Mathematical Sciences: Topological Methods in RepresentationTheory and Automorphic Forms

本课题的主要目标是发展拓扑学的方法，并将其应用于群和自同构形式的表示理论。第一部分是从微观局部的角度研究倒行逆施捆的程序。这包括将反常束的范畴扩展到任意齐次辛流形，以及对该理论的混合和不规则版本的研究。第二部分尝试表征理论的应用，特别是表征的几何构造和表征特征的微观局部研究。第三部分从不同角度对几何朗兰兹猜想进行了研究。群是描述对称的自然代数结构。这是一个经典的方法来理解一个群实现（或表示）它作为一个向量空间的变换集合。表征理论已经高度发展，甚至成为量子力学研究中不可或缺的一部分，但它绝不是一个完整的理论和一本封闭的书。研究者正在运用一些最新的拓扑和几何方法来回答表征理论的突出问题。与代数和分析中的其他主题的联系是预期的。

The main goal of this project is to develop methods from topology and apply them to representation theory of groups and automorphic forms. The first part is a program to study perverse sheaves from the micro-local point of view. This includes an extension of the category of perverse sheaves to arbitrary homogeneous symplectic manifolds as well as the study of the mixed and irregular versions of the theory. The second part attempts applications to representation theory, in particular, geometric constructions of representations and micro-local study of characters of representations. The third part is work on the geometric Langlands conjecture from various points of view. Groups are the natural algebraic structures to describe symmetry. It is a classical method of understanding a group to realize (or represent) it as a collection of transformations of a vector space. The theory of representations has become highly evolved and even made itself indispensable in the study of quantum mechanics, but it is by no means a completed theory and a closed book. The investigator is bringing to bear some of the latest topological and geometric methods in answering outstanding questions of representation theory. Connections with other topics in algebra and analysis are to be expected.