Invariant Theory, Moduli Space, and Automorphic Representations

不变理论、模空间和自同构表示

• 批准号: 2201314
• 负责人: Bao Chau Ngo
• 金额: $ 40万
• 依托单位: University of Chicago
• 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=2201314&HistoricalAwards=false
• 关键词: Invariant; Theory; Moduli; Space; Automorphic

The Langlands program envisions a deep relation between basic questions about integers, like how the number of solutions to equations modulo p varies as the prime number p varies, and the structure of certain infinite dimensional representations of groups of matrices. This project will investigate how representations of different groups of matrices are related to one another, known as Langlands's functoriality, from a geometric perspective. In the course of the project, the principal investigator will train graduate and undergraduate students in cutting edge areas of mathematics including algebraic geometry, representation theory, and number theory. The Arthur-Selberg trace formula and its relative variants are among the main tools at our disposal to address Langlands's functoriality conjecture. In this project, the principal investigator, Dr. Ngo Bao Chau, will investigate the structure of certain moduli spaces appearing naturally in the study of the geometric side of the (relative) trace formula. These new moduli spaces can be seen as a generalization of the Hitchin fibration, which was instrumental in the proof of the fundamental lemma by Dr. Ngo. Their investigation will require a new understanding of invariant theory which is a classical topic in algebraic geometry. Dr. Ngo expects these results in invariant theory will shed light not only on the trace formula but also on related problems, including the determination of the kernel of the nonabelian Fourier transform responsible for the functional equation of automorphic L-functions.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.

朗兰兹纲领设想了关于整数的基本问题之间的深层关系，例如模 p 的方程解的数量如何随着素数 p 的变化而变化，以及矩阵组的某些无限维表示的结构。该项目将从几何角度研究不同矩阵组的表示如何相互关联，即朗兰兹函子性。在该项目过程中，首席研究员将在数学前沿领域（包括代数几何、表示论和数论）对研究生和本科生进行培训。 Arthur-Selberg 迹公式及其相关变体是我们可以用来解决朗兰兹函数性猜想的主要工具之一。在这个项目中，首席研究员Ngo Bao Chau博士将研究在（相对）迹公式的几何边的研究中自然出现的某些模空间的结构。这些新的模空间可以被视为希钦纤维的推广，这对于 Ngo 博士证明基本引理起到了重要作用。他们的研究需要对不变理论有新的理解，这是代数几何中的一个经典主题。 Ngo 博士预计，不变理论中的这些结果不仅会阐明迹公式，还会阐明相关问题，包括确定负责自守 L 函数的函数方程的非阿贝尔傅里叶变换的核。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。