Number Theory, Algebraic Geometry & Representation Theory

摘要奖DMS-0401525 GunnellsThis建议涉及的问题，在数论，algebraicgeometry，并表示理论在三个广泛的领域。 首先，PI专注于与算术群和相关领域的上同调相关的主题，例如局部对称空间的几何及其紧化，模曲线和模形式的环，以及L-函数的特殊值。 其次，他正在研究仿射Weyl群中的典型Kazhdan-Lusztig胞腔的几何。Hirzebruch和D. Zagier在更新他们的书，Atiyah-Singer定理和初等数论。这一建议涉及数论，代数几何，和表示理论。 数论是研究整数性质的学科，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。 表示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对称性的基本不可约部分。 如今，这些学科所解决的问题和现象成为大部分当代数学研究的驱动力。 此外，这些学科在编码和数据传输、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.