Michael Griffin - Moonshine and mock modular forms

February 4, 2016
Korea Institute for Advanced Study

Monstrous Moonshine creates a bridge between the fields of representation theory and modular forms. Here we will review the theory of harmonic Maass forms and mock modular forms which extends the classical theory and provides a useful framework with which to study the McKay?Thompson series. Although the McKay?Thompson series are true modular functions, we may construct them as harmonic Maass forms by way of the Maass Poincare’ series. This construction yields exact formulas and asymptotics for the coefficients. These formulas in turn allow us to answer Witten’s question concerning distribution of the irreducible representations in Monstrous Moonshine. We will also consider the construction of weight 1/2 vector valued Maass--Poincare’ series. These forms have appeared in recently observed variations of the moonshine phenomena?notably the Umbral Moonshine conjectures of Cheng, Duncan, and Harvey, however convergence in this case is particularly delicate and relies on the convergence of the Selberg-Kloosterman zeta function.

Video source: media.kias.re.kr