Number Theory and Algebraic Geometry

这项建议涉及四个特定领域的数论和代数几何问题。首先，在与Avner Ash和Mark McConnell的合作中，Gunnells正在通过研究Hecke算子在不同上下文中算术群上同调上同调的作用来探索自同构形式的猜想。其次，Gunnells与RobertSczech共同研究了单模群的L-函数的特殊值与群上循环之间的关系。第三，与列夫·鲍里索夫一起，Gunnells正在研究使用Toric簇构造模形式。最后，与EricBabson和Richard Scott一起，Gunnells正在研究表象理论中自然产生的某些构型变体的几何。这项建议涉及数论和代数几何。数论是研究整数性质的学科，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许多数学研究的驱动力。此外，这些学科的结合在代码和数据传输、机器人学和理论计算机科学等不同领域做出了许多应用。

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.