RUI: Harmonic Maass Forms and Quantum Modular Forms

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This project studies questions about modular forms, broadly speaking, which are certain symmetric complex functions that have played key roles in mathematics and number theory. For example, the theory of modular forms was important in the 1995 proof of Fermat's Last Theorem, a conjecture which remained unsolved for centuries. Modular forms are also connected to the Riemann Hypothesis, a major unsolved conjecture, and yield applications to combinatorics, mathematical physics, cryptography, and more. Research goals include studying the theory and applications of mock modular forms, harmonic Maass forms, quantum modular forms, and related functions, which are more modern relatives to modular forms. While these subjects have seen substantial developments within the last 20 years, a comprehensive theory is still lacking. Some research projects within this RUI award are specifically designed for undergraduate research. Undergraduate research mentoring and training and writing for broad mathematical audiences are also major components of this project. An overarching goal of this RUI project is to understand roles played by the holomorphic parts of harmonic Maass forms (mock modular forms) and related functions, for example, holomorphic quantum modular forms, torus knots and quantum modularity, partial theta functions, quantum Jacobi forms, mock Jacobi forms, and combinatorial applications. Methods and goals build on the P.I.’s earlier work in these areas, and include tools and results from the growing theory of harmonic Maass forms, the developing subject of quantum modular forms, and connections between these two areas, among others.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.