Mathematical Sciences: Permutation Representations, Monodromy and Matrix Classes

该奖项支持两个领域的研究。首先，首席研究员将研究排列表示。他将对黎曼曲面上所有在属上足够大的原始分支覆盖的单群进行分类。他还将考虑有限域及其代数闭包和数域上的类似问题。此外，他将考虑涉及非共轭子群的问题，这些问题诱导出相同的置换表示。第二个主要研究领域是矩阵理论。特别地，主要研究者将决定关于整数上对称矩阵的最小多项式可以说些什么。他还打算研究矩阵的可交换r元组的各种结构。群是具有单一运算的代数结构。它出现在数学的许多领域，以及物理和化学。有限群的基本组成部分是有限单群。过去十年数学的主要成果之一是有限单群的分类，证明它需要1万到1.5万页的期刊。本研究旨在将此分类应用于任意有限群及其他代数结构的研究。

This award supports research in two areas. First the principal investigator will study permutation representations. He will classify all monodromy groups of primitive branched coverings of Riemann surfaces which are sufficiently large in terms of the genus. He will also consider similar problems over finite fields and their algebraic closures, and over number fields. In addition, he will consider problems involving nonconjugate subgroups which induce the same permutation representation. The second major area of investigation is matrix theory. In particular, the principal investigator will determine what can be said about the minimal polynomial of a symmetric matrix over the integers. He also intends to investigate the structure of the variety of commuting r-tuples of matrices. 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 and other algebraic structures.