Geometry, Representation theory, and the Langlands program

本项目利用和发展几何方法来解决几个长期存在的问题：微分方程组、李群的表示理论和朗兰兹程序。很长一段时间以来，人们都知道（特殊）函数，在广义上，可以根据它们所满足的微分方程组来研究。为此，京都的佐藤学派发展了线性微分方程组的一般理论。这种观点，以其不同的形式，现在在数学中无处不在。PI与Masaki Kashiwara共同解决了该领域长期存在的关键问题，即余维三猜想。柏原和PI将继续合作，以全面了解完整的规则微差分系统。这些系统最常出现在其他领域的应用中。朗兰兹程序提供了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。这个建议的一个主要领域是真实群体理论。实群是数论和物理学中出现的基本对称性。实群表示理论中一个关键的突出问题是确定一类特别重要的酉表示。威尔弗里德·施密德和PI做出了深远的猜想，把这个问题放在一般的数学背景下，用霍奇理论来表述这个问题。他们正在解决这些猜想和一元对偶的结构。PI还提出，在与Roman Bezrukavnikov的联合工作中，证明实群的绝对朗兰兹对偶性。微分方程被用来模拟自然界的各种现象。这个项目的一个主要方面旨在全面理解一类重要的微分方程系统。李群是自然界中存在的基本对称性。它们在物理学和数论中都很重要。一个关键的问题是理解这些对称在哪里发生，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，根据基本的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.