Research in Algebraic Combinatorics

Wachs 9701407 PI打算继续她目前在代数组合学方面的研究，重点是偏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，它在组合学和其他数学分支（如拓扑、代数和几何）之间提供了深刻而基本的联系。一个主要的推动力，这项研究已经来自斯坦利的考虑组行动的同源性偏序集在1982年的一个开创性的文件，另一个来自比约纳的开创性文件shellable偏序集，还有一个来自其基本的联系与理论的超平面安排所开发的Orlik和所罗门。一个更近的和极其重要的发展理论的偏序集拓扑是Goresky和麦克弗森的联系理论的子空间安排。这导致了复杂性理论的应用，由于Bjorner，Lovasz和姚明。这些最近的发展导致比约纳和PI扩展理论的壳性从纯非纯单纯复体和偏序集。 可壳性是一种组合工具，用于建立单纯复形和偏序集的某些拓扑和代数性质。非纯粹设置提供了一个更丰富和更强大的理论，PI计划继续探索和发展。 这项研究是在组合数学的一般领域。组合数学的目标之一是找到有效的方法来研究如何安排对象的离散集合。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，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。

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.