Theta functions in differential and arithmetic geometry

微分几何和算术几何中的 Theta 函数

• 批准号: RGPIN-2017-04959
• 负责人: Sankaran, Siddarth
• 金额: $ 3.06万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758574
• 关键词: Theta; functions; differential; arithmetic; geometry

My research lies in the field of arithmetic geometry, at the interface of two mathematical subfields: number theory and geometry. Number theory is the study of integers, which are essentially discrete and rigid in nature. On the other hand, geometry deals with objects that are continuous, that can be stretched and pulled and deformed in a fluid manner. Arithmetic geometry marries these two points of view, applying tools and intuitions from the world of geometry to gain greater insight into number theoretic phenomena, and vice versa. I'm particularly interested in Shimura varieties, which are geometric objects that have fascinated mathematicians for decades, in part because they seem to carry deep information about number theory, and are an ideal proving ground for the tools and techniques of arithmetic geometry. Indeed, many of the major recent successes in number theory, including the spectacular resolution of Fermat's last theorem, can be viewed in these terms. In some cases, there is a nesting phenomenon whereby one Shimura variety contains many sub-Shimura varieties called special cycles. In recent years, evidence has emerged that special cycles possess very subtle and mysterious symmetries, which can be expressed precisely in terms of a mathematical property known as modularity, and which mirror, in a sense, the behaviour of the classical theta functions that have been studied for well over 150 years. However, despite a wealth of beautiful mathematics inspiring deep conjectures around this phenomenon, at present a complete conceptual account is quite out of reach. The research described in this proposal is aimed towards closing this gap. In particular, I hope to make significant strides on the geometric aspects of modularity questions, in part by leveraging recent joint work with Stephan Ehlen that develops certain conceptual tools in this context. At the same time, there are interesting, and interrelated, problems in the arithmetic setting that I intend to study, assisted by a team of three graduate students. This work would provide compelling evidence for the conjectural picture described above. As a whole, the outcome of the proposed research will advance the state of the art in this area, and point the way towards a systematic understanding of this fascinating circle of ideas.

我的研究在于两个数学子场的界面：数字理论和几何形状的界面。数字理论是整数的研究，本质上是离散和僵化的。另一方面，几何形状涉及连续的物体，这些物体可以以流畅的方式拉伸，拉动和变形。算术几何形状结合了这两种观点，应用了几何世界的工具和直觉，以更深入地了解数量的理论现象，反之亦然。我对Shimura品种特别感兴趣，Shimura品种是数学家着迷数十年的几何对象，部分原因是它们似乎携带了有关数字理论的深刻信息，并且是算术几何工具和技术的理想证据。的确，可以用这些术语来查看数量理论的许多主要成功，包括Fermat的最后一个定理的壮观解决。在某些情况下，有一个嵌套现象，其中一个shimura品种包含许多称为特殊周期的亚simura品种。近年来，有证据表明，特殊周期具有非常微妙和神秘的对称性，可以准确地以一种称为模块化的数学特性来表达，并且从某种意义上讲，哪种镜像是经典的theta函数的行为研究了150多年。然而，尽管大量美丽的数学启发了这一现象的深刻猜想，但目前，一个完整的概念叙述是遥不可及的。该提案中描述的研究旨在缩小这一差距。特别是，我希望在模块化问题的几何方面取得了长足的进步，部分原因是利用与Stephan Ehlen的最新联合合作，在这种情况下开发了某些概念工具。同时，在我打算学习的算术环境中，有有趣且相互关联的问题，并由三个研究生团队的协助。这项工作将为上述猜想提供令人信服的证据。总体而言，拟议的研究的结果将推动该领域的最新技术，并为对这个迷人的思想圈子的系统理解指向。