Richard Borcherds | Monstrous Moonshine: From Group Theory to String Theory

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
• 02:51 : Success in mathematics
• 04:04 : Monstrous Moonshine overview and John Conway
• 09:44 : Technical overview

II. Group Theory

• 11:31 : Classification of finite-simple groups + history of the monster group
• 18:03 : Conway groups + Leech lattice
• 22:13 : Why was the monster conjectured to exist + more history 28:43 : Centralizers and involutions
• 32:37: Griess algebra

III. Modular Forms

• 36:42 : Definitions
• 40:06 : The elliptic modular function
• 48:58 : Subgroups of SL_2(Z)

IV. Monstrous Moonshine Conjecture Statement

• 57:17: Representations of the monster
• 59:22 : Hauptmoduls
• 1:03:50 : Statement of the conjecture
• 1:07:06 : Atkin-Fong-Smith's first proof
• 1:09:34 : Frenkel-Lepowski-Meurman's work + significance of Borcherd's proof

V. Sketch of Proof

• 1:14:47: Vertex algebra and monster Lie algebra
• 1:21:02 : No ghost theorem from string theory
• 1:25:24 : What's special about dimension 26?
• 1:28:33 : Monster Lie algebra details
• 1:32:30 : Dynkin diagrams and Kac-Moody algebras
• 1:43:21 : Simple roots and an obscure identity
• 1:45:13: Weyl denominator formula, Vandermonde identity
• 1:52:14 : Chasing down where modular forms got smuggled in
• 1:55:03 : Final calculations

VI. Epilogue

• 1:57:53 : Your most proud result?
• 2:00:47 : Monstrous moonshine for other sporadic groups?
• 2:02:28 : Connections to other fields. Witten and black holes and mock modular forms.

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

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