Geometry, Representation theory, and Langlands duality

这项数学研究项目属于表示论和几何的一般领域。它涉及到对自然界中出现的基本对称性的研究，即由索菲斯·李在1880年的S中引入的被称为李群的连续对称。该项目的主要目标之一是用几何方法来理解该理论的基本构件，即李群的不可约酉表示。此外，PI将利用几何方法来解决几个长期存在的问题，涉及微分方程组、模表示理论和李群的对偶。更详细地说，圆周率派和威尔弗里德·施密德提出了影响深远的猜想，把寻找不可约么正表示的问题放在一般的数学背景下。猜想本身超越了它们对表示论的应用，并涉及到斋藤森彦的混合Hodge模理论。朗兰兹计划提供了一种将数学领域联系起来的方法，这些领域往往没有直接的直接关系。它通过理论的对称性来实现这种关系，通过展示它们的表示之间的关系。本着这种精神，PI发起了与Geordie Williamson的合作，后者的目标是理解模表示理论。特别是，他们想要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建议，在与罗曼·别兹鲁卡夫尼科夫的联合工作中，证明实群的绝对朗兰兹对偶。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限度地超定的系统。PI与柏原正明合作，解决了这一领域长期存在的关键问题--协维3猜想。Kashiwara和PI将继续致力于全面理解完整正则微微分系统(这些系统在其他领域的应用中最经常出现)。为了更好地理解这些问题，PI还计划开发几个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.