Research in Algebraic Combinatorics

摘要：PI继续她目前在代数组合学方面的研究，专注于偏序集和图复合体的拓扑和代数方面。 这涉及到进一步发展强大的工具，如可壳性和纤维定理。 这些工具已经被PI和她的合作者使用和增强，用于研究一些重要的例子，包括与划分格相关的偏序集和与匹配复合体相关的图复合体。 研究拓扑方面的偏序集成长的著名1964年论文罗塔的莫比乌斯功能的偏序集。 它提供了一个深刻的和基本的联系之间的组合数学和其他分支的数学，如拓扑，代数和几何。 它在计算机科学中也有应用。 复杂性理论、纽结理论、群论和离散几何的研究激发了人们对研究图复形拓扑的兴趣，这是一个与偏序集拓扑密切相关的主题。 组合数学的目标之一是找到有效的方法来安排，枚举和操纵离散的对象集合。 离散系统的行为对现代通信和计算机系统极其重要。 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。

ABSTRACT:The PI continues her current research in algebraic combinatorics, focusing on topological and algebraic aspects of partially ordered sets and graph complexes. This involves further development of powerful tools such as shellability and fiber theorems. Such tools have been used and enhanced by the PI and her collaborators for the purpose of studying some important examples which include partially ordered sets related to the partition lattice and graph complexes related to the matching complex. The study of topological aspects of partially ordered sets grew out of the famous 1964 paper of Rota on the Moebius function of a partially ordered set. It provides a deep and fundamental link between combinatorics and other branches of mathematics such as topology, algebra and geometry. It also has applications in computer science. Research in complexity theory, knot theory, group theory and discrete geometry has inspired interest in studying the topology of graph complexes, a subject closely related to the topology of partially ordered sets.This research is in the general area of combinatorics. One of the goals of combinatorics is to find efficient methods for arranging, enumerating and manipulating discrete collections of objects. The behavior of discrete systems is extremely important to modern communications and computer systems. 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.