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.

Langlands计划设想了有关整数的基本问题之间的深厚关系，例如方程式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度表示的结构。该项目将从几何学角度研究不同矩阵的表示形式如何相互关联，称为兰兰兹的功能。在项目的过程中，主要研究者将在数学的最前沿领域培训研究生和本科生，包括代数几何，代表理论和数字理论。 Arthur-Selberg Trace Formula及其相对变体是我们应付解决Langlands功能性猜想的主要工具。在该项目中，主要研究员NGO BAO CHAU博士将研究某些模量空间的结构，这些模量空间自然出现在（相对）痕量公式的几何侧。这些新的模量空间可以看作是Hitchin纤维化的概括，这在NGO博士的基本引理方面具有重要作用。他们的调查将需要对不变理论的新理解，这是代数几何学中的经典主题。非政府组织博士在不变理论中期望这些结果不仅会阐明痕量公式，还将阐明相关问题，包括确定诺比尔傅里叶傅立叶转换的内核，负责自动l-功能的功能方程。这项奖项反映了NSF的法定任务，并通过使用基金会的范围进行评估，并通过评估了基金会的范围。