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

这个项目自然地处于表征理论和几何的交叉点。表征理论是数学的一个分支，致力于研究贯穿数学和科学的对称性。例如，对三维空间对称性的研究，或者更一般地说，对数学对象和结构的连续对称性的研究，被称为李群的表示理论，或者对多项式方程解的对称性的研究，被称为伽罗瓦理论。几何方法在解决表征理论中的问题上已经非常成功。该项目的主要目标之一是利用几何方法研究表征理论中的问题。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：希格斯束和基本群的表示，李群和对称变体的表示，以及朗兰兹计划。该项目的教育部分将在明尼苏达大学为表示理论、数论和代数几何创造一个垂直整合的学习环境，所有层次的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、课程开发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)高维变量的Hitchin态射，(2)实群，对称变量和相对朗兰兹对偶，以及(3)傅里叶变换，自同构l -函数和Braverman-Kazhdan计划。在项目(1)中，PI将发展高维品种的希钦态射理论。这个项目与不变量理论、代数几何和基本群表示的深层问题密切相关。在项目(2)中，PI将探索实约群几何与对称变体代数几何之间的联系，并将其应用于实群的几何Satake等价和几何Langlands对应以及相关Langlands对偶猜想的研究。在项目(3)中，PI将系统地研究关于自同构l函数亚纯延拓和泛函方程的Braverman-Kazhdan规划。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。

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.