Problems in number theory and representation theory

Gunnells建议在三个广泛的领域研究数论和表示论中的问题。 首先，他将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和算术几何的连接。 其次，他将研究附在Weyl群和仿射Weyl群上的多个Dirichlet级数，以及它们与组合学、表示论和自守形式的联系。第三，他将研究Coxeter群中Kazhdan-Lusztig细胞的几何结构，以及它们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Hirzebruch和Zagier合作，更新他们的书“TheAtiyah-Singer定理和初等数论”。“这个提议涉及数论和表示论。数论是研究整数性质的学科，是数学最古老的分支。 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。 该提案的一个主要目的是探索这两个主题之间的关系在“朗格兰哲学”的精神，它预测了数论和表示理论之间的深刻联系。 今天，这些学科所解决的问题和现象成为当代数学研究的驱动力。 此外，这些学科在编码和数据传输、化学、物理和理论计算机科学等不同领域都有许多应用。

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.