Mathematical Sciences: Cohomology of Arithmetic Groups

本文主要研究算术群的上同调以及与之相关的自同构型。它将集中在三个领域。第一种是广义模符号和鬼类。我们将通过某些有限群的表示理论来研究它们。二是具有不同系数模的上同调类之间以及Hecke代数的相关本征值之间的同余。这导致了由尖点形式构造扭上同调。有理数的伽罗瓦群的大型表示与猜想有联系。第三个领域是上同调群的计算机辅助计算。这项研究包含了代数和分析的有趣结合，用于研究数论中的问题。这是一项重要且非常深入的研究，在过去，P.I.非常成功地进行了这项研究。毫无疑问，他将在这一奖项的任期内继续取得令人瞩目的成果。

This research will be in the cohomology of arithmetic groups and the related automorphic forms. It will concentrate on three areas. The first is the generalized modular symbols and the ghost classes. They will be studied via the representation theory of certain finite groups. The second is the congruences between cohomology classes with different coefficient modules and between the associated eigenvalues of the Hecke algebra. This leads to construction of torsion cohomology from cusp forms. There is a conjectural link with large representations of the Galois group of the rational numbers. The third area is a computer-aided calculation of cohomology groups. This research contains an interesting blend of algebra and analysis which is used to study questions in number theory. This is an important and very deep study which the P.I. has been very successful at in the past. He will undoubtedly continue to produce striking results during the tenure of this award.