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. Originally published on June 9, 2022 on YouTube: https://youtu.be/O6k0JRpA2mg Corrections: 01:14:24 : The inequalities are reversed here. It is corrected at 01:16:16. Timestamps: 00:00:00 : Introduction 00:04:30 : Being a professional mathematician and academia vs industry 00:09:41 : John’s taste in mathematics 00:13:00 : Outline 00:17:23 : Braess’s Paradox: ”Opening a highway can increase traffic congestion.” 00:25:34 : Prisoner’s Dilemma. We need social forcing mechanisms to avoid undesirable outcomes (traffic jams). 00:31:20 : What is a graph 00:36:33 : Graph bottlenecks. Practical situations: Task assignment, the economy, organizational management. 00:42:44 : Quantifying bottlenecks: Cheeger’s constant 00:46:43 : Cheeger’s constant sample computations 00:52:07 : NP Hardness 00:55:48 : Graph Laplacian 00:58:30 : Graph Laplacian: Relation to Laplacian from calculus 01:00:27 : Graph Laplacian: 1-dimensional example 01:01:22 : Graph Laplacian: Analyst’s Laplacian vs Geometer’s Laplacian (they differ by a minus sign) 01:04:44 : Graph Laplacian: Some history 01:07:35 : Cheeger’s Inequality: Statement 01:09:37 : ***Cheeger’s Inequality: A great example of beautiful mathematics*** 01:10:46 : Cheeger’s Inequality: Computationally tractable approximation of Cheeger’s constant 01:14:57 : Rayleigh quotient, discussion of proof of Cheeger’s inequality 01:19:16 : Harmonic oscillators: Springs heuristic for lambda_2 and Cheeger’s inequality 01:22:20 : Interlude: Tutte’s Spring Embedding Theorem (planar embeddings in terms of springs) 01:26:23 : Harmonic oscillators: Resume lambda_2 discussion and spring tension 01:29:45 : Interlude: Graph drawing using eigenfunctions 01:33:17 : Harmonic oscillators: Resume lambda_2 discussion: complete graph example and bottleneck is a perturbation of two disconnected components 01:38:26 : Summary thus far and graph bisection 01:42:44 : Graph bisection: Large eigenvalues for PCA vs low eigenvalues for spectral bisection 01:43:40 : Graph bisection: 1-dimensional intuition 01:44:40 : Graph bisection: Nodal domains 01:46:29 : Graph bisection: Aha, the 1-d example now makes sense. Splitting by level set of second eigenfunction is a good way to partition domain. 01:47:43 : Spectral graph clustering (complementary to graph bisection) 01:51:50 : Ng-Jordan-Weiss paper 01:52:10 : PageRank: Google’s algorithm for ranking search results 01:53:44 : PageRank: Markov chain (Markov matrix) 01:57:32 : PageRank: Stationary distribution 02:00:20 : Perron-Frobenius Theorem 02:06:10 : Spectral gap: Analogy between bottlenecks for graphs and bottlenecks for Markov chain mixing 02:07:56 : Conclusion: State of the field, Urschel’s recent results 02:10:28 : 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”