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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：
2022303
负责人：
Tsao-Hsien Chen
金额：
$ 2.58万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2022303&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.

这项研究项目自然处于表象理论和几何的交叉点上。表示论是数学的一个分支，致力于使用线性代数中的技术来研究自然界中出现的对称性，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(称为李群理论)。几何方法在解决表象理论中的问题方面取得了很大的成功。这个项目的主要目标是用几何方法研究表象理论中的各种问题。PI将解决关于李群的对偶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。系数为模为素数的余项的微分方程和可积系统是研究项目的其他部分的主题。更详细地说，将进行三个项目。在第一个项目中，我们将建立对称空间的广义Springer对应。这个项目与代数几何、实群和p-ady群上的调和分析中的深层问题密切相关。在第二个项目中，将利用对称空间的奇妙紧化的几何来证明Soerel关于实群的Koszul对偶猜想。在第三个项目中，我们将发展高维变种的Hitchin纤颤理论，其目的是建立高维变种的非阿贝尔Hodge理论和建立高维变种的正特征几何Langland对应。