Topological Methods in Representation Theory and Automorphic Forms

9803862 Vilonen该研究计划由三个独立但密切相关的项目组成。第一个问题是与Wilfred Schmidd联合提出的，它涉及到约化李群表示理论中的一个重要遗留问题：确定它们的所有酉表示。他们将使用他们为证明Barbasch-Vogan猜想而开发的几何技术来解决这个问题。第二个项目与E.Frenkel、D.Gaitsgory和D.Kazhdan合作，处理朗兰兹猜想。他们将继续研究几何朗兰兹猜想和惠特克函数的几何理论。与Mirkovic合作的第三个项目的目的是开发代数群表示理论的几何技术。事实证明，这个项目对几何朗兰兹猜想的研究也很重要。目标是将它们的方法应用于正特征代数群的表示理论以及整数上的代数群表示理论。数学最有趣的方面之一是发现和理解物体之间的联系，而这些联系一开始似乎是无关的。朗兰兹提出了一种非常深入的这种联系，并被称为朗兰兹通信。这个研究项目的一个基本主题是通过使用几何学来建立这种联系。当对象本身不是几何对象，但它们之间的联系是通过几何方法建立的时，这种情况最有趣。与威尔弗里德·施密德就Barbasch-Vogan猜想进行的合作就有这种味道。这项研究计划的另外两个项目与朗兰兹通信直接相关。***

9803862 Vilonen This research program consists of three separate, but intimately related, projects. The first one, joint with Wilfried Schmid, concerns one of the important remaining questions in representation theory of reductive Lie groups: determining all of their unitary representations. They will attack this problem using the geometric techniques they developed to prove the Barbasch-Vogan conjecture. The second project, joint with E.Frenkel, D.Gaitsgory, and D.Kazhdan, deals with the Langlands conjectures. They will continue their work on the geometric Langlands conjecture and the geometric theory of Whittaker functions. The purpose of the third project, joint with Mirkovic, is to develop geometric techniques for representation theory of algebraic groups. This project has also turned out to be important for work on the geometric Langlands conjecture. The goal is to apply their methods to representation theory of algebraic groups in positive characteristic as well as over the integers. One of the most interesting aspects of mathematics is to discover and understand connections between objects that from the outset seem unrelated. A very deep such connection has been proposed by Langlands and has become known as the Langlands correspondence. A fundamental theme of this research program is to establish such connections by using geometry. The situation is most interesting when the objects themselves are not geometric but the connection between them is established by geometric means. The joint work with Wilfried Schmid on the Barbasch-Vogan conjecture has this flavour. The other two projects of this research program are directly related to the Langlands correspondence. ***