Number Theory and Algebraic Geometry

这个建议涉及数论和代数几何中四个具体领域的问题。 首先，在联合工作与阿夫纳阿什和马克麦康奈尔，Gunnells是探索automorphic形式的行动研究的Hecke运营商上同调的算术群在不同的情况下。 其次，Gunnells与RobertSchzech一起研究了幺模群的L-函数的特殊值与群上循环之间的关系。 第三，Gunnells与Lev Borisov一起研究了使用环面变体的模块形式的构造。 最后，与埃里克巴布森和理查德斯科特，冈内尔正在研究几何的某些配置品种自然出现在representationtheory.This建议涉及数论和代数几何。 数论是研究整数的性质，是数学最古老的分支。 代数几何学研究的是可以用最简单的方程，即多项式来定义的几何图形。 结合这两门学科所产生的问题和现象是当代许多哲学研究的驱动力。 此外，这些学科的结合在诸如编码和数据传输、机器人技术和理论计算机科学等不同领域中产生了许多应用。

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.