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.

朗兰兹纲领设想了整数基本问题之间的深层关系，比如模p方程的解的个数如何随着素数p的变化而变化，以及矩阵群的某些无限维表示的结构。这个项目将从几何的角度研究不同矩阵组的表示是如何相互关联的，称为朗兰兹函数性。在该项目的过程中，主要研究者将培训研究生和本科生在数学的前沿领域，包括代数几何，表示论和数论。阿瑟-塞尔伯格迹公式及其相关变体是我们处理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）迹公式的几何方面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看作是希钦纤维化的推广，希钦纤维化在Ngo博士的基本引理的证明中起了重要作用。他们的调查将需要一个新的理解不变的理论，这是一个经典的主题代数几何。Ngo博士希望这些不变量理论的结果不仅能阐明迹公式，还能解决相关问题，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，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。