Problems in number theory and representation theory

冈内尔斯建议在三个广泛的领域研究数论和表象理论中的问题。首先，他将集中于与算术群和相关区域的上同调有关的主题，例如局部对称空间的几何以及与算术几何的联系。其次，他将研究附着于Weyl群和仿射Weyl群的多重Dirichlet级数及其与组合学、表示论和自同构型的联系；第三，他将研究Coxeter群中Kazhdan-Lusztig胞胞的几何及其与组合学和表示论的联系。作为一个主要的写作项目，他将继续与赫兹布鲁赫和扎吉尔合作更新他们的著作《阿提亚-辛格定理和初等数论》。这项建议涉及数论和表示论。数论是研究整体数的性质的，是数学中最古老的分支。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片段。该提案的一个主要目的是在“语言哲学”的精神下探索这两个学科之间的关系，该哲学预言了数论和表象理论之间的深刻联系。今天，这些学科所提出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、化学、物理和理论计算机科学等不同领域做出了许多应用。

Gunnells proposes to work on problems in number theory andrepresentation theory in three broad areas. First, he will focus ontopics related to cohomology of arithmetic groups and allied areas,such as geometry of locally symmetric spaces and connections witharithmetic geometry. Second, he will study multiple Dirichlet seriesattached to Weyl groups and affine Weyl groups and their connectionsto combinatorics, representation theory, and automorphic forms.Third, he will investigate the geometry of Kazhdan-Lusztig cells inCoxeter groups and their connections with combinatorics andrepresentation theory. As a major writing project, he will continueto work with Hirzebruch and Zagier on the updating of their book "TheAtiyah-Singer theorem and elementary number theory."This proposal deals with number theory and representation theory.Number theory is the study of the properties of the whole numbers, andis the oldest branch of mathematics. Representation theory is thesystematic study of symmetry, through the development of simplemathematical objects that encode the fundamental irreducible pieces ofsymmetry. A principal aim of the proposal is to explore relationshipsbetween these two subjects in the spirit of the "Langlandsphilosophy," which predicts deep connections between number theory andrepresentation theory. Today the questions and phenomena addressed bythese subjects serve as driving forces in much of contemporarymathematics research. Moreover, the subjects have contributed manyapplications in such diverse areas as codes and data transmission,chemistry, physics, and theoretical computer science.