Number Theory and Algebraic Geometry

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This proposal concerns problems in number theory and algebraicgeometry in four specific areas. First, in joint work with Avner Ashand Mark McConnell, Gunnells is exploring conjectures in automorphic formsby studying the action of the Hecke operators on the cohomology ofarithmetic groups in different contexts. Second, jointly with RobertSczech, Gunnells is studying the connections between special values ofL-functions and group cocycles for the unimodular group. Third,jointly with Lev Borisov, Gunnells is investigating the constructionof modular forms using toric varieties. Finally, jointly with EricBabson and Richard Scott, Gunnells is studying the geometry of certainconfiguration varieties naturally arising in representation theory.This proposal deals with number theory and algebraic geometry. Numbertheory is the study of the properties of the whole numbers and is theoldest branch of mathematics. Algebraic geometry studies geometricfigures that can be defined by the simplest of equations, namelypolynomials. The questions and phenomena which arise from combining these two subjects serve as driving forces in much of contemporary mathematicsresearch. Moreover, the combination of these subjects has contributed many applicationsin such diverse areas as codes and data transmission, robotics, andtheoretical computer science.