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/