RUI: Mock modular forms and quantum modular forms with applications

This project is to study aspects of modular forms and related functions, which are, loosely speaking, highly symmetric complex functions. Modular forms have played significant 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. They are also fundamentally connected to the Riemann Hypothesis, a major unsolved conjecture in mathematics, and yield applications to other areas including combinatorics, mathematical physics, cryptography, and others. Research goals include studying the theory and applications of newer relatives to modular forms including mock modular forms, harmonic Maass forms, quantum modular forms, and related functions. Research and applications of these areas has developed within the last 20-25 years, however, a comprehensive theory is lacking. Certain research projects in this RUI project are specifically designed for undergraduate research, and undergraduate research mentoring and training are among the main components of the project. An overarching question in this project is to understand the various roles played by the holomorphic parts of harmonic Maass forms ("mock modular forms") and related objects in mathematics and number theory. One specific area of focus is the subject of quantum modular forms, including theory as well as applications to other fields such as topology and knot theory, and partition theory at the intersection of combinatorics and number theory. Some projects in this proposal are specifically designed for undergraduate research advised by the PI. Methods and goals build from the PI's existing research program in these areas, and include the growing theories of harmonic Maass forms, quantum modular forms, combinatorial q-hypergeometric series, and more. 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.