Problems in number theory and representation theory

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Gunnells proposes to work on problems in number theory andrepresentation theory in three broad areas. First, he will focus ontopics related to cohomology of arithmetic groups and allied areas,such as geometry of locally symmetric spaces and connections witharithmetic geometry. Second, he will study multiple Dirichlet seriesattached to Weyl groups and affine Weyl groups and their connectionsto combinatorics, representation theory, and automorphic forms.Third, he will investigate the geometry of Kazhdan-Lusztig cells inCoxeter groups and their connections with combinatorics andrepresentation theory. As a major writing project, he will continueto work with Hirzebruch and Zagier on the updating of their book "TheAtiyah-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, andis the oldest branch of mathematics. Representation theory is thesystematic study of symmetry, through the development of simplemathematical objects that encode the fundamental irreducible pieces ofsymmetry. A principal aim of the proposal is to explore relationshipsbetween these two subjects in the spirit of the "Langlandsphilosophy," which predicts deep connections between number theory andrepresentation theory. Today the questions and phenomena addressed bythese subjects serve as driving forces in much of contemporarymathematics research. Moreover, the subjects have contributed manyapplications in such diverse areas as codes and data transmission,chemistry, physics, and theoretical computer science.