Theorems at the interface of number theory and representation theory
October 21, 11:45 — 12:35pm at LIT 305.
We will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities 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 joint work with Ono, and Warnaar, we obtain 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, we revisit the theory of Monstrous Moonshine which asserts that the coefficients of the modular j-function 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 Ono and Duncan, we obtain exact formulas for these distributions. Moonshine have phenomena have also been observed connecting other sporadic groups such as the Mathieu group M24 to certain mock modular forms. We will also discuss recent developments in this theory of ”Umbral Moonshine.”