Group Representations and Applications

数学中群的概念是从对称性的概念发展而来的。自然界中的物体或理论构造的对称性可以用群来编码，群本身携带着关于物体结构的重要信息。表示论允许人们通过向量空间上的行为以统一的方式研究群体，向量空间模拟了群体在应用中的常见行为方式。表示论是数学的中心议题已经超过世纪，并且在物理学和化学中有重要的应用，特别是在量子力学和基本粒子理论中。有限群及其表示在编码理论和密码学中有着重要的应用，在现代计算和数字通信中扮演着重要的角色。该项目旨在促进对有限群的表示理论和一些应用的理解。更详细地说，这个项目集中在群表示理论及其应用中的几个重要问题。它将不同的数学领域联系在一起，如有限群和代数群，置换群，概率群论，群上同调，组合学，顶点算子代数和代数几何，表示论是主要的统一成分。该项目中解决的许多问题在群体表征的研究中自然出现，而其他问题则是由应用引起的。调查人员将研究几个问题沿着线的全球本地的原则，包括acutures的麦凯，Alperin，布劳尔，和其他人的扩展和推广经典的结果表示理论的有限群。研究者还打算研究低维模表示的分类，并发展特征标水平理论，目标是建立有限拟单群特征标值的强界。预期这些结果可用于研究子群格上的Aschbacher猜想、外幂上的Kollar-Larsen问题、Miyamoto问题、随机游动与反集中问题、Fuchsian群的表示变种、拟单群的字映射分布与Waring型问题、Holt猜想在二上同调群上的改进、和具有特殊性质的拟单群的表示。

The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or of a theoretical construct can be encoded by a group, which itself carries important information about the structure of the object. Representation theory allows one to study groups in a uniform way via their actions on vector spaces, which model common ways groups act in applications. Representation theory has been a central topic in mathematics for more than a century and has important applications in physics and chemistry, particularly in quantum mechanics and the theory of elementary particles. Finite groups and their representations have important applications in coding theory and cryptography, and play an important role in modern computation and digital communications. This project aims to advance understanding in the representation theory of finite groups and in a number of applications. In more detail, this project focuses on several important questions in group representation theory and its applications. It ties together different areas of mathematics, such as finite groups and algebraic groups, permutation groups, probabilistic group theory, group cohomology, combinatorics, vertex operator algebras, and algebraic geometry, with representation theory as the main unifying ingredient. Many of the questions addressed in the project come up naturally in the study of group representations, and others are motivated by applications. The investigator will study several problems along the lines of the global-local principle, including conjectures of McKay, Alperin, Brauer, and others that extend and generalize classical results in the representation theory of finite groups. The investigator also intends to study classification of modular representations of low dimension and to develop a theory of character level, with the goal of establishing strong bounds on character values for finite quasisimple groups. It is anticipated that the results can be applied to study Aschbacher's conjecture on subgroup lattices and the Kollar-Larsen problem on exterior powers, Miyamoto's problem, problems on random walks and anti-concentration, representation varieties of Fuchsian groups, word map distributions and Waring-type problems for quasisimple groups, refinements of Holt's conjecture on second cohomology groups, and representations of quasisimple groups with special properties.