Number Theory, Algebraic Geometry & Representation Theory

Gunnell奖DMS-0401525摘要这项建议涉及数论、代数几何和表示论三个广泛领域的问题。首先，PI关注与算术群和相关区域的上同调有关的主题，如局部对称空间及其紧化的几何，模形式的模曲线和模环，以及L函数的特殊值。第二，他正在研究仿射Weyl群中标准Kazhdan-Lusztig胞胞的几何。最后，他正在与F.Hirzebruch和D.Zagier一起更新他们的书，Atiyah-Singer定理和初等数论。这项建议涉及数论，代数几何和表示论。数论是研究全体数的性质的学科，是数学中最古老的分支。代数几何研究的几何图形可以用最简单的方程，即多项式来定义。表象理论是对对称性的系统研究，通过发展简单的数学对象来编码基本的不可约对称性片段。今天，这些课题所涉及的问题和现象成为当代许多数学研究的驱动力。此外，这些学科还在代码和数据传输、机器人学、化学、物理和理论计算机科学等不同领域做出了许多应用。

Abstract for award DMS-0401525 of GunnellsThis proposal concerns problems in number theory, algebraicgeometry, and representation theory in three broad areas. First, the PI is focusing on topics related to cohomology of arithmetic groups andallied areas, such as geometry of locally symmetric spaces and theircompactifications, modular curves and rings of modular forms, andspecial values of L-functions. Second, he is investigating thegeometry of canonical Kazhdan-Lusztig cells in affine Weyl groups.Finally, he is joining F. Hirzebruch and D. Zagier in the updating oftheir book, The Atiyah-Singer theorem and elementary number theory.This proposal deals with number theory, algebraic geometry, andrepresentation theory. Number theory is the study of the propertiesof the whole numbers, and is the oldest branch of mathematics.Algebraic geometry studies geometric figures that can be defined bythe simplest of equations, namely polynomials. Representation theoryis the systematic study of symmetry, through the development of simplemathematical objects that encode the fundamental irreducible pieces ofsymmetry. Today the questions and phenomena addressed by thesesubjects serve as driving forces in much of contemporary mathematicsresearch. Moreover, the subjects have contributed many applicationsin such diverse areas as codes and data transmission, robotics,chemistry, physics, and theoretical computer science.