Geometry, Representation theory, and Langlands duality

这个数学研究项目是在表示论和几何的一般领域。 它涉及研究自然界中发生的基本对称性，即由Sophus Lie在19世纪80年代引入的称为李群的连续对称性。 该项目的主要目标之一是理解理论的基本组成部分，李群的不可约酉表示，使用几何方法。 此外，PI将利用几何方法来攻击几个长期存在的问题，涉及微分方程系统，模表示理论和李群对偶。更详细地说，PI和Wilfried Schmid已经做出了意义深远的解释，将寻找不可约酉表示的问题放在了一般的数学背景下。这些结构本身超越了它们在表示论中的应用，涉及到斋藤守彦的混合霍奇模理论。 朗兰兹纲领提供了一种将数学领域联系起来的方法，这些领域通常没有直接的关系。 它通过理论的对称性来实现这种关系，表现出它们的表示之间的关系。 本着这种精神，PI已经开始与Geordie威廉姆森合作，其目标是理解模块化表示理论。 特别是，他们想解决理解不可约特征标的长期问题。 也在这个方向上，PI建议，在联合工作与罗马Bezrukavnikov，以证明一个明确的朗兰兹对偶的真实的群体。 数学中的许多结构都可以用微分方程组来模拟。 特别令人感兴趣的是最大超定系统。 PI与Masaki Kashiwara一起解决了这一领域长期存在的关键问题，即余维3猜想。 Kashiwara和PI将继续致力于全面了解完整规则微微分系统（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还计划开发多复变量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.