Group Representations and Applications

团体代表和申请

基本信息

批准号：
1665014
负责人：
Pham Tiep
金额：
$ 41万
依托单位：
University of Arizona
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2018-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665014&HistoricalAwards=false
关键词：
Group Representations Applications

项目摘要

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.

数学群体的概念是从对称的概念中得出的。对象在性质或理论构造中的对称性可以由一个组编码，该组本身包含有关对象结构的重要信息。表示理论允许一个人通过对向量空间的行动以统一的方式研究小组，这模拟了群体在应用中的作用。代表理论在一个多世纪以来一直是数学中的核心主题，并且在物理和化学方面具有重要的应用，尤其是在量子力学和基本颗粒理论中。有限群体及其表示在编码理论和密码学中具有重要的应用，并在现代计算和数字通信中发挥重要作用。该项目旨在提高在有限群体和许多应用中的代表理论中的理解。更详细地，该项目重点介绍小组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（例如有限群和代数群，置换群体，概率群体理论，群体共同体学，组合术，顶点操作员代数和代数几何形状和代表理论）与主要统一成分相关联。该项目中解决的许多问题自然出现在小组代表的研究中，而其他问题则是出于应用程序的动机。研究人员将研究几个问题，包括全球本地原理的界限，包括麦凯，阿尔珀林，布劳尔的猜想以及其他在有限群体的表示理论中扩展和推广经典结果的其他问题。研究者还打算研究低维度的模块化表示形式的分类并发展一个特征级别的理论，其目的是建立有限的准准群体的性格价值界限。预计结果可以应用于研究阿克巴赫对亚组晶格的猜想以及外部能力上的Kollar-Lass问题，宫本的问题，随机步行和反集中的问题和抗浓度，抗浓度，代表性，五及其五及五个群体的表现形式，单词图分布和战争类型的组合，以造成Quasisimime type type type type type type type type type type type type type type type type type type type type type type type。和具有特殊特性的准群组的表示。