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

拟议中的研究旨在了解问题在于数论，组合学和李理论的接口。 具体来说，PI旨在确定弱马斯形式，模拟theta函数，q-超几何级数和仿射李超代数的表示理论之间更精确的相互作用。 这些问题的起源可以追溯到著名的数学家S。Ramanujan和G.沃森（c. 1920年）定义了一个称为“mock theta函数”的有限函数列表，继续认识到它们的重要性，并宣布它们的理解和表征为“最终问题”。 这个问题在近90年后的今天仍然存在，只是在过去的8年里才取得了重大进展，并形成了一个更统一的弱马斯形式理论（由于小野，布林曼，茨韦格斯，扎吉尔等人的工作）。积极的结果包括：（1）对模拟θ函数及其在更大的群论框架中的位置有了更一般的理解，在这个框架中，可以理解它们与弱马斯形式的关系;（2）实现了模拟θ函数和弱马斯形式不仅在数论中，而且在数学和科学的其他领域中发挥的作用。 尽管这些最近的事态发展，一个完整的理论弱马斯形式仍然缺乏。 一个问题的PI将着手沿着这些线路包括进一步最近的结果PI和Bringmann-Ono，有关弱马斯形式和模拟θ函数的字符公式仿射李超代数由于卡茨和Wakimoto。另一个目标是通过研究这些序列的变体和更一般的家族，建立更统一的结果，将q-超几何级数与模形式和Maass形式联系起来。例如，目前，关于q-超几何级数所起的作用，存在着大量零碎的结果，直到最近，我们才开始更精确地理解与仿射李超代数的表示理论有关的弱马斯形式理论。所提出的研究领域，数论，是数学最古老的分支之一，并且在当今仍然是广泛和活跃的研究领域。 在经典中，模形式扮演了许多基本角色;它们是费马大定理、朗兰兹纲领、黎曼假设、伯奇和斯温纳顿-戴尔猜想等证明的核心，并在弦理论、组合学、密码学、数学物理以及许多其他领域产生应用。PI的中心研究对象，模拟θ函数和马斯形式，是经典模块化形式的自然亲戚，拟议的研究旨在促进理解它们的作用，不仅在数论和模块化形式，而且组合学和李理论。 模拟θ函数的重要性不太受上述拉马努金和沃森的原始背景的约束，正如现在已知它们发挥重要作用的学科数量所证明的那样。 此外，缺乏一个全面的理论，这两个激励进一步的研究。

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.