Mathematical Sciences: Cohomology of Arithmetic Groups

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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.