Combinatorial Problems in Algebra, Topology and Geometry

调查研究组合问题出现在不同领域的数学。 与R. Guralnick，研究了Riemann曲面的分支覆盖，其单值群是作用在n-集的固定大小子集上的对称交错群S_n和A_n。这个项目的主要目标是：1）证明在已知的少数例外情况下，覆盖曲面的亏格必须随着覆盖的片数和分支点的数目而增长; 2）确定覆盖空间的亏格至多为1的所有覆盖。此外，研究者继续研究V.Vassiliev的纽结和装饰的有限型不变量理论中的单调图和超图性质，最后，研究者继续研究有限群的子群格的序复形。他试图用拓扑方法来区分有限群的子群格与任意有限格中的区间。 与韦尔克，他调查了拓扑结构的顺序复杂的子群格的有限simplegroups.The调查员的主要利益是在组合学，这是研究离散的，通常是有限的数学对象。 组合对象出现在应用数学和计算机科学的各个领域，包括通信和算法复杂性理论。 此外，还有复杂的非离散组合对象，可以更好地理解，通过检查相关的组合对象。 研究者研究以这种方式出现的组合对象。

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.