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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.

这个研究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，致力于研究自然界中使用线性代数技术的对称性，例如研究三维空间中的对称性，或者更一般地研究数学对象和结构的连续对称性（称为李群理论）。几何方法在解决表征理论中的问题方面非常成功。本课题的主要目的是利用几何方法研究表征理论中的各种问题。PI将解决关于李群对偶性的几个长期存在的问题。PI还将研究表示理论在代数几何、数论和相关领域的应用。微分方程和系数为残模a素数的可积系统是研究项目其他部分的主题。更详细地说，将进行三个项目。在第一个项目中，将开发对称空间的广义施普林格对应。本课题与代数几何、实群、p进群的调和分析等深层问题密切相关。在第二个项目中，所谓的对称空间奇妙紧化的几何将被用来证明实数群的Soergel的Koszul对偶猜想。在第三个项目中，将发展高维品种的Hitchin振动理论，目标是构建非阿贝尔Hodge理论，并建立高维品种正特征的几何Langlands对应。