Problems in arithmetic groups and multiple Dirichlet series.

这个项目的重点是算术群和多重狄利克雷级数的问题，这些问题与自同构形式密切相关。其中一部分讨论了算术群的上同调及其相关领域，如局部对称空间的几何以及将上同调与算术几何和伽罗瓦表示联系起来的猜想。第二部分研究了Weyl群、仿射Weyl群和主齐次向量空间上的多重Dirichlet级数，以及它们与组合学、表示理论、自同构形式和数学物理的联系。作为写作项目，首席研究员将继续与F. Hirzebruch和D. Zagier合作，更新他们的书“the Atiyah-Singer定理和初等数论”。他还将与W. Stein合作，准备修订版的《模块化形式，一种计算方法》。本文讨论数论与表征理论之间的相互作用。数论是对整数性质的研究，是数学中最古老的分支。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。该提案的主要目的是在“朗兰兹哲学”的精神下探索这两个学科之间的关系，该哲学预测了数论和表征理论之间的深层联系。今天，这些学科提出的问题和现象成为当代数学研究的驱动力。此外，这些学科在代码和数据传输、化学、物理和理论计算机科学等不同领域都有许多应用。

This project focuses on problems in arithmetic groups and multiple Dirichlet series, problems that are intimately related to automorphic forms. One part addresses topics related to cohomology of arithmetic groups and allied areas, such as the geometry of locally symmetric spaces and the conjectures linking cohomology to arithmetic geometry and Galois representations. The second part studies multiple Dirichlet series attached to Weyl groups, affine Weyl groups, and principal homogeneous vector spaces, and their connections to combinatorics, representation theory, automorphic forms, and mathematical physics. As writing projects, the principal investigator will continue to work with F. Hirzebruch and D. Zagier on updating their book "The Atiyah-Singer theorem and elementary number theory." He will also work with W. Stein to prepare a revised version of "Modular forms, a computational approach."This proposal deals with the interactions between number theory and representation theory. Number theory is the study of the properties of the whole numbers, and is the oldest branch of mathematics. Representation theory is the systematic study of symmetry, through the development of simple mathematical objects that encode the fundamental irreducible pieces of symmetry. A principal aim of the proposal is to explore relationships between these two subjects in the spirit of the "Langlands philosophy," which predicts deep connections between number theory and representation theory. Today the questions and phenomena addressed by these subjects serve as driving forces in much of contemporary mathematics research. Moreover, the subjects have contributed many applications in such diverse areas as codes and data transmission, chemistry, physics, and theoretical computer science.