This award supports research at the intersection of number theory, combinatorics, and Lie theory. In particular, the P.I. seeks to determine more precise relationships and interplay between weak Maass forms and their generalizations, (non-holomorphic) Jacobi forms, and combinatorial q-hypergeometric series. The major project objectives include a study of quantum modular forms, vertex operator algebra trace functions and graded dimensions, and the automorphic properties of combinatorial q-series. The P.I. will additionally integrate a number of educational and outreach programs at all levels into the award objectives. Namely, the P.I. has begun a collaboration with the New Haven Public Schools, and will continue to develop a mathematics enrichment program for elementary and middle school students. The P.I. will also act as faculty advisor to the Yale University-New Haven chapter of MATHCOUNTS Outreach, an undergraduate arm of the national organization which promotes mathematics in New Haven Public Schools. The P.I. also seeks to enhance research and educational opportunities for graduate students, undergraduate students, and postdoctoral fellows, including women and girls in mathematics at all levels, through research collaboration, mentoring, and outreach programs.
Number theory is one of the oldest branches of mathematics, and continues to be a field of extensive and active research today. Modular forms have played many fundamental roles; they are central to the proof of Fermat's Last Theorem, the Langlands program, the Riemann hypothesis, and the Birch and Swinnerton-Dyer conjecture, for example, and yield applications in combinatorics, cryptography, mathematical physics, and many other areas. The P.I. will study natural relatives of modular forms, namely weak Maass forms and their generalizations. While recent developments have been made, a comprehensive theory is lacking. The proposed research seeks to contribute to the understanding of the roles of these functions not only within number theory and modular forms, but also combinatorics and Lie theory.
