# Michael Griffin : On the distribution of Moonshine and the Umbral Moonshine conjectures.

Monstrous Moonshine asserts that the coefficients of the modular j-function are given in terms of ''dimensions'' of virtual character for the Monster group. There are 194 irreducible representations of the Monster, the largest of the sporadic simple groups, and it has been a longstanding open problem to determine their distribution in Moonshine. Witten and others have demonstrated deep connections between Monstrous Moonshine and quantum physics. The distributions of the Monster representations in Moonshine can be interpreted as the distributions of black hole states in 3 dimensional quantum gravity. In joint work with Ono and Duncan, we obtain exact formulas for these distributions. Moonshine type-phenomena have been observed for other finite simple groups besides the Monster. The Umbral Moonshine conjectures of Cheng, Duncan, and Harvey asserts that the Moonshine extends to 24 isomorphism classes of even unimodular positive-definite rank 24 lattices. Monstrous Moonshine can be regarded as the case of the Leech lattice. In 2013, Gannon proved the case for the Mathieu group M24. We offer a method of proof for the remaining 22 cases.

**Category**: Algebraic Geometry**Duration**: 01:34:50**Date**: January 21, 2015 at 4:25 PM**Views**: 115-
**Tags:**seminar, Algebraic Geometry Seminar

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