Springer Theory for Symmetric Spaces, Real Groups, Hitchin Fibrations, and Geometric Langlands

对称空间、实群、希钦纤维和几何朗兰兹的施普林格理论

基本信息

批准号：
1702337
负责人：
Tsao-Hsien Chen
金额：
$ 15.3万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702337&HistoricalAwards=false
关键词：
Springer Theory Symmetric Spaces Real

项目摘要

This research project naturally sits at the intersection of representation theory and geometry. Representation theory is a branch of mathematics devoted to the study of symmetries that occur in nature using techniques from linear algebra, 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 theory of Lie groups). Geometric methods have been very successful in solving problems in representation theory. The main goal of this project is to study various questions in representation theory using geometric methods. The PI will attack several longstanding problems concerning dualities for Lie groups. The PI will also investigate applications of representation theory to algebraic geometry, number theory, and related areas. Differential equations and integrable systems whose coefficients are residues modulo a prime are the subject of the other parts of the research project.In more detail, three projects will be pursued. In the first project, a generalized Springer correspondence will be developed for symmetric spaces. This project is closely related to deep questions in algebraic geometry, real groups, and harmonic analysis on p-adic groups. In the second project, the geometry of the so-called wonderful compactification of symmetric spaces will be used to prove Soergel's Koszul duality conjecture for real groups. In the third project, a theory of Hitchin fibrations for higher-dimensional varieties will be developed with the goals of constructing a non-abelian Hodge theory and establishing the geometric Langlands correspondence in positive characteristic for higher-dimensional varieties.

该研究项目自然位于表示理论与几何形状的交集。表示理论是使用线性代数的技术（例如，研究三维空间中的对称性）或更一般地研究数学对象和结构的连续对称对称对称性（称为数学对象和结构（称为称为称为称为）（称为称为）的对称性（称为称为），代表理论是专门研究自然界中对称对称性的数学分支。谎言群体的理论）。几何方法在解决表示理论中的问题方面非常成功。该项目的主要目标是使用几何方法研究表示理论中的各种问题。 PI将攻击有关谎言群体二重性的几个长期问题。 PI还将调查表示理论对代数几何，数理论及相关领域的应用。系数为残基模型的微分方程和可集成的系统是研究项目其他部分的主题。更详细地说，将追求三个项目。在第一个项目中，将为对称空间开发广义的Springer对应关系。该项目与P-ADIC组的代数几何形状，真实组和谐波分析中的深层问题密切相关。在第二个项目中，所谓的对称空间的奇妙紧凑型的几何形状将用于证明Soergel的Koszul双重性猜想。在第三个项目中，将开发针对高维品种的Hitchin纤维纤维理论，其目标是构建非亚伯式霍奇理论并确定具有高维品种的积极特征的几何兰兰兹对应关系。