Geometry, Representation theory, and the Langlands program

该项目利用和发展几何方法来解决几个长期存在的问题：微分方程系统，李群表示论和朗兰兹程序。长期以来，人们一直认为（特殊）函数在广义上可以用它们所满足的微分方程组来研究。为此，一般理论系统的线性微分方程是由佐藤学校在京都。这种观点，以其各种形式，现在在数学中无处不在。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。柏原和PI将继续他们的合作，以全面了解完整的定期微微分系统。这些系统在其他领域的应用中最常出现。朗兰兹纲领提供了联系数学领域的方法，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。这一建议的一个主要领域是理论的真实的团体。真实的群是数论和物理学中的基本对称。真实的群的表示论中一个关键的突出问题是确定特别重要的酉表示类。Wilfried Schmid和PI已经做出了意义深远的解释，将这个问题置于一般的数学背景下，用Hodge理论来表述这个问题。他们正在解决这些问题和单一对偶的结构。PI还建议，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。微分方程用于模拟自然界中的各种现象。这个项目的一个主要方面旨在全面了解一类重要的微分方程系统。李群是自然界中最基本的对称性。它们在物理学和数论中都很重要。一个关键的问题是要理解这些对称性在哪里发生，以及它们以什么形式发生。这个项目的第二个主要方面将回答这个问题的基本几何结构称为霍奇结构。

This project makes use of and develops geometric methods to attack several longstanding problems on: systems of differential equations, representation theory of Lie groups, and the Langlands program. It has been understood for a long time that (special) functions, in generalized sense, can be studied in terms of the systems of differential equations that they satisfy. To this end, a general theory of systems of linear differential equations was developed by the Sato school in Kyoto. This point of view, in its various incarnations, is now ubiquitous in mathematics. 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 their collaboration towards a comprehensive understanding of holonomic regular microdifferential systems. These are the systems that most often come up in applications to other areas. The Langlands program provides 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. A major area of this proposal is the theory of real groups. Real groups are the fundamental symmetries that occur both in number theory and physics. A key outstanding question in representation theory of real groups is the determination of the particularly important class of unitary representations . Wilfried Schmid and the PI have made far-reaching conjectures which put this question in a general mathematical context phrasing the problem in terms of Hodge theory. They are on their way to settling these conjectures and the structures of the unitary dual. The PI also proposes, in joint work with Roman Bezrukavnikov, to prove a categorical Langlands duality for real groups. Differential equations are used to model various phenomena in nature. One major aspect of this project aims at a comprehensive understanding of an important class of systems of differential equations. Lie groups are the fundamental symmetries that occur in nature. They are important both in physics and number theory. A key question is to understand where these symmetries occur and it what form they occur. The second major aspect of this project will answer this question in terms of basic geometric structures called Hodge structures.