Research in Algebraic Combinatorics

Wachs9701407 The PI打算继续她目前在代数组合学方面的研究，专注于偏序集的拓扑和代数性质。偏序集拓扑理论起源于1964年Rota关于偏序集的Moebius函数的著名论文，它在组合学和其他数学分支(如拓扑学、代数和几何学)之间提供了深层次和基本的联系。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关于可壳偏序集的开创性论文，还有一个来自它与Orlik和所罗门发展的超平面排列理论的基本联系。偏序集拓扑理论的一个最新、最重要的发展是Goresky和MacFherson对子空间排列理论的链接。这导致了比约纳、洛瓦兹和姚在复杂性理论中的应用。这些最新的发展使得Bjorner和PI将可壳性理论从纯单纯复形和偏序集扩展到非纯单纯复形和偏序集。可壳性是建立单纯复形和偏序集的某些拓扑和代数性质的组合工具。非纯粹的背景提供了一个更丰富、更强大的理论，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有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

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.