza, Leibniz, Kant, and Weyl. arxiv:2206.14711

    Twitter:@iamtimnguyen

    Webpage:http://www.timothynguyen.org

  • Marcus Hutter | Universal Artificial Intelligence and Solomonoff Induction

    Marcus Hutter | Universal Artificial Intelligence and Solomonoff Induction

    · The Cartesian Cafe

    Marcus Hutter is an artificial intelligence researcher who is both a Senior Researcher at Google DeepMind and an Honorary Professor in the Research School of Computer Science at Australian National University. He is responsible for the development of the theory of Universal Artificial Intelligence, for which he has written two books, one back in 2005 and one coming right off the press as we speak. Marcus is also the creator of the Hutter prize, for which you can win a sizable fortune for achieving state of the art lossless compression of Wikipedia text.

    Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen

    In this technical conversation, we cover material from Marcus’s two books “Universal Artificial Intelligence” (2005) and “Introduction to Universal Artificial Intelligence” (2024). The main goal is to develop a mathematical theory for combining sequential prediction (which seeks to predict the distribution of the next observation) together with action (which seeks to maximize expected reward), since these are among the problems that intelligent agents face when interacting in an unknown environment. Solomonoff induction provides a universal approach to sequence prediction in that it constructs an optimal prior (in a certain sense) over the space of all computable distributions of sequences, thus enabling Bayesian updating to enable convergence to the true predictive distribution (assuming the latter is computable). Combining Solomonoff induction with optimal action leads us to an agent known as AIXI, which in this theoretical setting, can be argued to be a mathematical incarnation of artificial general intelligence (AGI): it is an agent which acts optimally in general, unknown environments. The second half of our discussion concerning agents assumes familiarity with the basic setup of reinforcement learning.

    I. Introduction

    00:38 : Biography01:45 : From Physics to AI03:05 : Hutter Prize06:25 : Overview of Universal Artificial Intelligence11:10 : Technical outline

    II. Universal Prediction

    18:27 : Laplace’s Rule and Bayesian Sequence Prediction40:54 : Different priors: KT estimator44:39 : Sequence prediction for countable hypothesis class53:23 : Generalized Solomonoff Bound (GSB)57:56 : Example of GSB for uniform prior1:04:24 : GSB for continuous hypothesis classes1:08:28 : Context tree weighting1:12:31 : Kolmogorov complexity1:19:36 : Solomonoff Bound & Solomonoff Induction1:21:27 : Optimality of Solomonoff Induction1:24:48 : Solomonoff a priori distribution in terms of random Turing machines1:28:37 : Large Language Models (LLMs)1:37:07 : Using LLMs to emulate Solomonoff induction1:41:41 : Loss functions1:50:59 : Optimality of Solomonoff induction revisited1:51:51 : Marvin Minsky

    III. Universal Agents

    1:52:42 : Recap and intro1:55:59 : Setup2:06:32 : Bayesian mixture environment2:08:02 : AIxi. Bayes optimal policy vs optimal policy2:11:27 : AIXI (AIxi with xi = Solomonoff a priori distribution)2:12:04 : AIXI and AGI. Clarification: ASI (Artificial Super Intelligence) would be a more appropriate term than AGI for the AIXI agent.2:12:41 : Legg-Hutter measure of intelligence2:15:35 : AIXI explicit formula2:23:53 : Other agents (optimistic agent, Thompson sampling, etc)2:33:09 : Multiagent setting2:39:38 : Grain of Truth problem2:44:38 : Positive solution to Grain of Truth guarantees convergence to a Nash equilibria2:45:01 : Computable approximations (simplifying assumptions on model classes): MDP, CTW, LLMs2:56:13 : Outro: Brief philosophical remarks

    Further Reading:M. Hutter, D. Quarrel, E. Catt. An Introduction to Universal Artificial IntelligenceM. Hutter. Universal Artificial IntelligenceS. Legg and M. Hutter. Universal Intelligence: A Definition of Machine Intelligence

    Twitter: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

  • Richard Borcherds | Monstrous Moonshine: From Group Theory to String Theory

    Richard Borcherds | Monstrous Moonshine: From Group Theory to String Theory

    · The Cartesian Cafe

    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: Biography02:51 : Success in mathematics04:04 : Monstrous Moonshine overview and John Conway09:44 : Technical overview

    II. Group Theory

    11:31 : Classification of finite-simple groups + history of the monster group18:03 : Conway groups + Leech lattice22:13 : Why was the monster conjectured to exist + more history 28:43 : Centralizers and involutions32:37: Griess algebra

    III. Modular Forms

    36:42 : Definitions40:06 : The elliptic modular function48:58 : Subgroups of SL_2(Z)

    IV. Monstrous Moonshine Conjecture Statement

    57:17: Representations of the monster59:22 : Hauptmoduls1:03:50 : Statement of the conjecture1:07:06 : Atkin-Fong-Smith's first proof1:09:34 : Frenkel-Lepowski-Meurman's work + significance of Borcherd's proof

    V. Sketch of Proof

    1:14:47: Vertex algebra and monster Lie algebra1:21:02 : No ghost theorem from string theory1:25:24 : What's special about dimension 26?1:28:33 : Monster Lie algebra details1:32:30 : Dynkin diagrams and Kac-Moody algebras1:43:21 : Simple roots and an obscure identity1:45:13: Weyl denominator formula, Vandermonde identity1:52:14 : Chasing down where modular forms got smuggled in1:55:03 : 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

  • Announcements for 2024 and a Message to Viewers

    Announcements for 2024 and a Message to Viewers

    · The Cartesian Cafe

    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 | Bell’s Theorem and Beyond: Nobody Understands Quantum Mechanics

    Tim Maudlin | Bell’s Theorem and Beyond: Nobody Understands Quantum Mechanics

    · The Cartesian Cafe

    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: Biography05:26: Interdisciplinary work11:54 : Physicists working on the wrong things16:47 : Bell's Theorem soft overview24:14: Common misunderstanding of "God does not play dice."25:59: Technical outline

    II. EPR Paradox / Argument

    29:14 : EPR is not a paradox34:57 : Criterion of reality43:57 : Mathematical formulation46:32 : Locality: No spooky action at a distance49:54 : Bertlmann's socks53:17 : EPR syllogism summarized54:52 : Determinism is inferred not assumed1:02:18 : Clarifying analogy: Coin flips1:06:39 : Einstein's objection to determinism revisited

    III. Bohm Segue

    1:11:05 : Introduction1:13:38: Bell and von Neumann's error1:20:14: Bell's motivation: Can I remove Bohm's nonlocality?

    IV. Bell's Theorem and Related Examples

    1:25:13 : Setup1:27:59 : 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 inequality1:49:42: GHZ Example

    V. Miscellany

    2:06:23 : Statistical independence assumption2:13:18: The 2022 Nobel Prize2:17:43: Misconceptions and hidden variables2:22:28: The assumption of local realism? Repeat: Determinism is a conclusion not an assumption.

    VI. Interpretations of Quantum Mechanics

    2:28:44: Interpretation is a misnomer2:29:48: 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 Padilla | Fantastic Numbers, Naturalness, and Anthropics in Physics

    Antonio Padilla | Fantastic Numbers, Naturalness, and Anthropics in Physics

    · The Cartesian Cafe

    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 : Introduction01:06 : Math and or versus physics12:09 : Backstory behind Tony's book14:12 : Joke about theoreticians and numbers16:18 : Technical outline

    Part II. Size, Age, and Quantity in the Universe

    21:42 : Size of the observable universe22:32 : Standard candles27:39 : Hubble rate29:02 : Measuring distances and time37:15 : Einstein and Minkowski40:52 : Definition of Hubble parameter42:14 : Friedmann equation47:11 : Calculating the size of the observable universe51:24 : Age of the universe56:14 : Number of atoms in the observable universe1:01:08 : Critical density1:03:16: 10^80 atoms of hydrogen1:03:46 : Universe versus observable universe

    Part III. Extreme Physics and Doppelgangers

    1:07:27 : Long-term fate of the universe1:08:28 : Black holes and a googol years1:09:59 : Poincare recurrence1:13:23 : Doppelgangers in a googolplex meter wide universe1:16:40 : Finitely many states and black hole entropy1:25:00 : Black holes have no hair1:29:30 : Beckenstein, Christodolou, Hawking1:33:12 : Susskind's thought experiment: Maximum entropy of space1:42:58 : Estimating the number of doppelgangers1:54:21 : 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 discrepancy2:07:29 : Interlude: Energy shift clarification. Gravity is key.2:15:34 : Corrections to the cosmological constant2:18:47 : String theory landscape: 10^500 possibilities2:20:41 : Anthropic selection2:25:59 : 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 | Cryptography: The Art of Mathematical Secrecy

    Boaz Barak | Cryptography: The Art of Mathematical Secrecy

    · The Cartesian Cafe

    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 Industry10:07 : Military service12:53 : Technical overview17:01 : Whiteboard outline

    II. Warmup

    24:42 : Substitution ciphers27:33 : Viginere cipher29:35 : Babbage and Kasiski31:25 : Enigma and WW233:10 : Alan Turing

    III. Private Key Cryptography: Perfect Secrecy

    34:32 : Valid encryption scheme40:14 : Kerckhoffs's Principle42:41 : Cryptography = steelman your adversary44:40 : Attempt #1 at perfect secrecy49:58 : Attempt #2 at perfect secrecy56:02 : Definition of perfect secrecy (Shannon)1:05:56 : Enigma was not perfectly secure1:08:51 : Analogy with differential privacy1:11:10 : Example: One-time pad (OTP)1:20:07 : Drawbacks of OTP and Soviet KGB misuse1:21:43 : Important: Keys cannot be reused!1:27:48 : Shannon's Impossibility Theorem

    IV. Computational Secrecy

    1:32:52 : Relax perfect secrecy to computational secrecy1:41:04 : What computational secrecy buys (if P is not NP)1:44:35 : Pseudorandom generators (PRGs)1:47:03 : PRG definition1:52:30 : PRGs and P vs NP1:55:47: PRGs enable modifying OTP for computational secrecy

    V. Public Key Cryptography

    2:00:32 : Limitations of private key cryptography2:09:25 : Overview of public key methods2:13:28 : Post quantum cryptography

    VI. Applications

    2:14:39 : Bitcoin2:18:21 : Digital signatures (authentication)2:23:56 : Machine learning and deepfakes2:30:31 : A conceivable doomsday scenario: P = NP

    Further reading: Boaz Barak. An Intensive Introduction to Cryptography

    Twitter: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

  • Sean Carroll | The Many Worlds Interpretation & Emergent Spacetime

    Sean Carroll | The Many Worlds Interpretation & Emergent Spacetime

    · The Cartesian Cafe

    Sean Carroll is a theoretical physicist and philosopher who specializes in quantum mechanics, cosmology, and the philosophy of science. He is the Homewood Professor of Natural Philosophy at Johns Hopkins University and an external professor at the Sante Fe Institute. Sean has contributed prolifically to the public understanding of science through a variety of mediums: as an author of several physics books including Something Deeply Hidden and The Biggest Ideas in the Universe, as a public speaker and debater on a wide variety of scientific and philosophical subjects, and also as a host of his podcast Mindscape which covers topics spanning science, society, philosophy, culture, and the arts.

    www.patreon.com/timothynguyen

    In this episode, we take a deep dive into The Many Worlds (Everettian) Interpretation of quantum mechanics. While there are many philosophical discussions of the Many Worlds Interpretation available, ours marries philosophy with the technical, mathematical details. As a bonus, the whole gamut of topics from philosophy and physics arise, including the nature of reality, emergence, Bohmian mechanics, Bell's Theorem, and more. We conclude with some analysis of Sean's speculative work on the concept of emergent spacetime, a viewpoint which naturally arises from Many Worlds. This video is most suitable for those with a basic technical understanding of quantum mechanics.

    Part I: Introduction

    00:00:00 : Introduction00:05:42 : 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 review00:24:25 : The measurement problem00:25:28 : Einstein: "God does not play dice"00:27:49 : The reality problem

    Part III: Many Worlds

    00:31:53 : How MW comes in00:34:28 : EPR paradox (original formulation)00:40:58 : Simpler to work with spin00:42:03 : Spin entanglement00:44:46 : Decoherence00:49:16 : System, observer, environment clarification for decoherence00:53:54 : Density matrix perspective (sketch)00:56:21 : Deriving the Born rule00:59:09 : 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 credences01:11:28 : Observer-system split: pointer-state problem01:13:11 : Schrodinger's cat and decoherence01:18:21 : Consciousness and perception01:21:12 : Emergence and MW01:28:06 : Sorites Paradox and are there infinitely many worlds01:32:50 : Bad objection to MW: "It's not falsifiable."

    Part IV: Additional Topics

    01:35:13 : Bohmian mechanics01:40:29 : Bell's Theorem. What the Nobel Prize committee got wrong01:41:56 : David Deutsch on Bohmian mechanics01:46:39 : Quantum mereology01:49:09 : Path integral and double slit: virtual and distinct worlds

    Part V. Emergent Spacetime

    01:55:05 : Setup02:02:42 : Algebraic geometry / functional analysis perspective02:04:54 : Relation to MW

    Part VI. Conclusion

    02:07:16 : Distribution of QM beliefs02:08:38 : 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 | Introduction to Thermal Physics

    Daniel Schroeder | Introduction to Thermal Physics

    · The Cartesian Cafe

    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 : Introduction00:01:54 : Writing Books00:06:51 : Academic Track: Research vs Teaching00:11:01 : Charming Book Snippets00:14:54 : Discussion Plan: Two Basic Questions00:17:19 : Temperature is What You Measure with a Thermometer00:22:50 : Bad definition of Temperature: Measure of Average Kinetic Energy00:25:17 : Equipartition Theorem00:26:10 : Relaxation Time00:27:55 : Entropy from Statistical Mechanics00:30:12 : Einstein solid00:32:43 : Microstates + Example Computation00:38:33: Fundamental Assumption of Statistical Mechanics (FASM)00:46:29 : Multiplicity is highly concentrated about its peak00:49:50 : Entropy is Log(Multiplicity)00:52:02 : The Second Law of Thermodynamics00:56:13 : FASM based on our ignorance?00:57:37 : Quantum Mechanics and Discretization00:58:30 : More general mathematical notions of entropy01:02:52 : Unscrambling an Egg and The Second Law of Thermodynamics01:06:49 : Principle of Detailed Balance01:09:52 : How important is FASM?01:12:03 : Laplace's Demon01:13:35 : The Arrow of Time (Loschmidt's Paradox)01:15:20 : Comments on Resolution of Arrow of Time Problem01:16:07 : Temperature revisited: The actual definition in terms of entropy01:25:24 : Historical comments: Clausius, Boltzmann, Carnot01:29:07 : Final Thoughts: Learning Thermodynamics

    Further Reading:

    Daniel Schroeder. An Introduction to Thermal PhysicsL. Landau & E. Lifschitz. Statistical Physics.

    Twitter: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

  • Ethan Siegel | Demystifying Dark Matter

    Ethan Siegel | Demystifying Dark Matter

    · The Cartesian Cafe

    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 writing00:07:26 : Keeping up with the field outside academia00:11:42 : If you have a bone to pick with Ethan...00:12:50 : On looking like a scientist and words of wisdom00:18:24 : Understanding dark matter = one of the most important open problems00:21:07 : Technical outline

    Part II. Ordinary Matter

    23:28 : Matter and radiation scaling relations29:36 : Hubble constant31:00 : Components of rho in Friedmann's equations34:14 : Constituents of the universe41:21 : Big Bang nucleosynthesis (BBN)45:32 : eta: baryon to photon ratio and deuterium formation53:15 : Mass ratios vs eta

    Part III. Dark Matter

    1:01:02 : rho = radiation + ordinary matter + dark matter + dark energy1:05:25 : nature of peaks and valleys in cosmic microwave background (CMB): need dark matter1:07:39: Fritz Zwicky and mass mismatch among galaxies of a cluster1:10:40 : Kent Ford and Vera Rubin and and mass mismatch within a galaxy1:11:56 : Recap: BBN tells us that only about 5% of matter is ordinary1:15:55 : Concordance model (Lambda-CDM)1:21:04 : Summary of how dark matter provides a common solution to many problems1:23:29 : Brief remarks on modified gravity1:24:39 : Bullet cluster as evidence for dark matter1:31:40 : Candidates for dark matter (neutrinos, WIMPs, axions)1:38:37 : Experiment vs theory. Giving up vs forging on1:48:34 : Conclusion

    Image Credits: http://timothynguyen.org/image-credits/

    Further learning:

    E. Siegel. Beyond the GalaxyEthan 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-vraVtCzwTesting Eric Weinstein's and Stephen Wolfram's Theories of Everythinghttps://youtu.be/DPvD4VnD5Z4

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

  • Alex Kontorovich | Circle Packings and Their Hidden Treasures

    Alex Kontorovich | Circle Packings and Their Hidden Treasures

    · The Cartesian Cafe

    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: Biography11:08: Lean and Formal Theorem Proving13:05: Competitiveness and academia15:02: Erdos and The Book19:36: I am richer than Elon Musk21:43: Overview

    II. Setup

    24:23: Triangles and tangent circles27:10: The Problem of Apollonius28:27: 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 packing43:22: History: Notebooks of Leibniz45:05: Orientations (inside and outside of packing)45:47: Asymptotics of circle packings48:50: Fractals50:54: Metacomment: Mathematical intuition51:42: 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 Theorem1:05:58: Definition: bend = 1/radius1:11:31: Computing the two bends in the Apollonian problem1:15:00: Why integral bends?1:15:40: Frederick Soddy: Nobel laureate in chemistry1:17:12: 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: Example1:30:45: Coxeter groups: Definition1:37:20: Poincare: Dynamics on hyperbolic space1:39:18: Video demo: flows in hyperbolic space and circle packings1:42:30: Integral representation of the Coxeter group1:46:22: Indefinite quadratic forms and integer points of orthogonal groups1:50:55: Admissible residue classes of bends1:56:11: Why these residues? Answer: Strong approximation + Hasse principle2:04:02: Major conjecture2:06:02: The conjecture restores the "Local to Global" principle (for thin groups instead of orthogonal groups)2:09:19: Confession: What a rich subject2:10:00: Conjecture is asymptotically true2:12:02: M. C. Escher

    VI. Dimension Three: Sphere Packings

    2:13:03: Setup + what Soddy built2:15:57: Local to Global theorem holds

    VII. Conclusion

    2:18:20: Wrap up2:19:02: Russian school vs Bourbaki

    Image Credits: http://timothynguyen.org/image-credits/

  • Greg Yang | Large N Limits: Random Matrices & Neural Networks

    Greg Yang | Large N Limits: Random Matrices & Neural Networks

    · The Cartesian Cafe

    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 : Biography00:02:45 : Harvard hiatus 1: Becoming a DJ00:07:40 : I really want to make AGI happen (back in 2012)00:09:09 : Impressions of Harvard math00:17:33 : Harvard hiatus 2: Math autodidact00:22:05 : Friendship with Shing-Tung Yau00:24:06 : Landing a job at Microsoft Research: Two Fields Medalists are all you need00:26:13 : Technical intro: The Big Picture00:28:12 : Whiteboard outline

    Part II. Classical Probability Theory

    00:37:03 : Law of Large Numbers00:45:23 : Tensor Programs Preview00:47:26 : Central Limit Theorem00:56:55 : Proof of CLT: Moment method1:00:20 : Moment method explicit computations

    Part III. Random Matrix Theory

    1:12:46 : Setup1:16:55 : Moment method for RMT1:21:21 : Wigner semicircle law

    Part IV. Tensor Programs

    1:31:03 : Segue using RMT1:44:22 : TP punchline for RMT1:46:22 : The Master Theorem (the key result of TP)1:55:04 : Corollary: Reproof of RMT results1:56:52 : General definition of a tensor program

    Part V. Neural Networks and Machine Learning

    2:09:05 : Feed forward neural network (3 layers) example2:19:16 : Neural network Gaussian Process2:23:59 : Many distinct large N limits for neural networks2:27:24 : abc parametrizations (Note: "a" is absorbed into "c" here): variance and learning rate scalings2:36:54 : Geometry of space of abc parametrizations2:39:41: Kernel regime2:41:32 : Neural tangent kernel2:43:35: (No) feature learning2:48:42 : Maximal feature learning2:52:33 : Current problems with deep learning2:55:02 : 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 | Quantum Computing: Dismantling the Hype

    Scott Aaronson | Quantum Computing: Dismantling the Hype

    · The Cartesian Cafe

    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: Biography01:02: Shtetl Optimized and the ways of blogging09:56: sabattical at OpenAI, AI safety, machine learning10:54: "I study what we can't do with computers we don't have"

    Part II. Introduction (Technical)

    22:57: Overview24:13: SMBC Cartoon: "The Talk". Summary of misconceptions of the field33:09: How all quantum algorithms work: choreograph pattern of interference34:38: Outline

    Part III. Setup

    36:10: Review of classical bits40:46: Tensor product and computational basis42:07: Entanglement44:25: What is not spooky action at a distance46:15: Definition of qubit48:10: bra and ket notation50:48: Superposition example52:41: Measurement, Copenhagen interpretation

    Part IV. Working with qubits

    57:02: Unitary operators, quantum gates1:03:34: Philosophical aside: How to "store" 2^1000 bits of information.1:08:34: CNOT operation1:09:45: quantum circuits1:11:00: Hadamard gate1:12:43: circuit notation, XOR notation1:14:55: Subtlety on preparing quantum states1:16:32: Building and decomposing general quantum circuits: Universality1:21:30: Complexity of circuits vs algorithms1:28:45: How quantum algorithms are physically implemented1:31:55: 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 box1:45:30: Efficient classical f yields efficient U_f1:47:26: Toffoli gate1:50:07: Garbage and quantum uncomputing1:54:45: Implementing (-1)^f(x))1:57:54: Deutsch-Jozsa algorithm: Where quantum beats classical2:07:08: The point: constructive and destructive interference

    Part VI. Complexity Classes

    2:08:41: Recap. History of Simon's and Shor's Algorithm2:14:42: BQP2:18:18: EQP2:20:50: P2:22:28: NP2:26:10: P vs NP and NP-completeness2:33:48: P vs BQP2:40:48: NP vs BQP2:41:23: Where quantum computing explanations go off the rails

    Part VII. Quantum Supremacy

    2:43:46: Scalable quantum computing2:47:43: Quantum supremacy2:51:37: Boson sampling2:52:03: What Google did and the difficulties with evaluating supremacy3:04:22: Huge open questionTwitter: @IAmTimNguyen

    Homepage: www.timothynguyen.org

  • Grant Sanderson (3Blue1Brown) | Unsolvability of the Quintic

    Grant Sanderson (3Blue1Brown) | Unsolvability of the Quintic

    · The Cartesian Cafe

    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:Introduction00:52: 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 quintic15:26: Problem formulation17:42: History of solving polynomial equations19:50: Po-Shen Loh

    Part II. Working Up to the Quintic

    28:06: Quadratics34:38 : Cubics37:20: Viete’s formulas48:51: Math duels over solving cubics: del Ferro, Fiorre, Tartaglia, Cardano, Ferrari53:24: Prose poetry of solving cubics54:30: Cardano’s Formula derivation1:03:22: 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 unsolvable1:17:20: Origins of group theory?1:23:29: History’s First Whiff of Galois Theory1:25:24: Fundamental Theorem of Symmetric Polynomials1:30:18: Solving the quartic from the resolvent1:40:08: Recap of overall logic

    Part IV. Unsolvability of the Quintic

    1:52:30: S_5 and A_5 group actions2:01:18: Lagrange’s approach fails!2:04:01: Abel’s proof2:06:16: Arnold’s Topological Proof2:18:22: Closing Remarks

    Further Reading on Arnold's Topological Proof of Unsolvability of the Quintic:

    L. Goldmakher. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdfB. Katz. https://www.youtube.com/watch?v=RhpVSV6iCko

    Twitter: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

  • John Baez | The Algebra of Grand Unified Theories

    John Baez | The Algebra of Grand Unified Theories

    · The Cartesian Cafe

    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: Introduction05:50: Climate change09:40: Crackpot index14:50: Eric Weinstein, Brian Keating, Geometric Unity18:13: Overview of “The Algebra of Grand Unified Theories” paper25:40: Overview of Standard Model and GUTs34:25: SU(2), spin, isospin of nucleons 40:22: SO(4), Spin(4), double cover44:24: three kinds of spin

    Part II. Zoology of Standard Model

    49:35: electron and neutrino58:40: quarks1:04:51: the three generations of the Standard Model1:08:25: isospin quantum numbers1:17:11: U(1) representations (“charge”)1:29:01: hypercharge1:34:00: strong force and color1:36:50: SU(3)1:40:45: antiparticles

    Part III. SU(5) numerology

    1:41:16: 32 = 2^5 particles1:45:05: Mapping SU(3) x SU(2) x U(1) to SU(5) and hypercharge matching2:05:17: Exterior algebra of C^5 and more hypercharge matching2:37:32: SU(5) rep extends Standard Model rep

    Part IV. How the GUTs fit together

    2:41:42: SO(10) rep: brief remarks2:46:28: Pati-Salam rep: brief remarks2:47:17: Commutative diagram: main result2:49:12: What about the physics? Spontaneous symmetry breaking and the Higgs mechanism

    Twitter: @iamtimnguyen

    Webpage: http://www.timothynguyen.org

  • Tai-Danae Bradley | Category Theory and Language Models

    Tai-Danae Bradley | Category Theory and Language Models

    · The Cartesian Cafe

    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 : Introduction00:03:07 : How did you get into category theory?00:06:29 : Outline of podcast00:09:21 : Motivating category theory00:11:35 : Analogy: Object Oriented Programming00:12:32 : Definition of category00:18:50 : Example: Category of sets00:20:17 : Example: Matrix category00:25:45 : Example: Preordered set (poset) is a category00:33:43 : Example: Category of finite-dimensional vector spaces00:37:46 : Forgetful functor00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity!00:40:06 : Definition of functor00:42:01 : Example: API change between programming languages is a functor00:44:23 : Example: Groups, group homomorphisms are categories and functors00:47:33 : Resume definition of functor00:49:14 : Example: Functor between poset categories = order-preserving function00:52:28 : Hom Functors. Things are getting meta (no not the tech company)00:57:27 : Category theory is beautiful because of its rigidity01:00:54 : Contravariant functor01:03:23 : Definition: Presheaf01:04:04 : 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 theory01:15:44 : Definition: Natural transformation01:19:21 : Example: Indexing category01:21:54 : Example: Change of currency as natural transformation01:25:35 : Isomorphism and natural isomorphism01:27:34 : Notion of isomorphism in different categories01:30:00 : Yoneda Lemma01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation01:42:33 : Analogy between Yoneda Lemma and linear algebra01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors01:50:40 : Yoneda embedding is fully faithful. Reasoning about this.01:55:15 : Language Category02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics"

    Further Reading:

    Tai-Danae's Blog: https://www.math3ma.com/categoriesTai-Danae Bradley. "What is applied category theory?" https://arxiv.org/pdf/1809.05923.pdfTai-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 | Tackling Graph Theory

    John Urschel | Tackling Graph Theory

    · The Cartesian Cafe

    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: Introduction04:30: Being a professional mathematician and academia vs industry09:41: John's taste in mathematics13:00: Outline17:23: 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 graph36:33: Graph bottlenecks. Practical situations: Task assignment, the economy, organizational management.42:44: Quantifying bottlenecks: Cheeger's constant46:43: Cheeger's constant sample computations52:07: NP Hardness55:48: Graph Laplacian1:00:27: Graph Laplacian: 1-dimensional example

    III. Cheeger's Inequality and Harmonic Oscillators

    1:07:35: Cheeger's Inequality: Statement1:09:37: Cheeger's Inequality: A great example of beautiful mathematics1:10:46: Cheeger's Inequality: Computationally tractable approximation of Cheeger's constant1:19:16: Harmonic oscillators: Springs heuristic for lambda_2 and Cheeger's inequality1:22:20: 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 bisection1:42:44: Graph bisection: Large eigenvalues for PCA vs low eigenvalues for spectral bisection1:43:40: Graph bisection: 1-dimensional intuition1:47:43: Spectral graph clustering (complementary to graph bisection)

    V. Markov chains and PageRank

    1:52:10: PageRank: Google's algorithm for ranking search results1:53:44: PageRank: Markov chain (Markov matrix)1:57:32: PageRank: Stationary distribution2:00:20: Perron-Frobenius Theorem2:06:10: Spectral gap: Analogy between bottlenecks for graphs and bottlenecks for Markov chain mixing2:07:56: Conclusion: State of the field, Urschel's recent results2:10:28: 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 | The Big Bang, Inflation, and Gravitational Waves

    Richard Easther | The Big Bang, Inflation, and Gravitational Waves

    · The Cartesian Cafe

    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 : Introduction00:02:42 : Astronomy must have been one of the earliest sciences00:03:57 : Eric Weinstein and Geometric Unity00:13:47 : Outline of podcast00:15:10 : Brian Keating, Losing the Nobel Prize, Geometric Unity00:16:38 : Big Bang and General Relativity00:21:07 : Einstein's equations00:26:27 : Einstein and Hilbert00:27:47 : Schwarzschild solution (typo in video)00:33:07 : Hubble00:35:54 : One galaxy versus infinitely many00:36:16 : Olbers' paradox00:39:55 : Friedmann and FRLW metric00:41:53 : Friedmann metric was audacious?00:46:05 : Friedmann equation00:48:36 : How to start a fight in physics: West coast vs East coast metric and sign conventions.00:50:05 : Flat vs spherical vs hyperbolic space00:51:40 : Stress energy tensor terms00:54:15 : Conservation laws and stress energy tensor00:58:28 : Acceleration of the universe01:05:12 : Derivation of a(t) ~ t^2/3 from preceding computations01:05:37 : a = 0 is the Big Bang. How seriously can we take this?01:07:09 : Lemaitre01:11:51 : 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 observable01:24:54 : Planck length01:26:33 : Physics down to the Big Bang singularity01:28:07 : Density of photons vs matter01:33:41 : Inflation and Alan Guth01:36:49 : No magnetic monopoles?01:38:30 : Constant density requires negative pressure01:42:42 : Is negative pressure contrived?01:49:29 : Marrying General Relativity and Quantum Mechanics01:51:58 : Symmetry breaking01:53:50 : How to corroborate inflation?01:56:21 : Sabine Hossenfelder's criticisms02:00:19 : Gravitational waves02:01:31 : LIGO02:04:13 : CMB (Cosmic Microwave Background)02:11:27 : Relationship between detecting gravitational waves and inflation02:16:37 : BICEP202:19:06 : Brian Keating's Losing the Nobel Prize and the problem of dust02:24:40 : BICEP302:26:26 : 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
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