Combinatorial Problems in Algebra, Topology and Geometry

研究人员研究数学各个领域中出现的组合问题。在与R.Guralnick的合作中，研究者研究了Riemann曲面的分支覆盖，其单线群是作用于n-集合中固定大小的子集上的对称交错群S_n和A_n。这个项目的主要目的是：1)证明在一个已知的小例外列表中，覆盖曲面的亏格必须随着覆盖的片数和分支点的数量而增长；2)确定覆盖空间亏格至多为1的所有这样的覆盖。这个项目是Guralnick和J.Thompson发起的程序的最后步骤之一。此外，作者继续研究了V.Vassiliev关于纽结和装饰的有限类型不变量理论中出现的单调图和超图的性质，最后继续考察了有限群的子群格的序复。他试图用拓扑方法来区分有限群子群格中的区间和任意有限格。与V.Welker一起，他研究了有限单群的子群格的序复形的拓扑学。他的主要兴趣是组合学，这是对离散的、通常是有限的数学对象的研究。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，还有复杂的非离散数学对象，通过检查相关的组合对象可以更好地理解这些对象。研究人员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.