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

一个多世纪以来，数学家们已经知道，从整数运算中产生的某些数量（例如将自然数表示为四个平方和的方法的数量）与从表示理论和谐波分析中产生的数量（傅里叶系数）之间存在微妙的关系，谐波分析是理解函数图如何被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θ对应来研究这些关系的几何，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。这些广泛的猜想在爱森斯坦级数和某些志村变分的内部几何之间建立了深刻的关系，志村变分后来成为现代算术几何的基本对象。首席研究员打算在全局设置中使用派生代数几何的方法，并在局部设置中使用最新发展的方法，以获得对支撑这些猜想真理的几何的概念理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.