RUI: Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms: Theory and Applications

Amherst College, Amherst MA

Investigators

Abstract

This is an RUI award in the area of number theory, one of the oldest branches of pure mathematics, which continues to be actively and extensively researched today. The P.I. will primarily study functions which are natural relatives to modular forms called mock modular and quantum modular forms, and harmonic Maass forms. Modular forms are among the most fundamental objects in the area of number theory in mathematics. For example, they are central to the Riemann Hypothesis (1859), so significant a conjecture that the Clay Mathematics Institute is offering one million dollars to anyone who can solve it. Major developments in the theory of modular forms were also crucial to the 1995 proof of Fermat's Last Theorem (1637), a conjecture which defeated mathematicians for 358 years. Beyond number theory, modular forms also yield applications to the diverse areas of combinatorics, mathematical physics, cryptography, and more. The newer related theories of harmonic Maass forms and mock modular forms have evolved substantially over the last 10-15 years, and have opened the door to many new applications and significant results; a comprehensive theory is still lacking, however. Moreover, within the last few years, the relatively new subject of quantum modular forms has shown intriguing connections to other areas including topology, mathematical physics, representation theory, combinatorics, and more. The P.I. will study the theory and applications of harmonic Maass forms, mock modular forms, quantum modular forms and related functions. An overarching question is to understand the various roles played by the holomorphic parts of harmonic Maass forms and related automorphic objects in mathematics and number theory. Methods include tools from the theory of harmonic Maass forms, aspects of the developing theory of quantum modular forms, transformation and other properties of mock modular and mock Jacobi forms, zeta functions, analytic properties of Mordell and Eichler integrals, theta functions, and combinatorial q-hypergeometric series. The relatively new subject of quantum modular forms in particular has shown rich connections to other areas including harmonic Maass forms, Eichler integrals, radial limits and mock modular forms, combinatorics, partial theta functions, Jacobi forms, topology, and vertex algebras, many of which are included within the scope of this proposal. Some projects within this RUI proposal are specifically designed for undergraduate research. Undergraduate research mentoring and training, and (expository) writing for broad mathematical audiences, are also components of this proposal. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.