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.

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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