Representation Theory and Automorphic Forms

Barbasch教授将研究约化群的单幂表示的性质，如它们的唯一性、它们与自同构形式的关系以及轨道方法。他将继续与A. Moy和S. Evens就相关主题进行合作。Speh教授将继续与J. Rohlfs在爱森斯坦上同调方面的合作，旨在计算Tamagawa数并获得周期积分的合理性结果。与Dan Barbasch一起，她将继续致力于从扭曲的内窥镜群中提升自同构形式。Birgit Speh还将继续她在Seiberg-Witten方程和扭曲扭转方面的工作。这个建议属于李群的表示理论及其在自同构形式中的应用的一般领域。李群（为纪念挪威数学家索夫斯·李而命名）及其表示理论一直是20世纪数学的主要主题之一。李群表示理论作为利用系统固有对称性的数学工具，在分析和理论物理，特别是量子力学和基本粒子物理中产生了深远的影响。现代形式的自同构形式理论是由Gelfand、Piatetski-Shapiro、Langlands和Borel等人提出的，在很大程度上依赖于李群的表征理论。它对经典数论中出现的问题产生了重大影响，并有望继续这样做。这些理论非常抽象，但在理论计算机科学和编码理论等领域有着惊人的重要应用。

AbstractSpeh/BarbaschProfessor Barbasch will investigate properties of unipotent representations of reductive groups such as their unitarity and their relation to automorphic forms and the orbit method. He will continue his joint work with A. Moy and S. Evens on related topics. Professor Speh will continue her joint work with J. Rohlfs on Eisenstein cohomology aiming to compute Tamagawa numbers and obtain rationality results for periods integrals. Jointly with Dan Barbasch she will continue work on lifting automorphic forms from twisted endoscopic groups. Birgit Speh will also continue her work on Seiberg-Witten equations and twisted torsion.This proposal falls in the general area of representation theory of Lie groups and its applications to automorphic forms. Lie groups (named in honor of the Norwegian mathematician Sophus Lie) and their representation theory, have been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact in analysis and theoretical physics, especially quantum mechanics and elementary particle physics. The theory of automorphic forms in its modern form was initiated by Gelfand, Piatetski-Shapiro, Langlands and Borel and relies heavily on the representation theory of Lie groups. It has had a significant impact on problems arising in classical number theory and is expected to continue to do so. These theories are very abstract, but have surprising important applications in areas like theoretical computer science and coding theory.