Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory

Abstract

Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and $q$-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular $L$-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.