Research in Algebraic Combinatorics

该PI打算继续她目前在代数组合学方面的研究，重点是部分有序集的拓扑和代数性质。偏序集拓扑理论起源于1964年罗塔关于偏序集的莫比乌斯函数的著名论文，它为组合学和其他数学分支（如拓扑学、代数和几何）之间提供了深刻而基本的联系。这项研究的一个主要推动力来自Stanley在1982年的一篇开创性论文中对偏序集同调的群作用的考虑，另一个来自Bjorner关于可套序集的开创性论文，还有一个来自它与Orlik和Solomon发展的超平面排列理论的基本联系。偏序集拓扑理论最近的一个极其重要的发展是Goresky和MacPherson与子空间排列理论的联系。这导致了Bjorner， Lovasz和Yao在复杂性理论中的应用。这些最近的发展使得Bjorner和PI将可壳性理论从纯的扩展到非纯的简单复合物和偏序集。壳性是一种组合工具，用于建立简单复合体和偏序集的某些拓扑和代数性质。非纯环境提供了一个更丰富、更有力的理论，PI计划继续探索和发展。这项研究属于组合学的一般领域。组合学的目标之一是找到研究离散对象集合如何排列的有效方法。离散系统的行为对现代通信极为重要。例如，大型网络的设计，比如那些出现在电话系统中的网络，以及计算机科学中处理离散对象集的算法设计，这就利用了组合研究。

Wachs 9701407 The PI intends to continue her current research in algebraic combinatorics, focusing on topological and algebraic properties of partially ordered sets. The theory of poset topology, which grew out of the famous 1964 paper of Rota on the Moebius function of a partially ordered set, provides a deep and fundamental link between combinatorics and other branches of mathematics such as topology, algebra and geometry. One major impetus for this study has come from Stanley's consideration of group actions on the homology of posets in a seminal 1982 paper, another comes from Bjorner's pioneering paper on shellable posets and still another comes from its fundamental connection with the theory of hyperplane arrangements as developed by Orlik and Solomon. A more recent and extremely important development in the theory of poset topology is Goresky and MacPherson's link to the theory of subspace arrangements. This has lead to applications in complexity theory due to Bjorner, Lovasz and Yao. These recent developments have lead Bjorner and the PI to extend the theory of shellability from pure to nonpure simplicial complexes and posets. Shellability is a combinatorial tool for establishing certain topological and algebraic properties of simplicial complexes and posets. The nonpure setting provides for a richer and more powerful theory that the PI plans to continue to explore and develop. This research is in the general area of Combinatorics. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.