Theta functions in differential and arithmetic geometry

我的研究领域是算术几何，处于数论和几何这两个数学子领域的交界处。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，几何学处理的对象是连续的，可以以流动的方式拉伸、拉动和变形。算术几何将这两种观点结合在一起，应用几何界的工具和直觉来更好地洞察数论现象，反之亦然。*我对下村变种特别感兴趣，它们是几十年来吸引数学家的几何对象，部分原因是它们似乎携带着关于数论的深刻信息，是算术几何工具和技术的理想试验场。事实上，最近在数论中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簇包含许多称为特殊圈的子Shimura簇。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，这种对称性可以精确地用一种称为模数的数学性质来表示，在某种意义上，它反映了研究了150多年的经典theta函数的行为。然而，尽管围绕这一现象有丰富的美丽数学引发了深刻的猜测，但目前还很难有一个完整的概念性解释。*本提案中描述的研究旨在弥合这一差距。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的联合工作，该工作在这一背景下开发了某些概念工具。与此同时，我打算在一个由三名研究生组成的团队的帮助下，研究算术环境中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进这一领域的最新水平，并为系统地理解这一迷人的思想圈指明了方向。

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.