This talk was the introduction to the Berkeley graduate number theory discussion seminar on 2020-10-28, and the aim was to explain why number theorists might be interested in sporadic simple groups.
We give a brief summary of monstrous moonshine relating sporadic groups to modular functions, and explain that a central theme is trying to find algebraic structures (vertex algebras, Lie algebras) acted on by sporadic groups. We give several examples of the Weyl denominator functions of Lie algebras, pointing out that many of them unexpectedly turn out to be automorphic forms. Finally we mention umbral moonshine, relating M24 to mock theta functions, for which no Lie algebras are yet known.
Correction: I forgot to include Ooguri and Tachikawa among those who noticed the connection
between mock theta functions and M24.
Негізгі бет Math talk: Sporadic groups and number theory
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