Applications of automorphic forms and hypergeometric q-series

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The primary aim of this research program is to study the applications of modular and automorphic forms, and hypergeometric q-series. These applications include a wide selection of different areas of mathematics and mathematical physics, such as the theory of integer partitions, combinatorial probability and Markov processes, bootstrap percolation models, affine Lie superalgebras, and (quadratic) Hurwitz class numbers. The automorphic objects of interest include modular and Jacobi forms, as well as mock modular and Jacobi forms, which have seen a great deal of recent interest thanks to work of Borcherds, Bringmann, Bruinier, Funke, Ono, Zagier, and Zwegers. Mock modular forms are of particular interest due both to their connections with harmonic Maass forms, as well as their famous history as objects of mystery dating back to Ramanujan and Watson. As an example of the interplay between automorphic forms and other topics, the PI and Bringmann recently used combinatorial probability bounds for gap-avoiding sequences (that first arose in Holroyd, Liggett, and Romik's study of finite-size scaling in bootstrap percolation) in order to prove a cuspidal asymptotic expansion for a family of hypergeometric q-series considered by Andrews. Due to its scope, this research program has the potential for wide-ranging applications. An underlying theme is the universality of the tools and techniques of modern number theory, whose best-known uses include cryptography and cellular communication. This research will also illustrate applications to high-energy physics (where wall-crossings and black holes are described by mock theta functions), and biological 'growth' processes (where the the large-scale behavior of cellular automata is determined by the asymptotic expansions of modular forms).