Combinatorial Problems in Algebra, Topology and Geometry

University Of Miami, Coral Gables FL

Investigators

Abstract

The investigator studies combinatorial problems which arise in various areas of mathematics. In joint work with R. Guralnick, the investigator examines branched coverings of Riemann surfaces whose monodromy groups are the symmetric and alternating groups S_n and A_n acting on subsets of a fixed size from the n-set. The main goals of this project are 1) to show that with a small and known list of exceptions, the genus of the covering surface must grow with both the number of sheets of the covering and the number of branch points, and 2) to determine all such coverings for which the genus of the covering space is at most one. This project is one of the final steps in a program initiated by Guralnick and J. Thompson. In addition, the investigator continues his study of monotone graph and hypergraph properties which arise in V. Vassiliev's theory of finite type invariants of knots and ornaments. Finally, the investigator continues his examination of order complexes of subgroup lattices of finite groups. He attempts to use topological methods to distinguish intervals in subgroup lattices of finite groups from arbitrary finite lattices. With V. Welker, he investigates the topology of the order complexes of subgroup lattices of finite simple groups. The investigator's main interests are in combinatorics, which is the study of discrete, usually finite mathematical objects. Combinatorial objects arise in various areas of applied mathematics and computer science, including communications and the theory of algorithmic complexity. Also, there are complicated nondiscrete mathematical objects which can be better understood by examining associated combinatorial objects. The investigator studies combinatorial objects which arise in this manner.