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/

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

Part II. Classical Probability Theory

• 00:37:03 : Law of Large Numbers

Part III. Random Matrix Theory

• 1:12:46 : Setup

Part IV. Tensor Programs

• 1:31:03 : Segue using RMT

Part V. Neural Networks and Machine Learning

• 2:09:05 : Feed forward neural network (3 layers) example

Further Reading:

Tensor Programs I, II, III, IV, V by Greg Yang and coauthors.

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Scott Aaronson is a professor of computer science at University of Texas at Austin and director of its Quantum Information Center. Previously he received his PhD at UC Berkeley and was a faculty member at MIT in Electrical Engineering and Computer Science from 2007-2016. Scott has won numerous prizes for his research on quantum computing and complexity theory, including the Alan T Waterman award in 2012 and the ACM Prize in Computing in 2020. In addition to being a world class scientist, Scott is famous for his highly informative and entertaining blog Schtetl Optimized, which has kept the scientific community up to date on quantum hype for nearly the past two decades.

In this episode, Scott Aaronson gives a crash course on quantum computing, diving deep into the details, offering insights, and clarifying misconceptions surrounding quantum hype.

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

Correction: 59:03: The matrix denoted as "Hadamard gate" is actually a 45 degree rotation matrix. The Hadamard gate differs from this matrix by a sign flip in the last column. See 1:11:00 for the Hadamard gate.

Part I. Introduction (Personal)

• 00:00: Biography

Part II. Introduction (Technical)

• 22:57: Overview

Part III. Setup

• 36:10: Review of classical bits

Part IV. Working with qubits

• 57:02: Unitary operators, quantum gates

Part V. Quantum Speedup

• 1:35:48: Query complexity (black box / oracle model)

Part VI. Complexity Classes

• 2:08:41: Recap. History of Simon's and Shor's Algorithm

Part VII. Quantum Supremacy

• 2:43:46: Scalable quantum computing

Homepage: www.timothynguyen.org

Grant Sanderson is a mathematician who is the author of the YouTube channel “3Blue1Brown”, viewed by millions for its beautiful blend of visual animation and mathematical pedagogy. His channel covers a wide range of mathematical topics, which to name a few include calculus, quaternions, epidemic modeling, and artificial neural networks. Grant received his bachelor's degree in mathematics from Stanford University and has worked with a variety of mathematics educators and outlets, including Khan Academy, The Art of Problem Solving, MIT OpenCourseWare, Numberphile, and Quanta Magazine.

In this episode, we discuss the famous unsolvability of quintic polynomials: there exists no formula, consisting only of finitely many arithmetic operations and radicals, for expressing the roots of a general fifth degree polynomial in terms of the polynomial's coefficients. The standard proof that is taught in abstract algebra courses uses the machinery of Galois theory. Instead of following that route, Grant and I proceed in barebones style along (somewhat) historical lines by first solving quadratics, cubics, and quartics. Along the way, we present the insights obtained by Lagrange that motivate a very natural combinatorial question, which contains the germs of modern group theory and Galois theory and whose answer suggests that the quintic is unsolvable (later confirmed through the work of Abel and Galois). We end with some informal discussions about Abel's proof and the topological proof due to Vladimir Arnold.

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

Part I. Introduction

• 00:00:Introduction

Part II. Working Up to the Quintic

• 28:06: Quadratics

Part III. Thinking More Systematically

• 1:12:25: Takeaways and Lagrange’s insight into why quintic might be unsolvable

Part IV. Unsolvability of the Quintic

• 1:52:30: S_5 and A_5 group actions

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

1. L. Goldmakher. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

John Baez is a mathematical physicist, professor of mathematics at UC Riverside, a researcher at the Centre for Quantum Technologies in Singapore, and a researcher at the Topos Institute in Berkeley, CA. John has worked on an impressively wide range of topics, pure and applied, ranging from loop quantum gravity, applications of higher categories to physics, applied category theory, environmental issues and math related to engineering and biology, and most recently on applying network theory to scientific software.Additionally, John is a prolific writer and blogger. This first began with John’s column This Week's Finds in Mathematical Physics, which ran 300 issues between 1993 and 2010, which then continued in the form of his ongoing blog Azimuth. Last but not least, John is also a host and contributor of the popular blog The n-category Cafe.

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:

1. 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!

Part I. Introduction

• 00:00: Introduction

Part II. Zoology of Standard Model

• 49:35: electron and neutrino

Part III. SU(5) numerology

• 1:41:16: 32 = 2^5 particles

Part IV. How the GUTs fit together

• 2:41:42: SO(10) rep: brief remarks

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

Tai-Danae Bradley is a mathematician who received her Ph.D. in mathematics from the CUNY Graduate Center. She was formerly at Alphabet and is now at Sandbox AQ, a startup focused on combining machine learning and quantum physics. Tai-Danae is a visiting research professor of mathematics at The Master’s University and the executive director of the Math3ma Institute, where she hosts her popular blog on category theory. She is also a co-author of the textbook Topology: A Categorical Approach that presents basic topology from the modern perspective of category theory.

In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions.

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

Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc

Timestamps:

• 00:00:00 : Introduction

• Tai-Danae's Blog: https://www.math3ma.com/categories

John Urschel received his bachelors and masters in mathematics from Penn State and then went on to become a professional football player for the Baltimore Ravens in 2014. During his second season, Urschel began his graduate studies in mathematics at MIT alongside his professional football career. Urschel eventually decided to retire from pro football to pursue his real passion, the study of mathematics, and he completed his doctorate in 2021. Urschel is currently a scholar at the Institute for Advanced Study where he is actively engaged in research on graph theory, numerical analysis, and machine learning. In addition, Urschel is the author of Mind and Matter, a New York Times bestseller about his life as an athlete and mathematician, and has been named as one of Forbes 30 under 30 for being an outstanding young scientist.

In this episode, John and I discuss a hodgepodge of topics in spectral graph theory. We start off light and discuss the famous Braess's Paradox in which traffic congestion can be increased by opening a road. We then discuss the graph Laplacian to enable us to present Cheeger's Theorem, a beautiful result relating graph bottlenecks to graph eigenvalues. We then discuss various graph embedding and clustering results, and end with a discussion of the PageRank algorithm that powers Google search.

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

Originally published on June 9, 2022 on YouTube: https://youtu.be/O6k0JRpA2mg

Timestamps:

I. Introduction

• 00:00: Introduction

II. Spectral Graph Theory Basics

• 31:20: What is a graph

III. Cheeger's Inequality and Harmonic Oscillators

• 1:07:35: Cheeger's Inequality: Statement

IV. Graph bisection and clustering

• 1:38:26: Summary thus far and graph bisection

V. Markov chains and PageRank

• 1:52:10: PageRank: Google's algorithm for ranking search results

Further Reading:

• A. Ng, M. Jordan, Y. Weiss. "On Spectral Clustering: Analysis and an algorithm"

Richard Easther is a scientist, teacher, and communicator. He has been a Professor of Physics at the University of Auckland for over the last 10 years and was previously a professor of physics at Yale University. As a scientist, Richard covers ground that crosses particle physics, cosmology, astrophysics and astronomy, and in particular, focuses on the physics of the very early universe and the ways in which the universe changes between the Big Bang and the present day.

In this episode, Richard and I discuss the details of cosmology at large, both technically and historically. We dive into Einstein's equations from general relativity and see what implications they have for an expanding universe alongside a discussion of the cast of characters involved in 20th century cosmology (Einstein, Hubble, Friedmann, Lemaitre, and others). We also discuss inflation, gravitational waves, the story behind Brian Keating's book Losing the Nobel Prize, and the current state of experiments and cosmology as a field.

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

Originally published on May 3, 2022 on YouTube: https://youtu.be/DiXyZgukRmE

Timestamps:

• 00:00:00 : Introduction

Notes:

• Easther's blogpost on Eric Weinstein: http://excursionset.com/blog/2013/5/25/trainwrecks-i-have-seen

Further learning:

• Matts Roos. "Introduction to Cosmology"