Representations of Finite Groups and Applications

本文主要讨论有限群表示论及其应用中的几个重要问题。这些问题中的许多自然出现-一些长期存在并在群表示理论中发挥核心作用，而其他问题则是由各种应用引起的。该建议将不同的数学领域联系在一起，例如有限群和代数群，有限置换群理论，群上同调，组合学，算子代数和代数几何，主要统一成分是表示论。PI将沿着局部-全局原理研究沿着几个问题，包括Alperin重量猜想，Brauer的零高度猜想，以及一些关于有限群的复和Brauer特征的合理性和整除性的进一步假设。PI还将继续他的长期项目，分类模块表示有限quasisimple组的低维。然后，他将应用他的结果，以实现一些应用程序的重大进展，包括华林型问题的quasisimple集团，Aschbacher的猜想子群格，Kollar-Larsen问题的外部权力（在代数几何中的应用），以及有限群的二阶上同调群的Guralnick-Holt猜想及其表示，和具有特殊性质的有限拟单群的表示（在有限单群的子群结构中的应用）。本建议的主要研究领域是群表示理论。数学中群的概念是由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，这个群携带了许多关于物体本身结构的重要信息。表示论允许人们通过向量空间上的行为来研究群体，向量空间模拟了群体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，并在物理和化学中有许多重要的应用，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已经被证明是有价值的，并且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。研究人员的研究将导致理解有限群的表示理论的重要进展，并有助于在其应用中取得重大进展。

This proposal focuses on several important problems in representation theory of finite groups and its applications. Many of these problems come up naturally -- some long-standing and playing a central role -- in group representation theory, and others are motivated by various applications. The proposal ties together different areas of mathematics, such as finite groups and algebraic groups, finite permutation group theory, group cohomology, combinatorics, operator algebras, and algebraic geometry, with the main unifying ingredient being the representation theory. The PI will study several problems along the lines of the local-global principle, including the Alperin weight conjecture, Brauer's height zero conjecture, and some further conjectures concerning rationality and divisibility properties of complex and Brauer characters of finite groups. The PI will also continue his long-term project to classify modular representations of finite quasisimple groups of low dimension. He will then apply his results to achieve significant progress on a number of applications, including Waring-type problems for quasisimple groups, Aschbacher's conjecture on subgroup lattices, the Kollar-Larsen problem on exterior powers (with application in algebraic geometry), and the Guralnick-Holt conjecture on second cohomology groups for finite groups and their presentations, and representations of finite quasisimple groups with special properties (with application in the subgroup structure of finite simple groups).The main area of research in this proposal is group representation theory. The concept of a group in mathematics grew out ofthe notion of symmetry. The symmetries of an object in nature or science are encoded by a group, and this group carries a lot of important information about the structure of the object itself. The 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 many important applications in physics and chemistry, particularly in quantum mechanics and in the theory of elementary particles. Finite groups and their representations have already proved valuable in coding theory and cryptography, and are expected to continue to play an important role in the modern world of computers and digital communications. The investigator's research will lead to important advances in understanding the representation theory of finite groups and help achieve significant progress in a number of its applications.