Geometrization of the Local and Global Theta Correspondence with Applications to the Kudla Program

局部和全局 Theta 对应的几何化及其在 Kudla 程序中的应用

基本信息

批准号：
2200804
负责人：
Keerthi Madapusi
金额：
$ 28.8万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200804&HistoricalAwards=false
关键词：
Geometrization Local Global Theta Correspondence

项目摘要

For well over a century now, mathematicians have known that certain quantities arising from the arithmetic of the integers (such as the number of ways of representing a natural number as a sum of four squares) have subtle relationships with quantities (Fourier coefficients) arising from representation theory and harmonic analysis, which is the process of understanding how graphs of functions can be realized as superpositions of simpler waveforms. The overarching goal of this project is to study the geometry of these relationships, via the so-called theta correspondence, thereby making progress towards resolving longstanding conjectures in arithmetic geometry. The project provides training opportunities for graduate students. These wide-ranging conjectures posit a deep relationship between Eisenstein series and the internal geometry of certain Shimura varieties, which have turned out to be fundamental objects in modern arithmetic geometry. The principal investigator intends to use methods from derived algebraic geometry in the global setting and recent developments in the local setting to gain a conceptual understanding of the geometry undergirding the truth of these conjectures.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.

一个多世纪以来，数学家们已经知道，由整数算术产生的某些量（例如将自然数表示为四个平方和的方式的数量）与由以下因素产生的量（傅里叶系数）有着微妙的关系：表示论和谐波分析，这是理解如何将函数图实现为更简单波形的叠加的过程。该项目的总体目标是通过所谓的 theta 对应关系来研究这些关系的几何形状，从而在解决算术几何中长期存在的猜想方面取得进展。该项目为研究生提供培训机会。这些广泛的猜想提出了爱森斯坦级数与某些志村簇的内部几何之间的深刻关系，这些簇已被证明是现代算术几何的基本对象。首席研究员打算在全球环境中使用派生代数几何的方法和当地环境中的最新发展，以获得对支撑这些猜想真相的几何的概念性理解。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。