Mathematical Sciences: Topological Methods in Representation Theory and Automorphic Forms

小行星9504299 调查员将在以下主题上开展工作， 范围使用拓扑方法：李群的表示理论， 代数群的表示理论，自守层和 几何朗兰兹猜想，以及反常层和微局部 反常的束。 我们的目标是进一步发展几何 他和施密德在解决Barbasch-Vogan问题时使用的技术 另一方面，他的技术 联合工作与Mirkovic模块和积分表示 代数群 数学问题的一个典型类型是发现和 理解从一开始看起来不相关的事物之间的联系。 这种联系中最深刻、最神秘的一种是 由朗兰兹提出，现在通常称为朗兰兹 通信。 调查员工作的一个主要特点是 发展几何技术来发现和建立这种联系。 情况确实是最有趣的时候两件事要 相关的不是几何的，而是建立了它们之间的联系 通过几何学 这是调查员与下列人员联合工作的性质： Schmid证明了Barbasch-Vogan猜想。 他的其他作品 涉及上面提到的朗兰兹对应。 在他的工作中， 米尔科维奇，他设法在理解一个方面上走得很远， 通过发展新的几何技巧来实现朗兰兹对偶。 ***

9504299 Vilonen The investigator will work on the following subjects, to a large extent using topological methods: representation theory of Lie groups, representation theory of algebraic groups, automorphic sheaves and the geometric Langlands conjecture, and perverse sheaves and micro-local perverse sheaves. The goal is to develop further the geometric techniques that he used with Schmid in solving the Barbasch-Vogan conjecture on the one hand and on the other hand the techniques in his joint work with Mirkovic on modular and integral representations of algebraic groups. One typical type of a problem in mathematics is to discover and understand connections beteen things that from the outset seem unrelated. One of the deepest and most mysterious types of such connections has been suggested by Langlands and is now generally called a Langlands correspondence. A main feature of the investigator's work is to develop geometric techniques to discover and establish such connections. The situation is indeed the most interesting when the two things to be related are not geometric but the connection between them is established by geometry. This is the nature of the investigator's joint work with Schmid that proved the Barbasch-Vogan conjecture. His other work involves the Langlands correspondence alluded to above. In his work with Mirkovic he managed to get quite far in understanding one aspect of Langlands duality by developing new geometric techniques. ***