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: $ 1.53万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=678534
• 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.

我的研究属于算术几何领域，处于两个数学子领域的交汇处：数论和几何。数论是对整数的研究，整数本质上是离散的和刚性的。另一方面，几何学处理的是连续的物体，可以以流体的方式拉伸、拉动和变形。算术几何将这两种观点结合起来，应用几何世界中的工具和直觉来更深入地了解数论现象，反之亦然。*** 我对 Shimura 品种特别感兴趣，它们是令人着迷的几何对象几十年来，数学家们一直在研究它们，部分原因是它们似乎携带着有关数论的深刻信息，并且是算术几何工具和技术的理想试验场。事实上，数论中最近取得的许多重大成功，包括费马大定理的惊人解决，都可以用这些术语来看待。****** 在某些情况下，存在一种嵌套现象，即一个 Shimura 簇包含许多子簇。 -志村品种称为特殊周期。近年来，有证据表明特殊循环具有非常微妙和神秘的对称性，这种对称性可以用称为模性的数学属性精确地表达，并且在某种意义上反映了经典 theta 函数的行为。研究了150多年。然而，尽管大量美丽的数学激发了人们对这一现象的深入猜想，但目前完整的概念性解释还相当遥不可及。****** 本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重大进展，部分是通过利用最近与 Stephan Ehlen 的合作，在这方面开发了某些概念工具。与此同时，我打算在三名研究生团队的协助下研究算术设置中存在有趣且相互关联的问题。这项工作将为上述推测提供令人信服的证据。总体而言，拟议研究的结果将推动该领域的最新技术发展，并为系统地理解这一迷人的思想圈指明道路。