Geometry, Representation theory, and Langlands duality

这个数学研究项目是在表示理论和几何的一般领域。它涉及到对自然界中基本对称性的研究，即被称为李群的连续对称性，是由苏菲斯·李在19世纪80年代引入的。该项目的主要目标之一是利用几何方法理解理论的基本组成部分，即李群的不可约酉表示。此外，PI将利用几何方法来解决有关微分方程组、模表示理论和李群对偶性的几个长期存在的问题。更详细地说，PI和威尔弗里德·施密德做出了影响深远的猜想，把寻找不可约的酉表示的问题放在了一般的数学背景下。这些猜想本身超越了它们在表征理论中的应用，并且涉及到Morihiko Saito的混合Hodge模块理论。朗兰兹纲领提供了一种联系数学领域的方法，这些领域通常没有直接的直接关系。它通过展示理论表示之间的关系，通过理论的对称性来实现这种关系。本着这种精神，PI已经开始与Geordie Williamson合作，其目标是理解模块化表示理论。特别是，他们希望解决长期存在的理解不可约字符的问题。同样在这个方向上，PI提议与Roman Bezrukavnikov合作，证明实群的绝对朗兰兹对偶性。数学中的许多结构都可以用微分方程组来建模。特别令人感兴趣的是最大过度确定系统。PI与Masaki Kashiwara合作，解决了这个领域长期存在的关键问题——三维猜想。Kashiwara和PI将继续致力于全面理解完整规则微差分系统（这些系统最常出现在其他领域的应用中）。为了更好地理解这些问题，PI还计划开发一个关于几个复杂变量的经典理论的相对版本。

This mathematics research project is in the general area of representation theory and geometry. It involves the study of basic symmetries that occur in nature, namely the continuous symmetries known as Lie groups, introduced by Sophus Lie in the 1880's. One of the main goals of the project is to understand the basic building blocks of the theory, the irreducible unitary representations of Lie groups, using geometric methods. In addition the PI will make use of geometric methods to attack several longstanding problems concerning systems of differential equations, modular representation theory, and dualities for Lie groups. In more detail, the PI and Wilfried Schmid have made far-reaching conjectures which put the problem of finding the irreducible unitary representations in a general mathematical context. The conjectures themselves go beyond their application to representation theory and involve the theory of mixed Hodge modules of Morihiko Saito. The Langlands program provides a means of relating areas of mathematics that often do not have a straightforward direct relationship. It implements this relationship via the symmetries of the theories by exhibiting a relationship between their representations. In this spirit the PI has initiated a collaboration with Geordie Williamson, whose goal is to understand modular representation theory. In particular, they want to settle the longstanding problem of understanding the irreducible characters. Also in this direction, the PI proposes, in joint work with Roman Bezrukavnikov, to prove a categorical Langlands duality for real groups. Many structures in mathematics can be modeled by systems of differential equations. Of particular interest are the maximally over-determined systems. The PI, jointly with Masaki Kashiwara, has solved the key longstanding problem in this area, the codimension-three conjecture. Kashiwara and the PI will continue working towards a comprehensive understanding of holonomic regular microdifferential systems (these are the systems that most often come up in applications to other areas). To understand these issues better, the PI also plans to develop a relative version of the classical theory of several complex variables.