Mathematical Sciences: Topological Methods in Representation Theory and Automorphic Forms

研究者将在以下主题上工作，在很大程度上使用拓扑方法：李群的表示理论，代数群的表示理论，自同构束和几何朗兰兹猜想，反常束和微局部反常束。他的目标是进一步发展他和Schmid在解决Barbasch-Vogan猜想时使用的几何技术另一方面是他和Mirkovic在代数群的模和积分表示方面的合作技术。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。朗兰兹提出了这种联系中最深奥、最神秘的一种，现在一般称之为朗兰兹对应。研究者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。当两个相关的事物不是几何的，但它们之间的联系是由几何建立起来的，这种情况确实是最有趣的。这就是研究者与施密德共同工作的性质，证明了巴巴什-沃根猜想。他的其他工作包括上面提到的朗兰兹通信。在他与米尔科维奇的合作中，他通过发展新的几何技术，在理解朗兰兹对偶的一个方面取得了很大的进展。＊＊＊

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. ***