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

拟议的研究旨在理解位于数论，组合学和李论界面的问题。具体来说，PI试图确定弱质量形式、模拟函数、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。这类问题的起源可以追溯到杰出的数学人物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θ函数”，后来意识到它们的重要性，并宣布对它们的理解和表征为“最终问题”。近90年后的今天，这个问题仍然存在，直到最近8年才有了重大进展和更统一的弱质量形式理论（由于Ono、Bringmann、Zwegers、Zagier等人的工作）。积极的结果包括：(1)对模拟θ函数的更普遍的理解及其在更大的群论框架中的位置，在这个框架中，它们与弱质量形式的关系可以被理解；(2)实现模拟θ函数和弱质量形式的作用，不仅在数论中，而且在数学和科学的其他领域。尽管最近有了这些进展，但仍然缺乏一个完整的弱质量形式理论。PI将沿着这些路线着手解决的一个问题包括进一步推进PI和Bringmann-Ono最近的结果，将弱质量形式和模拟函数与由Kac和Wakimoto提出的仿射李超代数的特征公式联系起来。另一个目标是通过研究q-超几何级数的变异体和更一般的族，建立q-超几何级数与模形式和质量形式之间更统一的结果。目前，关于q-超几何级数所扮演的角色，存在着大量零碎的结果，例如，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相关的弱质量形式理论。所提出的研究领域，数论，是数学中最古老的分支之一，并且在今天仍然是一个广泛而活跃的研究领域。经典地，模形式扮演了许多基本的角色；它们是证明费马大定理、朗兰兹程序、黎曼假设、伯奇猜想和斯温纳顿-戴尔猜想的核心，并在弦理论、组合学、密码学、数学物理以及许多其他领域得到应用。PI的中心研究对象，模拟theta函数和质量形式，是经典模形式的自然亲戚，提出的研究旨在有助于理解它们不仅在数论和模形式，而且在组合学和李论中的作用。如上所述，模拟θ函数的重要性与拉马努金和沃森的原始背景联系较少，这一点可以从它们现在已知发挥重要作用的惊人数量的学科中得到证明。此外，缺乏一个全面的理论，这两者都激励着进一步的研究。

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.