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法。他将对Riemann曲面的原始分枝覆盖的所有单色群进行分类，这些覆盖在亏格上足够大。他还将在有限域及其代数闭包和数域上考虑类似的问题。此外，他还将考虑涉及诱导相同置换表示的非共轭子群的问题。研究的第二个主要领域是矩阵理论。特别是，首席研究员将确定关于整数上对称矩阵的极小多项式可以说些什么。他还打算研究矩阵的各种可交换r-字节组的结构。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。这项研究的目的是将这种分类用于研究任意有限群和其他代数结构。

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.