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 : Biography

II. Universal Prediction

• 18:27 : Laplace’s Rule and Bayesian Sequence Prediction

III. Universal Agents

• 1:52:42 : Recap and intro

Further Reading:

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

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

II. Group Theory

• 11:31 : Classification of finite-simple groups + history of the monster group

III. Modular Forms

• 36:42 : Definitions

IV. Monstrous Moonshine Conjecture Statement

• 57:17: Representations of the monster

V. Sketch of Proof

• 1:14:47: Vertex algebra and monster Lie algebra

VI. Epilogue

• 1:57:53 : Your most proud result?

Further reading: V Tatitschef. A short introduction to Monstrous Moonshine. https://arxiv.org/pdf/1902.03118.pdf

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

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

Patreon:

https://www.patreon.com/timothynguyen

Tim Maudlin is a philosopher of science specializing in the foundations of physics, metaphysics, and logic. He is a professor at New York University, a member of the Foundational Questions Institute, and the founder and director of the John Bell Institute for the Foundations of Physics.

Patreon (bonus materials + video chat):

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

II. EPR Paradox / Argument

• 29:14 : EPR is not a paradox

III. Bohm Segue

• 1:11:05 : Introduction

IV. Bell's Theorem and Related Examples

• 1:25:13 : Setup

V. Miscellany

• 2:06:23 : Statistical independence assumption

VI. Interpretations of Quantum Mechanics

• 2:28:44: Interpretation is a misnomer

Further Reading:

J. Bell. Speakable and Unspeakable in Quantum Mechanics

T. Maudlin. Quantum Non-Locality and Relativity

Wikipedia: Mermin's device, GHZ experiment

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Antonio (Tony) Padilla is a theoretical physicist and cosmologist at the University of Nottingham. He serves as the Associate Director of the Nottingham Centre of Gravity, and in 2016, Tony shared the Buchalter Cosmology Prize for his work on the cosmological constant. Tony is also a star of the Numberphile YouTube channel, where his videos have received millions of views and he is also the author of the book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity.

Patreon: https://www.patreon.com/timothynguyen

This episode combines some of the greatest cosmological questions together with mathematical imagination. Tony and I go through the math behind some oft-quoted numbers in cosmology and calculate the age, size, and number of atoms in the universe. We then stretch our brains and consider how likely it would be to find your Doppelganger in a truly large universe, which takes us on a detour through black hole entropy. We end with a discussion of naturalness and the anthropic principle to round out our discussion of fantastic numbers in physics.

Part I. Introduction

• 00:00 : Introduction

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

• 21:42 : Size of the observable universe

Part III. Extreme Physics and Doppelgangers

• 1:07:27 : Long-term fate of the universe

Part IV: Naturalness and Anthropics

• 1:54:34 : What is naturalness? Examples.

Further reading: Antonio Padilla. Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Boaz Barak is a professor of computer science at Harvard University, having previously been a principal researcher at Microsoft Research and a professor at Princeton University. His research interests span many areas of theoretical computer science including cryptography, computational complexity, and the foundations of machine learning. Boaz serves on the scientific advisory boards for Quanta Magazine and the Simons Institute for the Theory of Computing and he was selected for Foreign Policy magazine’s list of 100 leading global thinkers for 2014.

www.patreon.com/timothynguyen

Cryptography is about maintaining the privacy and security of communication. In this episode, Boaz and I go through the fundamentals of cryptography from a foundational mathematical perspective. We start with some historical examples of attempts at encrypting messages and how they failed. After some guesses as to how one might mathematically define security, we arrive at the one due to Shannon. The resulting definition of perfect secrecy turns out to be too rigid, which leads us to the notion of computational secrecy that forms the foundation of modern cryptographic systems. We then show how the existence of pseudorandom generators (which remains a conjecture) ensures that such computational secrecy is achievable, assuming P does not equal NP. Having covered private key cryptography in detail, we then give a brief overview of public key cryptography. We end with a brief discussion of Bitcoin, machine learning, deepfakes, and potential doomsday scenarios.

I. Introduction

• 00:17 : Biography: Academia vs Industry

II. Warmup

• 24:42 : Substitution ciphers

III. Private Key Cryptography: Perfect Secrecy

• 34:32 : Valid encryption scheme

IV. Computational Secrecy

• 1:32:52 : Relax perfect secrecy to computational secrecy

V. Public Key Cryptography

• 2:00:32 : Limitations of private key cryptography

VI. Applications

• 2:14:39 : Bitcoin

Further reading: Boaz Barak. An Intensive Introduction to Cryptography

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

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

www.patreon.com/timothynguyen

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

Part I: Introduction

• 00:00:00 : Introduction

Part II: Quantum Mechanics in a Nutshell

• 00:14:58 : Textbook QM review

Part III: Many Worlds

• 00:31:53 : How MW comes in

Part IV: Additional Topics

• 01:35:13 : Bohmian mechanics

Part V. Emergent Spacetime

• 01:55:05 : Setup

Part VI. Conclusion

• 02:07:16 : Distribution of QM beliefs

Further reading:

• Hugh Everett. The Theory of the Universal Wave Function, 1956.

Fragments of the IDW: Joe Rogan, Sam Harris, Eric Weinstein: https://youtu.be/jM2FQrRYyas

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Daniel Schroeder is a particle and accelerator physicist and an editor for The American Journal of Physics. Dan received his PhD from Stanford University, where he spent most of his time at the Stanford Linear Accelerator, and he is currently a professor in the department of physics and astronomy at Weber State University. Dan is also the author of two revered physics textbooks, the first with Michael Peskin called An Introduction to Quantum Field Theory (or simply Peskin & Schroeder within the physics community) and the second An Introduction to Thermal Physics. Dan enjoys teaching physics courses at all levels, from Elementary Astronomy through Quantum Mechanics.

In this episode, I get to connect with one of my teachers, having taken both thermodynamics and quantum field theory courses when I was a university student based on Dan's textbooks. We take a deep dive towards answering two fundamental questions in the subject of thermodynamics: what is temperature and what is entropy? We provide both a qualitative and quantitative analysis, discussing good and bad definitions of temperature, microstates and macrostates, the second law of thermodynamics, and the relationship between temperature and entropy. Our discussion was also a great chance to shed light on some of the philosophical assumptions and conundrums in thermodynamics that do not typically come up in a physics course: the fundamental assumption of statistical mechanics, Laplace's demon, and the arrow of time problem (Loschmidt's paradox) arising from the second law of thermodynamics (i.e. why is entropy increasing in the future when mechanics has time-reversal symmetry).

Patreon: https://www.patreon.com/timothynguyen

Outline:

• 00:00:00 : Introduction

Further Reading:

• Daniel Schroeder. An Introduction to Thermal Physics

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Ethan Siegel is a theoretical astrophysicist and science communicator. He received his PhD from the University of Florida and held academic positions at the University of Arizona, University of Oregon, and Lewis & Clark College before moving on to become a full-time science writer. Ethan is the author of the book Beyond The Galaxy, which is the story of “How Humanity Looked Beyond Our Milky Way And Discovered The Entire Universe” and he has contributed numerous articles to ScienceBlogs, Forbes, and BigThink. Today, Ethan is the face and personality behind Starts With A Bang, both a website and podcast by the same name that is dedicated to explaining and exploring the deepest mysteries of the cosmos.

In this episode, Ethan and I discuss the mysterious nature of dark matter: the evidence for it and the proposals for what it might be.

Patreon: https://www.patreon.com/timothynguyen

Part I. Introduction

• 00:00:00 : Biography and path to science writing

Part II. Ordinary Matter

• 23:28 : Matter and radiation scaling relations

Part III. Dark Matter

• 1:01:02 : rho = radiation + ordinary matter + dark matter + dark energy

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

• E. Siegel. Beyond the Galaxy

More Ethan Siegel & Timothy Nguyen videos:

• Brian Keating’s Losing the Nobel Prize Makes a Good Point but …
  

Twitter: @iamtimnguyen

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

II. Setup

• 24:23: Triangles and tangent circles

III. Circle Packings

• 41:49: Iterating tangent circles: Apollonian circle packing

IV. Simple Geometry and Number Theory

• 1:04:51: Descartes’s Theorem

V. Group Theory, Hyperbolic Dynamics, and Advanced Number Theory

• 1:22:02: Generating circle packings through repeated inversions (through dual circles)

VI. Dimension Three: Sphere Packings

• 2:13:03: Setup + what Soddy built

VII. Conclusion

• 2:18:20: Wrap up

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