Maass Forms in Algebra, Arithmetic Geometry, Combinatorics, Representation Theory, and String Theory

爱因斯坦的广义相对论是在一百多年前，也就是1915年提出的。尽管当时没有实际应用，但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。就在一年前，也就是1914年，印度数学天才、自学成才的拉马努扬来到英国，与剑桥数学家G·H·哈代合作。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，并为未来的应用奠定重要的基础。该项目致力于发展调和Maass形式理论，并将该理论应用于数论、表示论和数学物理等主题。调和Maass形式推广了模形式，现在被认为为Ramanujan的模拟theta函数提供了一个理论框架。弗里曼·戴森早在1987年就预见到弦理论中的模拟theta函数的作用，而目前正在发展中的本影月光理论正在产生强有力的证据来支持这一观点。在这些领域早期工作的基础上，研究人员的目标是开发用于调和Maass形式的代数、解析和组合工具，这些工具将揭示诸如椭圆曲线的阶数、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等问题。

Einstein's general relativity was formulated just over a hundred years ago, in 1915. Although there was no practical application available then, the theory now underpins the Global Positioning System, an almost ubiquitous feature of modern life. Just a year earlier, in 1914, the Indian mathematical genius and autodidact Ramanujan had 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. The project is dedicated to the development of the theory of harmonic Maass forms and the application of this theory to topics 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. Freeman Dyson anticipated a role for the mock theta functions in string theory in 1987, and the currently-developing umbral moonshine theory is producing strong evidence in support of this vision. Building on their earlier work in these areas, the investigators aim to develop algebraic, analytic, and combinatorial tools for harmonic Maass forms that will shed light upon problems such as ranks of elliptic curves, the relationship between monstrous and umbral moonshine, and the consequences of this relationship for physics.