Problems in number theory and representation theory

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Gunnells proposes to work on problems in number theory and representation theory in three broad areas. First, he will focus on topics related to cohomology of arithmetic groups and allied areas, such as geometry of locally symmetric spaces and connections with arithmetic geometry. Second, he will study multiple Dirichlet series attached to Weyl groups and affine Weyl groups and their connections to combinatorics, representation theory, and automorphic forms. Third, he will investigate the geometry of Kazhdan-Lusztig cells in Coxeter groups and their connections with combinatorics and representation theory. As a major writing project, he will continue to work with Hirzebruch and Zagier on the updating of their book "The Atiyah-Singer theorem and elementary number theory." This proposal deals with 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.