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.

Project Report

The primary research objectives of this proposal lie at the interface between Number Theory, Combinatorics, and Lie Theory. Number Theory is one of the oldest branches of mathematics, and continues to be a field of extensive research today. The central objects of study in this proposal are complex-valued functions called modular forms, and related functions, including weak Maass forms, mock theta functions, and quantum modular forms. Historically, modular forms are central to some of mathematics' greatest problems, including Fermat's Last Theorem, the Langlands program, and the Riemann hypothesis. Modular forms are also often naturally equipped with combinatorial structure, yielding applications in Combinatorics and Cryptography. The research in this proposal seeks to contribute to the understanding of the roles played by (mock) modular forms in the aforementioned areas. Three central projects resulting from this proposal are as follows. The first, "Almost harmonic Maass forms and Kac-Wakimoto characters" (joint with K. Bringmann), investigates the automorphic properties of affine Lie superalgebra characters due to Kac-Wakimoto pertaining to sl(m|n)^ highest weight modules. We prove, using a new approach, that while the characters are not ordinary modular forms, they are are essentially holomorphic parts of certain generalizations of weak Maass forms which we call "almost harmonic Maass forms". Additionally, we obtain their asymptotic expansions. In a second project, "l-adic properties of the partition function" (joint with Z. Kent and K. Ono), we use the theory of modular forms mod p due to J.P. Serre to show that generally, certain combinatorial partition functions are governed by "fractal" behavior, yielding a conceptual explanation of Ramanujan's original partition congruences. A third project, "q-series and quantum modular forms" (joint with K. Ono and R.C. Rhoades), focuses on the subject of quantum modular forms, defined by D. Zagier in 2010. Our work proves and generalizes Ramanujan's remaining conjecture from his last letter to Hardy, relating the combinatorial rank and crank functions, mock theta functions, and quantum modular forms. Integrated into this proposal are educational and outreach components. In 2011, with R. Howe, the PI co-led a summer seminar for public school teachers as part of the Yale National Initiative, which functions to strengthen teaching in public schools, and is an intensive and sustained collaboration among Yale faculty members and public school teachers from across the United States. In addition to research lectures, the P.I. delivered public and undergraduate lectures on research, education, science, and academia, and continues to mentor and advise students at various levels.

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Division of Mathematical Sciences (DMS)
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Andrew Pollington
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Yale University
New Haven
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