Representations of Finite Groups and Applications

该提案的主要研究领域是有限群的表示理论。数学中群的概念是从对称性的概念发展而来的。在自然界或科学中，物体的对称性是由一个群编码的，这个群携带着许多关于物体本身结构的重要信息。群表示理论允许人们通过它们在向量空间上的行为来研究群，向量空间模拟了它们在真实的世界中出现的方式。它已经吸引了数学家超过世纪，并在量子力学，基本粒子理论，编码理论和密码学中有许多重要的应用，预计将继续在现代计算机和数字通信世界中发挥重要作用。这些研究将有助于对有限群表示理论的理解和应用。学生的参与将是该项目的一个重要的科学组成部分。该项目集中在有限群的表示理论及其应用中的几个问题。许多这样的问题在群表示论中自然出现。其他人则是出于各种应用的动机。PI将研究几个全局-局部问题，包括McKay，Alperin和Brauer的结构，以及其他一些结构，这些结构推广了有限群表示论中的经典结果。PI还将继续他的长期项目，以发展一个理论的字符水平和建立强界的字符值有限群的李型，并分类模块表示有限quasisimple群的低维或特殊性质。然后，他将应用结果，以实现一些应用程序的重大进展，包括本地系统和他们的monodromy组，Aschbacher的猜想子群格，随机游走的凯莱和麦凯图和代表品种的Fuchsian组，字地图分布和汤普森的猜想简单的群体，宫本的问题与应用程序的顶点算子代数，这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The main area of research in this proposal is the representation theory of finite groups. The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or science are encoded by a group, which carries a lot of important information about the structure of the object itself. Group representation theory allows one to study groups via their actions on vector spaces which model the ways they arise in the real world. It has fascinated mathematicians for more than a century, and has had many important applications in quantum mechanics, the theory of elementary particles, coding theory, and cryptography, and is expected to continue to play an important role in the modern world of computers and digital communications. This research will contribute to advances in the understanding and applications of representation theory of finite groups. Student involvement will be a scientifically important component of the project.This project focuses on several problems in the representation theory of finite groups and its applications. Many of these problems come up naturally in the group representation theory. Others are motivated by various applications. The PI will study several global-local problems, including the conjectures of McKay, Alperin, and Brauer, and some other conjectures which generalize classical results in representation theory of finite groups. The PI will also continue his long-term project to develop a theory of character level and to establish strong bounds on character values of finite groups of Lie type, and to classify modular representations of finite quasisimple groups of low dimension or with special properties. He will then apply the results to achieve significant progress on a number of applications, including local systems and their monodromy groups, Aschbacher's conjecture on subgroup lattices, random walks on Cayley and McKay graphs and representation varieties of Fuchsian groups, word map distributions and Thompson's conjecture for simple groups, Miyamoto's problem with applications to vertex operator algebras, and bounds on cohomology groups and presentations of finite quasisimple groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.