Higgs Bundles, Real Quasi-Maps, and Automorphic L-Functions

这个数学研究项目自然位于表示论和几何的交叉点。表示论是数学的一个分支，研究数学对象和结构的对称性。几何方法在回答表示论问题上非常成功。该项目的主要目标是使用几何方法研究表征理论中的开放问题。具体来说，研究人员计划使用和开发几何工具来解决李群表示，希格斯束和基本群表示以及朗兰兹计划研究中的几个长期存在的问题。更详细地说，研究集中在三个项目：（1）Hitchin态射的高维品种，（2）真实的准映射和应用，（3）非线性傅立叶变换和Braverman-Kazhdan程序。在项目（1）中，研究者将发展高维簇的Hitchin态射理论，并将其应用于Simpson对应的研究。这个项目与不变量理论、高维代数几何和基本群的表示等深层次问题密切相关。在项目（2）中，研究者将探索真实的拟映射的几何结构，并建立应用于真实的群的表示理论、Kostant-Sekiguchi对应和对称空间的Springer理论的研究。在项目（3）中，研究者将研究亚纯延拓和自守L-函数的函数方程的Braverman-Kazhdan方法。该项目基于有限域和D模环境下的非线性傅立叶变换和Braverman-Kazhdan猜想的结果，旨在构建局部域环境下的非线性傅立叶核。该奖项反映了NSF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 mathematics research project naturally sits at the intersection of representation theory and geometry. Representation theory is a branch of mathematics studying symmetries of mathematical objects and structures. Methods from geometry have been very successful in answering questions in representation theory. The main goal of the project is to study open questions in representation theory using geometric methods. Specifically, the investigator plans to use and develop geometric tools to attack several longstanding problems in the study of representations of Lie groups, Higgs bundles and representations of fundamental groups, and the Langlands program. In more detail, the research centers on three projects: (1) Hitchin morphisms for higher dimensional varieties, (2) real quasi-maps and applications, and (3) nonlinear Fourier transforms and the Braverman-Kazhdan program. In project (1), the investigator will develop the theory of Hitchin morphisms for higher dimensional varieties and apply it to the study of the Simpson correspondence. This project is closely related to deep questions in invariant theory, higher-dimensional algebraic geometry, and representations of fundamental groups. In project (2), the investigator will explore the geometric structure of real quasi-maps and establish applications to the study of representation theory of real groups, the Kostant-Sekiguchi correspondence, and Springer theory for symmetric spaces. In project (3), the investigator will study the Braverman-Kazhdan approach to meromorphic continuation and functional equations of automorphic L-functions. Based on results on nonlinear Fourier transforms and the Braverman-Kazhdan conjecture in the finite field and D-modules setting, the project aims to construct nonlinear Fourier kernels in the local fields setting.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.