Problems in arithmetic groups and multiple Dirichlet series.

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This project focuses on problems in arithmetic groups and multiple Dirichlet series, problems that are intimately related to automorphic forms. One part addresses topics related to cohomology of arithmetic groups and allied areas, such as the geometry of locally symmetric spaces and the conjectures linking cohomology to arithmetic geometry and Galois representations. The second part studies multiple Dirichlet series attached to Weyl groups, affine Weyl groups, and principal homogeneous vector spaces, and their connections to combinatorics, representation theory, automorphic forms, and mathematical physics. As writing projects, the principal investigator will continue to work with F. Hirzebruch and D. Zagier on updating their book "The Atiyah-Singer theorem and elementary number theory." He will also work with W. Stein to prepare a revised version of "Modular forms, a computational approach."This proposal deals with the interactions between number theory and representation theory. Number theory is the study of the properties of the whole numbers, and is the oldest branch of mathematics. Representation theory is the systematic study of symmetry, through the development of simple mathematical objects that encode the fundamental irreducible pieces of symmetry. A principal aim of the proposal is to explore relationships between these two subjects in the spirit of the "Langlands philosophy," which predicts deep connections between number theory and representation theory. Today the questions and phenomena addressed by these subjects serve as driving forces in much of contemporary mathematics research. Moreover, the subjects have contributed many applications in such diverse areas as codes and data transmission, chemistry, physics, and theoretical computer science.