Research in Algebraic Combinatorics

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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.