RUI: Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms: Theory and Applications

这是数论领域的RUI奖，数论是纯数学中最古老的分支之一，今天仍在积极而广泛地研究。pi将主要研究与模形式自然相关的函数，称为模拟模形式和量子模形式，以及谐波质量形式。模形式是数论领域中最基本的数学对象之一。例如，它们是黎曼假设（1859年）的核心，这个猜想如此重要，以至于克莱数学研究所（Clay mathematical Institute）悬赏100万美元给任何能解决这个猜想的人。模形式理论的重大发展对1995年费马大定理（1637）的证明也至关重要，费马大定理是一个击败数学家358年的猜想。除了数论之外，模形式还可以应用于组合学、数学物理、密码学等不同领域。在过去的10-15年里，谐波质量形式和模拟模块形式的新相关理论已经有了很大的发展，并为许多新的应用和显著的结果打开了大门；然而，目前还缺乏一个全面的理论。此外，在过去几年中，量子模形式这一相对较新的主题已经显示出与其他领域的有趣联系，包括拓扑学、数学物理、表示理论、组合学等。该项目将研究谐波质量形式、模拟模形式、量子模形式和相关函数的理论和应用。一个首要的问题是理解谐波质量形式的全纯部分和相关的自同构对象在数学和数论中所起的各种作用。方法包括调和质量形式理论、量子模形式发展理论、模拟模和模拟Jacobi形式的变换和其他性质、zeta函数、Mordell和Eichler积分的解析性质、theta函数和组合q-超几何级数的工具。特别是量子模形式这一相对较新的主题已经显示出与其他领域的丰富联系，包括谐波质量形式、Eichler积分、径向极限和模拟模形式、组合学、偏函数、Jacobi形式、拓扑和顶点代数，其中许多都包括在本提案的范围内。这个RUI提案中的一些项目是专门为本科生研究设计的。本科研究指导和培训，以及（说明文）写作广泛的数学读者，也是这个建议的组成部分。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This is an RUI award in the area of number theory, one of the oldest branches of pure mathematics, which continues to be actively and extensively researched today. The P.I. will primarily study functions which are natural relatives to modular forms called mock modular and quantum modular forms, and harmonic Maass forms. Modular forms are among the most fundamental objects in the area of number theory in mathematics. For example, they are central to the Riemann Hypothesis (1859), so significant a conjecture that the Clay Mathematics Institute is offering one million dollars to anyone who can solve it. Major developments in the theory of modular forms were also crucial to the 1995 proof of Fermat's Last Theorem (1637), a conjecture which defeated mathematicians for 358 years. Beyond number theory, modular forms also yield applications to the diverse areas of combinatorics, mathematical physics, cryptography, and more. The newer related theories of harmonic Maass forms and mock modular forms have evolved substantially over the last 10-15 years, and have opened the door to many new applications and significant results; a comprehensive theory is still lacking, however. Moreover, within the last few years, the relatively new subject of quantum modular forms has shown intriguing connections to other areas including topology, mathematical physics, representation theory, combinatorics, and more. The P.I. will study the theory and applications of harmonic Maass forms, mock modular forms, quantum modular forms and related functions. An overarching question is to understand the various roles played by the holomorphic parts of harmonic Maass forms and related automorphic objects in mathematics and number theory. Methods include tools from the theory of harmonic Maass forms, aspects of the developing theory of quantum modular forms, transformation and other properties of mock modular and mock Jacobi forms, zeta functions, analytic properties of Mordell and Eichler integrals, theta functions, and combinatorial q-hypergeometric series. The relatively new subject of quantum modular forms in particular has shown rich connections to other areas including harmonic Maass forms, Eichler integrals, radial limits and mock modular forms, combinatorics, partial theta functions, Jacobi forms, topology, and vertex algebras, many of which are included within the scope of this proposal. Some projects within this RUI proposal are specifically designed for undergraduate research. Undergraduate research mentoring and training, and (expository) writing for broad mathematical audiences, are also components of this proposal.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.