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.
