Weak Maass forms, mock theta functions, q-hypergeometric series, and applications

Yale University, New Haven CT

Investigators

Abstract

The proposed research seeks to understand problems that lie at the interface of number theory, combinatorics, and Lie theory. Specifically, the PI seeks to determine a more precise interplay between weak Maass forms, mock theta functions, q-hypergeometric series, and the representation theory of affine Lie superalgebras. The origins of such problems date back to prominent mathematical figures S. Ramanujan and G. Watson (c. 1920) who defined a finite list of functions called ``mock theta functions", went on to realize their significance, and declared their understanding and characterization as ``the final problem". The problem remains current now nearly 90 years later, with major strides and a more unifying theory of weak Maass forms developed only within the last 8 years (due to work of Ono, Bringmann, Zwegers, Zagier, and others). Positive results include (1) a more general understanding of the mock theta functions and their placement within a larger group-theoretical framework in which their relationship to weak Maass forms may be understood, and (2) a realization of the roles of the mock theta functions and weak Maass forms played not only in number theory, but other areas of mathematics and science. Despite these recent developments, a complete theory of weak Maass forms is still lacking. One problem the PI will embark upon along these lines includes furthering recent results of the PI and Bringmann-Ono, relating weak Maass forms and mock theta functions to character formulas for affine Lie superalgebras due to Kac and Wakimoto. Another goal is to establish more unifying results relating q-hypergeometric series to modular forms and Maass forms by studying variants and more general families of such series. Currently, largely piecemeal results exist regarding the roles played by q-hypergeometric series, for example, and only very recently have we begun to understand more precisely the theory of weak Maass forms as related to the representation theory of affine Lie superalgebras. The proposed area of research, number theory, is one of the oldest branches of mathematics, and continues to be a field of extensive and active research in the present day. Classically, modular forms have played many fundamental roles; they are central to the proof of Fermat's Last Theorem, the Langlands program, the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, for example, and yield applications in string theory, combinatorics, cryptography, mathematical physics, as well as many other areas. The central objects of study of the PI, mock theta functions and Maass forms, are natural relatives of classical modular forms, and the proposed research seeks to contribute to the understanding of their roles not only within number theory and modular forms, but also combinatorics and Lie theory. The prominence of the mock theta functions is less bound to the original contexts of Ramanujan and Watson as described above, as evidenced by the striking number of disciplines in which they are now known to play significant roles. Moreover, a comprehensive theory is lacking, both motivating further research.