Mathematical Sciences: The Structure and Representations of Algebraic Groups and Associated Finite Groups

本研究主要研究代数群及其相关有限群的结构和表示。主要研究人员将建立关于单代数群的约化子群的结构和嵌入的结果；致力于将有限群的子群结构与代数群的子群结构联系起来；研究正特征标域上对称群的表示理论；以及研究定义特征标域上的李型有限群的表示。群是具有单一运算的代数结构。它出现在许多数学领域，也出现在物理和化学领域。有限群的基本构件是有限单群。过去十年数学中的主要结果之一是有限单群的分类，要证明这一点需要10,000到15,000页的期刊页。本研究的目的是将这种分类应用于任意有限群的研究。

This research is concerned with the structure and representations of algebraic groups and associated finite groups. The principal investigator will establish results on the structure and embedding of reductive subgroups of simple algebraic groups; work on linking the subgroup structure of finite groups to that of algebraic groups; study the representation theory of the symmetric groups over fields of positive characteristic; and study representations of finite groups of Lie type over a field of the defining characteristic. A group is an algebraic structure with a single operation. It appears in many areas of mathematics, as well as, physics and chemistry. The fundamental building blocks of finite groups are finite simple groups. One of the major results in mathematics of the past decade is the classification of the finite simple groups, the proof of which would require 10,000 to 15,000 journal pages. This research is aimed at using this classification in the study of arbitrary finite groups.