CAREER: Hitchin morphisms, relative Langlands duality, and automorphic L-functions

职业生涯：希钦态射、相对朗兰兹对偶性和自守 L 函数

• 批准号: 2143722
• 负责人: Tsao-Hsien Chen
• 金额: $ 42.5万
• 依托单位: University of Minnesota-Twin Cities
• 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: 2022-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143722&HistoricalAwards=false
• 关键词: CAREER; Hitchin; morphisms; relative; Langlands

This project sits naturally at the intersection of representation theory and geometry. Representation theory is a branch of mathematics devoted to the study of symmetries that occur throughout mathematics and science. For example, the study of symmetries in three-dimensional space or more generally the study of continuous symmetries of mathematical objects and structures, known as representation theory of Lie groups, or the study of symmetries of solutions of polynomial equations, known as Galois theory. Methods from geometry have been very successful in solving problems in representation theory. One of the main goals of the project is to study questions in representation theory using geometric methods. In this project the PI will develop and use geometric tools to attack several longstanding problems on: Higgs bundles and representations of the fundamental group, representations of Lie groups and symmetric varieties, and the Langlands program. The education component of the project will create a vertically integrated learning environment for representation theory, number theory, and algebraic geometry at the University of Minnesota, from which students and researchers at all levels will benefit. The plans include supports for outreach activities, development of courses and learning seminars, and summer programs. In more detail, the PI will conduct research on the following projects: (1) Hitchin morphisms for higher dimensional varieties, (2) Real groups, symmetric varieties, and the relative Langlands duality, and (3) Fourier transforms, automorphic L-functions, and the Braverman-Kazhdan program. In project (1), the PI will develop the theory of Hitchin morphisms for higher dimensional varieties. This project is closely related to deep questions in invariant theory, algebraic geometry, and representations of the fundamental group. In project (2), the PI will explore connections between the geometry of real reductive groups and the algebraic geometry of symmetric varieties and apply them to the study of geometric Satake equivalence and the geometric Langlands correspondence for real groups and the relative Langlands duality conjectures. In project (3), the PI will systematically study the Braverman-Kazhdan program on meromorphic continuation and functional equations 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.

这个项目自然地位于表象理论和几何的交叉点上。表示论是数学的一个分支，致力于研究贯穿整个数学和科学的对称性。例如，研究三维空间中的对称性，或者更广泛地说，研究数学对象和结构的连续对称性，称为李群表示理论，或研究多项式方程的解的对称性，称为伽罗瓦理论。几何方法在解决表象理论中的问题方面已经取得了很大的成功。该项目的主要目标之一是使用几何方法研究表示理论中的问题。在这个项目中，PI将开发和使用几何工具来解决几个长期存在的问题：希格斯丛和基本群的表示，李群和对称簇的表示，以及朗兰兹程序。该项目的教育部分将为明尼苏达大学的表示论、数论和代数几何创造一个垂直整合的学习环境，各级学生和研究人员都将从中受益。这些计划包括支持外展活动，开发课程和学习研讨会，以及暑期项目。更详细地说，PI将进行以下项目的研究：(1)高维簇的希钦态射，(2)实群，对称簇，以及相对朗兰兹对偶，(3)傅立叶变换，自同构L函数，以及Braverman-Kazhdan程序。在方案(1)中，PI将发展高维簇的希钦态射理论。这个项目与不变理论、代数几何和基本群表示中的深层次问题密切相关。在项目(2)中，PI将探索实还原群的几何与对称簇的代数几何之间的联系，并将它们应用于研究实群的几何Satake等价和几何Langland对应以及相对的Langlands对偶猜想。在项目(3)中，PI将系统地研究Braverman-Kazhdan计划关于亚纯延拓和自同构L函数的函数方程。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。