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

这项拟议的研究试图理解存在于数论、组合学和测谎论之间的问题。具体地说，PI试图确定弱Maass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人物S.Ramanujan和G.Watson(约1920年)，他们定义了一个有限的函数列表，称为‘’模拟theta函数‘’，进而认识到它们的意义，并宣布对它们的理解和表征是‘’最后的问题‘’。近90年后的今天，这个问题仍然存在，只有在过去的8年里(由于小野、布林曼、齐格斯、扎吉尔和其他人的工作)，才取得了重大进展，形成了更统一的弱Maass形式理论。积极的结果包括：(1)更广泛地理解了模拟theta函数及其在一个更大的群论框架中的位置，在这个框架中可以理解它们与弱Maass形式的关系，以及(2)认识到模拟theta函数和弱Maass形式不仅在数论中，而且在数学和科学的其他领域中所起的作用。尽管最近取得了这些进展，但弱Maass形式的完整理论仍然缺乏。PI将沿着这些思路着手解决的一个问题包括进一步发展PI和Bringmann-Ono的最新结果，将弱Maass形式和mock theta函数与仿射Lie超代数的特征标公式联系起来，这是由于Kac和Wakimoto的结果。另一个目的是通过研究q-超几何级数的变体和更一般的族，建立将q-超几何级数与模形式和Maass形式联系起来的更统一的结果。目前，关于q-超几何级数所起作用的结果很大程度上是零碎的，直到最近我们才开始更准确地理解与仿射李超代数的表示理论有关的弱Maass形式理论。经典上，模形式扮演了许多基本角色；例如，它们是证明费马最后定理、朗兰兹程序、黎曼假设、Birch和Swinnerton-Dyer猜想的核心，并在弦论、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theta函数和Maass形式，是经典模形式的天然亲属，所提出的研究试图不仅有助于理解它们在数论和模形式中的作用，而且还有助于理解组合学和Lie理论中的作用。如上所述，模拟theta功能的突出与Ramanujan和Watson的原始上下文约束较少，这从它们现在已知在其中发挥重要作用的惊人数量的学科中得到了证明。此外，还缺乏一个全面的理论，这两方面都推动了进一步的研究。

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.