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

• 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

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，从整数的算术中产生的某些量(例如将自然数表示为四个平方之和的方法的数量)与来自表示理论和调和分析的量(傅立叶系数)具有微妙的关系，调和分析是理解如何将函数的图形实现为更简单波形的叠加的过程。这个项目的首要目标是通过所谓的theta对应来研究这些关系的几何学，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了培训机会。这些广泛的猜想建立了Eisenstein级数和某些Shimura簇的内部几何之间的深层次联系，这些簇已被证明是现代算术几何中的基本对象。首席研究人员打算使用全球背景下的派生代数几何和当地背景下的最新发展来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，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。