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

这个研究项目自然处于表示论和几何的交叉点。表示论（英语：Representation theory）是数学的一个分支，致力于使用线性代数的技术来研究自然界中发生的对称性，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）。几何方法在解决表示论中的问题时非常成功。这个项目的主要目标是使用几何方法研究表示论中的各种问题。PI将解决几个长期存在的关于李群对偶的问题。PI还将研究表示论在代数几何，数论和相关领域的应用。本研究计划的其他部分，则以微分方程式及系数为模素数余数的可积系统为研究课题。在第一个项目中，一个广义的施普林格对应将开发对称空间。这个项目与代数几何、真实的群和p-adic群的调和分析中的深层问题密切相关。在第二个项目中，对称空间的所谓奇妙紧化的几何学将用于证明真实的群的Soergel的Koszul对偶猜想。在第三个项目中，将发展一个高维簇的Hitchin纤维化理论，目标是建立一个非交换的Hodge理论，并建立高维簇的正特征几何Langlands对应。

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.