Number Theory and Algebraic Geometry

本建议涉及数论和代数几何中四个特定领域的问题。首先，在与Avner Ashand Mark McConnell的合作中，gunnell通过研究Hecke算子在不同情况下对算术群上同调的作用来探索自同构形式的猜想。其次，gunnell与robertschzech共同研究了非模群的l -函数的特殊值与群环之间的联系。第三，与Lev Borisov合作，gunnell正在研究使用环向变体的模形式的构造。最后，与EricBabson和Richard Scott一起，gunnell正在研究表征理论中自然产生的某些构型变化的几何。这个建议涉及数论和代数几何。数论是对整数性质的研究，是数学中最古老的分支。代数几何研究可以用最简单的方程即多项式来定义的几何图形。将这两门学科结合起来所产生的问题和现象，是当代数学研究的动力。此外，这些学科的结合在代码和数据传输、机器人和理论计算机科学等不同领域贡献了许多应用。

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.