RUI: mock modular forms and quantum modular forms with applications

The National Science Foundation Division of Mathematical Sciences awarded Amherst College, Trustees Of, $254,952 on August 15, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to conduct theoretical research on mock modular forms, harmonic Maass forms, quantum modular forms, and related symmetric complex functions with applications to number theory, combinatorics, mathematical physics, cryptography, topology, knot theory, and partition theory.

The project emphasizes undergraduate research participation, with specific research projects designed for undergraduate mentoring and training alongside faculty-led investigation of how holomorphic parts of harmonic Maass forms and related objects function in mathematics and number theory. Research goals include developing a comprehensive theory of mock modular forms and quantum modular forms—newer relatives to classical modular forms that have emerged over the past 20–25 years but lack unified theoretical frameworks—and exploring their connections to established mathematical problems including the Riemann Hypothesis.

Performance is located in Amherst, Massachusetts, with an obligated amount of $254,952 and a period of performance extending from the award date through July 31, 2029. The project builds from the principal investigator's existing research program in harmonic Maass forms, quantum modular forms, and combinatorial q-hypergeometric series.

Abstract

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.- subawards are not planned for this award.