Number Theory and Algebraic Geometry

Barnard College, New York NY

Investigators

Abstract

This proposal concerns problems in number theory and algebraic geometry in four specific areas. First, in joint work with Avner Ash and Mark McConnell, Gunnells is exploring conjectures in automorphic forms by studying the action of the Hecke operators on the cohomology of arithmetic groups in different contexts. Second, jointly with Robert Sczech, Gunnells is studying the connections between special values of L-functions and group cocycles for the unimodular group. Third, jointly with Lev Borisov, Gunnells is investigating the construction of modular forms using toric varieties. Finally, jointly with Eric Babson and Richard Scott, Gunnells is studying the geometry of certain configuration varieties naturally arising in representation theory. This proposal deals with number theory and algebraic geometry. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. Algebraic geometry studies geometric figures that can be defined by the simplest of equations, namely polynomials. The questions and phenomena which arise from combining these two subjects serve as driving forces in much of contemporary mathematics research. Moreover, the combination of these subjects has contributed many applications in such diverse areas as codes and data transmission, robotics, and theoretical computer science.