Multiple Dirichlet series, Whittaker functions, and the cohomology of arithmetic groups

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

This research 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 individual areas themselves have contributed many applications in such diverse areas as coding and data transmission, chemistry, physics, and theoretical computer science. The problems that will be investigated in this project focus on arithmetic groups and multiple Dirichlet series, objects that are intimately related to automorphic forms. The first 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. A central goal of this part is the computational investigation of torsion in the cohomology of various arithmetic groups and its relation to arithmetic. The second part studies multiple Dirichlet series attached to Weyl groups. These are infinite series in several complex variables that satisfy a group of functional equations intermixing all the variables. The topics in this part emphasize connections between these series and combinatorics, representation theory, automorphic forms, and mathematical physics.