Enumerative and Topological Properties of Matroids

PI计划继续研究拟阵的枚举和拓扑性质。 拟阵是由Whitney于1935年提出的，它是数学基本结构之一线性独立性的组合抽象。 计数方面的工作涉及组合技术从交换代数，拓扑学和组合学，以研究h-向量的断路和独立复杂。 这些组合不变量在超平面排列、线性编码理论、网络可靠性和球的连续性等方面都有应用。 拟阵的拓扑性质将利用研究人员最近的发现，拟阵可以表示为同伦球的安排进行探讨。 本文的研究是在组合数学的一般领域中进行的，组合数学的中心课题之一是具有复杂结构的离散对象的计数问题。 例如，这个建议包括估计可能在电话或其他类似通信系统的设计中发现的大型网络的可靠性的方法。

The PI plans to continue his research into theenumerative and topological properties of matroids. Introduced in 1935 by Whitney, matroids are acombinatorial abstraction of one of the fundamentalstructures of mathematics, linear independence. Theenumerative aspect of the work involves combiningtechniques from commutative algebra, topology andcombinatorics in order to study h-vectors of brokencircuit and independence complexes. Thesecombinatorial invariants appear in applications tohyperplane arrangements, linear coding theory, networkreliability and quotients of spheres. Topologicalproperties of matroids will be explored using theinvestigator's recent discovery that matroids can berepresented as arrangements of homotopy spheres. Thispoint of view indicates possible applications tocombinatorial projective planes, oriented matroids andclassical bundle theories.This research is in the general area of combinatorics.One of the central themes in combinatorics is theenumeration of discrete objects with complicatedstructure. For instance, this proposal includesmethods for estimating the reliability of largenetworks of the type that might be found in the designof telephone or other similar communication systems.