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.
