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Mathematics Colloquium


Date:  Tuesday, November 18, 2014

Title:  Revisiting classical results at the interface of number theory and representation theory

Abstract:  The speaker will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities are two peculiar identities which express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent work with Griffin and Warnaar, the speaker has obtained a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood q-series. This work characterizes those integral units that arise from this theory. In a related direction, the speaker revisits the classical Moonshine Theorem which asserts that the coefficients of the modular j-functions are dimensions of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Griffin and Duncan, the speaker has obtained exact formulas for these distributions.

Speaker:  Ken Ono
Institution:  Emory University


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