Combinatorial Problems in Algebra, Topology and Geometry

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The investigator studies combinatorial problems which arisein various areas of mathematics. In joint work with R. Guralnick,the investigator examines branched coverings of Riemann surfaceswhose monodromy groups are the symmetric and alternating groupsS_n and A_n acting on subsets of a fixed size from the n-set. Themain 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 coveringspace is at most one.This project is one of the final steps in a program initiated byGuralnick and J. Thompson. In addition, the investigator continues hisstudy 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 complexesof subgroup lattices of finite groups. He attempts to use topologicalmethods to distinguish intervals in subgroup lattices of finite groupsfrom arbitrary finite lattices. With V. Welker, he investigates thetopology of the order complexes of subgroup lattices of finite simplegroups.The investigator's main interests are in combinatorics, which is thestudy of discrete, usually finite mathematical objects. Combinatorialobjects arise in various areas of applied mathematics and computerscience, including communications and the theory of algorithmiccomplexity. Also, there are complicated nondiscrete mathematicalobjects which can be better understood by examining associatedcombinatorial objects. The investigator studies combinatorial objectswhich arise in this manner.