The Cartesian Cafe podcast

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محتوای ارائه شده توسط Timothy Nguyen. تمام محتوای پادکست شامل قسمت‌ها، گرافیک‌ها و توضیحات پادکست مستقیماً توسط Timothy Nguyen یا شریک پلتفرم پادکست آن‌ها آپلود و ارائه می‌شوند. اگر فکر می‌کنید شخصی بدون اجازه شما از اثر دارای حق نسخه‌برداری شما استفاده می‌کند، می‌توانید روندی که در اینجا شرح داده شده است را دنبال کنید.https://fa.player.fm/legal

Biscuits & Jam

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The War and Treaty Are Getting Carried Away 46:55

۵ روز پیش46:55

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46:55

The War and Treaty’s Michael and Tanya Trotter grew up in Cleveland, Ohio, and Washington, DC, respectively, but both have family roots in the South. They also grew up in the musical traditions of their churches – Tanya in the Black Baptist Church and Michael in the Seventh Day Adventist Church – where they learned the power of song to move people. After becoming a father at a very young age, Michael eventually joined the armed forces and served in Iraq and Germany, where he took up songwriting as a way of dealing with his experiences there. Meanwhile Tanya embarked on a singing and acting career after a breakthrough appearance in Sister Act 2 alongside Whoopi Goldberg and Lauryn Hill. Now, after a long and sometimes traumatic journey, Michael and Tanya are married, touring, winning all sorts of awards, and set to release their fifth album together, and their fourth as The War and Treaty. Sid talks to Michael and Tanya about the new record, Plus One , as well as their collaboration with Miranda Lambert, what it was like to record at FAME studios in Muscle Shoals, and how they’re blending country, soul, gospel, and R&B. Learn more about your ad choices. Visit podcastchoices.com/adchoices…

The Cartesian Cafe

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The Cartesian Cafe is the podcast where an expert guest and Timothy Nguyen map out scientific and mathematical subjects in detail. This collaborative journey with other experts will have us writing down formulas, drawing pictures, and reasoning about them together on a whiteboard. If you’ve been longing for a deeper dive into the intricacies of scientific subjects, then this is the podcast for you. Topics covered include mathematics, physics, computer science, machine learning, and artificial intelligence. Content also viewable on YouTube: www.youtube.com/timothynguyen and Spotify. Timothy Nguyen is a mathematician and AI researcher working in industry. Homepage: www.timothynguyen.com, Twitter: @IAmTimNguyen Patreon: www.patreon.com/timothynguyen

22 قسمت

#Science #Math #Physics #Timothy Nguyen

The Cartesian Cafe

14 subscribers

updated حدود یک سال پیش

اشتراک گذاری

خانه سری•Feed

22 قسمت

#Science #Math #Physics #Timothy Nguyen

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The Cartesian Cafe

1
Justin Clarke-Doane | Mathematics, Reality, and Morality 2:34:12

13 weeks پیش2:34:12

2:34:12

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 : Preview 01:56 : Naturalism & Mathematical vs Moral Realism 05:34 : Outline of the Discussion Part II. Philosophy of Mathematics 13:25 : Mathematical Realism 18:36 : The Reality of Numbers 27:58 : Anti-Realist Positions in Mathematics 41:49 : Fictionalism in Mathematics 44:06 : Distinguishing Metaphysics from Epistemology 45:39 : The Role of Naturalism and Fictionalism Part III. Philosophy of Morality (vs Mathematics) 50:24 : Moral Realism and Anti-Realism 58:31 : Analogies Between Mathematical and Moral Realism 01:05:30 : Kant's Constructivism and Ethical Contextualism 01:10:40 : Error Theory in Ethics 01:16:02 : Mathematical Realism and Moral Anti-Realism 01:17:22 : Contextualism and Moral Realism Part IV. Select Topics from Justin's Book 01:19:11 : Justification and Self-Evidence 01:21:24 : The Practice of Axiomatization: Mathematics vs Ethics 01:24:51 : Pushback: Is there really controversy in math? 01:30:24 : Justification and Belief: Quinean Empiricism and Harman's Thesis 01:41:44 : Observations, Explanations, and Moral Facts 01:48:41 : Supervenience and High-Level Descriptions 02:00:43 : Justification vs Truth: Reliability Challenge in Mathematics and Morality 02:03:53 : 2+2 not equaling 4: Accidental Truth vs Truth per se 02:13:10 : Pluralism in Mathematics and Ethics 02:31:27 : Concluding Thoughts 02:32:49 : Correction: "relativism" should be "realism" Further reading: Justin Clarke-Doane. Morality and Mathematics. X: @iamtimnguyen Webpage: http://www.timothynguyen.org…

The Cartesian Cafe

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Jay McClelland | Neural Networks: Artificial and Biological 2:59:15

23 weeks پیش2:59:15

2:59:15

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 : Preview 01:10 : Cognitive psychology 07:14 : Interdisciplinary work and Jay's academic journey 12:39 : Context affects perception 13:05 : Chomsky and psycholinguists 8:03 : Technical outline Part II. The Brain 00:20:20 : Structure of neurons 00:25:26 : Action potentials 00:27:00 : Synaptic processes and neuron firing 00:29:18 : Inhibitory neurons 00:33:10 : Feedforward neural networks 00:34:57 : Visual system 00:39:46 : Various parts of the visual cortex 00:45:31 : Columnar organization in the cortex 00:47:04 : Colocation in artificial vs biological networks 00:53:03 : Sensory systems and brain maps Part III. Approaches to AI, PDP, and Learning Rules 01:12:35 : Chomsky, symbolic rules, universal grammar 01:28:28 : Neuroscience, Francis Crick, vision vs language 01:32:36 : Neuroscience = bottom up 01:37:20 : Jay’s path to AI 01:43:51 : James Anderson 01:44:51 : Geoff Hinton 01:54:25 : Parallel Distributed Processing (PDP) 02:03:40 : McClelland & Rumelhart’s reading model 02:31:25 : Theories of learning 02:35:52 : Hebbian learning 02:43:23 : Rumelhart’s Delta rule 02:44:45 : Gradient descent 02:47:04 : Backpropagation 02:54:52 : 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…

The Cartesian Cafe

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Michael Freedman | A Fields Medalist Panorama 2:52:45

33 weeks پیش2:52:45

2:52:45

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 : Preview 01:34 : Fields Medalist working in industry 03:24 : Academia vs industry 04:59 : Mathematics and art 06:33 : Technical overview II. Early Mike: The Poincare Conjecture (PC) 08:14 : Introduction, statement, and history 14:30 : Three categories for PC (topological, smooth, PL) 17:09 : Smale and PC for d at least 5 17:59 : Homotopy equivalence vs homeomorphism 22:08 : Joke 23:24 : Morse flow 33:21 : Whitney Disk 41:47 : Casson handles 50:24 : Manifold factors and the Whitehead continuum 1:00:39 : Donaldson’s results in the smooth category 1:04:54 : (Not) writing up full details of the proof then and now 1:08:56 : Why Perelman succeeded II. Mid Mike: Topological Quantum Field Theory (TQFT) and Quantum Computing (QC) 1:10:54: Introduction 1:11:42: Cliff Taubes, Raoul Bott, Ed Witten 1:12:40 : Computational complexity, Church-Turing, and Mike’s motivations 1:24:01 : Why Mike left academia, Microsoft’s offer, and Station Q 1:29:23 : Topological quantum field theory (according to Atiyah) 1:34:29 : Anyons and a theorem on Chern-Simons theories 1:38:57 : Relation to QC 1:46:08 : Universal TQFT 1:55:57 : Witten: Donalson theory cannot be a unitary TQFT 2:01:22 : Unitarity is possible in dimension 3 2:05:12 : Relations to a theory of everything? 2:07:21 : Where topological QC is now III. Present Mike: Social Economics 2:11:08 : Introduction 2:14:02 : Lionel Penrose and voting schemes 2:21:01 : Radical markets (pun intended) 2:25:45 : Quadratic finance/funding 2:30:51 : Kant’s categorical imperative and a paper of Vitalik Buterin, Zoe Hitzig, Glen Weyl 2:36:54 : Gauge equivariance 2:38:32 : Bertrand Russell: philosophers and differential equations IV: Outro 2:46:20 : Final thoughts on math, science, philosophy 2:51:22 : Career advice Some Further Reading: Mike’s Harvard lecture on PC4: https://www.youtube.com/watch?v=TSF0i6BO1Ig Behrens et al. The Disc Embedding Theorem. M. Freedman. Spinoza, Leibniz, Kant, and Weyl. arxiv:2206.14711 Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org…

The Cartesian Cafe

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Marcus Hutter | Universal Artificial Intelligence and Solomonoff Induction 3:01:55

43 weeks پیش3:01:55

3:01:55

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 : Biography 01:45 : From Physics to AI 03:05 : Hutter Prize 06:25 : Overview of Universal Artificial Intelligence 11:10 : Technical outline II. Universal Prediction 18:27 : Laplace’s Rule and Bayesian Sequence Prediction 40:54 : Different priors: KT estimator 44:39 : Sequence prediction for countable hypothesis class 53:23 : Generalized Solomonoff Bound (GSB) 57:56 : Example of GSB for uniform prior 1:04:24 : GSB for continuous hypothesis classes 1:08:28 : Context tree weighting 1:12:31 : Kolmogorov complexity 1:19:36 : Solomonoff Bound & Solomonoff Induction 1:21:27 : Optimality of Solomonoff Induction 1:24:48 : Solomonoff a priori distribution in terms of random Turing machines 1:28:37 : Large Language Models (LLMs) 1:37:07 : Using LLMs to emulate Solomonoff induction 1:41:41 : Loss functions 1:50:59 : Optimality of Solomonoff induction revisited 1:51:51 : Marvin Minsky III. Universal Agents 1:52:42 : Recap and intro 1:55:59 : Setup 2:06:32 : Bayesian mixture environment 2:08:02 : AIxi. Bayes optimal policy vs optimal policy 2:11:27 : 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 intelligence 2:15:35 : AIXI explicit formula 2:23:53 : Other agents (optimistic agent, Thompson sampling, etc) 2:33:09 : Multiagent setting 2:39:38 : Grain of Truth problem 2:44:38 : Positive solution to Grain of Truth guarantees convergence to a Nash equilibria 2:45:01 : Computable approximations (simplifying assumptions on model classes): MDP, CTW, LLMs 2:56:13 : Outro: Brief philosophical remarks Further Reading: M. Hutter, D. Quarrel, E. Catt. An Introduction to Universal Artificial Intelligence M. Hutter. Universal Artificial Intelligence S. Legg and M. Hutter. Universal Intelligence: A Definition of Machine Intelligence Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org…

The Cartesian Cafe

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Richard Borcherds | Monstrous Moonshine: From Group Theory to String Theory 2:05:15

1 year پیش2:05:15

2:05:15

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…

The Cartesian Cafe

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Announcements for 2024 and a Message to Viewers 1:13

1 year پیش1:13

1:13

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

The Cartesian Cafe

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Tim Maudlin | Bell’s Theorem and Beyond: Nobody Understands Quantum Mechanics 2:41:51

1 year پیش2:41:51

2:41:51

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…

The Cartesian Cafe

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Antonio Padilla | Fantastic Numbers, Naturalness, and Anthropics in Physics 2:34:13

1 year پیش2:34:13

2:34:13

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…

The Cartesian Cafe

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Boaz Barak | Cryptography: The Art of Mathematical Secrecy 2:33:29

۲ years پیش2:33:29

2:33:29

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…

The Cartesian Cafe

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Sean Carroll | The Many Worlds Interpretation & Emergent Spacetime 2:12:40

۲ years پیش2:12:40

2:12:40

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…

The Cartesian Cafe

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Daniel Schroeder | Introduction to Thermal Physics 1:33:14

۲ years پیش1:33:14

1:33:14

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…

The Cartesian Cafe

1
Ethan Siegel | Demystifying Dark Matter 1:49:00

۲ years پیش1:49:00

1:49:00

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 Everything https://youtu.be/DPvD4VnD5Z4 Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org…

The Cartesian Cafe

1
Alex Kontorovich | Circle Packings and Their Hidden Treasures 2:20:02

۲ years پیش2:20:02

2:20:02

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/…

The Cartesian Cafe

1
Greg Yang | Large N Limits: Random Matrices & Neural Networks 3:01:27

۲ years پیش3:01:27

3:01:27

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…

The Cartesian Cafe

1
Scott Aaronson | Quantum Computing: Dismantling the Hype 3:05:08

۲ years پیش3:05:08

3:05:08

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…

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