Number Theory, Algebraic Geometry & Representation Theory

摘要：gunnells的DMS-0401525奖。该提案涉及数论、代数几何和表示理论三个广泛领域的问题。首先，PI关注于算术群和相关领域的上同调，例如局部对称空间的几何及其紧化，模形式的模曲线和环，以及l函数的特殊值。其次，他正在研究仿射Weyl群中典型Kazhdan-Lusztig细胞的几何形状。最后，他将与F. Hirzebruch和D. Zagier一起更新他们的著作《Atiyah-Singer定理和初等数论》。这个建议涉及数论、代数几何和表示理论。数论研究的是整数的性质，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。表征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的对称性。今天，这些学科所探讨的问题和现象成为当代数学研究的推动力。此外，这些学科在代码和数据传输、机器人、化学、物理和理论计算机科学等不同领域都有许多应用。

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.