Group Representations and Applications

数学中群的概念源于对称的概念。一个物体在自然界或理论结构上的对称性可以被一个群体编码，这个群体本身就携带着关于物体结构的重要信息。表示理论允许人们通过群体在向量空间上的行为以统一的方式来研究群体，这模拟了群体在应用程序中的常见行为方式。一个多世纪以来，表征理论一直是数学的中心话题，在物理和化学，特别是量子力学和基本粒子理论中有着重要的应用。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发挥着重要的作用。这个项目旨在促进对有限群的表示理论的理解和在一些应用中。更详细地说，本项目着重于群表示理论及其应用中的几个重要问题。它将数学的不同领域联系在一起，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，以表示理论作为主要的统一成分。项目中解决的许多问题都是在研究群体表征时自然出现的，而其他问题则是由应用程序驱动的。研究者将沿着全局-局部原则的路线研究几个问题，包括McKay， Alperin， Brauer和其他人的猜想，这些猜想扩展和推广了有限群表示理论中的经典结果。研究者还打算研究低维模表示的分类，并发展字符水平理论，目标是建立有限拟单群的字符值的强界。研究结果可应用于研究子群格上的Aschbacher猜想和外幂上的kolar - larsen问题、Miyamoto问题、随机漫步和反集中问题、Fuchsian群的表示变化、拟单群的词映射分布和waring型问题、第二上同调群上Holt猜想的改进、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.