Harmonic Maass Forms, "Moonshine," and Arithmetic Statistics

爱因斯坦的广义相对论是在一百多年前提出的。这一理论现在支撑着全球定位系统，这是现代生活中几乎无处不在的特征。几乎就在同一时间，数学天才拉马努扬来到英国，与剑桥数学家G·H·哈代一起工作。他当时分享的观察和见解继续吸引着数学界，他的个人故事吸引了公众的想象力。近年来，有证据表明，爱因斯坦的工作和拉马努扬的工作是相关的。这项研究项目旨在发展这种联系背后的理论，并为未来在数学、信号处理和物理方面的应用奠定重要的基础。作为这个奖项的一部分，PI将继续培训研究生数学。这个项目致力于算术统计和调和数学形式理论的发展。这些工具将应用于数论、表示论和数学物理中的各种开放问题。调和Maass形式推广了模形式，现在被认为为Ramanujan的模拟theta函数以及目前正在发展的本影月光理论提供了一个理论框架。在PI早期在这些领域的工作的基础上，这个项目的目标是开发调和Maass形式的代数、解析和组合工具，这些工具将阐明统计问题，如椭圆曲线的阶数、数域类别组中的扭转、模形式的傅立叶系数的分布、怪物和本影月光之间的关系，以及这种关系对物理学的影响。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Einstein's general relativity was formulated just over a hundred years ago. This theory now underpins the Global Positioning System, an almost ubiquitous feature of modern life. Almost concurrently, the mathematical genius Ramanujan arrived in England to work with Cambridge mathematician G. H. Hardy. The observations and insights he shared then continue to fascinate the mathematical world, and his personal story has captured the public imagination. In recent years evidence has emerged that the work of Einstein and the work of Ramanujan are related. This research project aims to develop the theory underlying this connection and lay important groundwork for future applications to mathematics, signal processing, and physics. The PI will continue to train graduate students in mathematics as part of this award.This project is dedicated to arithmetic statistics and the development of the theory of harmonic Maass forms. These tools will be applied to various open problems in number theory, representation theory, and mathematical physics. Harmonic Maass forms generalize modular forms, and are now recognized as furnishing a theoretical framework for Ramanujan's mock theta functions, as well as the currently-developing umbral moonshine theory. Building on the PI’s earlier work in these areas, the goal of this project is to develop algebraic, analytic, and combinatorial tools for harmonic Maass forms that will shed light upon statistical problems such as ranks of elliptic curves, torsion in class groups of number fields, distribution of Fourier coefficients of modular forms, the relationship between monstrous and umbral moonshine, and the consequences of this relationship for physics.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.