The PI continues her investigation of algebraic and topological aspects of simplicial complexes associated with partially ordered sets (posets) and monotone graph properties. The theory of poset topology, which grew out of the famous 1964 paper of Rota on the Mobius 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. The significance of the topological study of monotone graph properties was first demonstrated in Kahn, Saks, and Sturtevant's 1984 proof of the prime power case of the evasiveness conjecture in algorithmic complexity theory. There are three parts to this project. In the first part, the PI explores connections between three fascinating combinatorial complexes, which have appeared in the literature in various contexts; namely the no-perfect matching complex, the 1 mod k partition poset and a generalization of the Whitehouse tree complex. In the second part, the PI continues her study of the matching complex, the chessboard complex and variations. These complexes arose in diverse settings such as group theory, discrete geometry and commutative algebra. In the third part, the PI continues her work on some intriguing conjectures of Hanlon dealing with Lie algebra homology. It is expected that the research in all three parts will involve the development of new techniques in topological and algebraic combinatorics. Algebraic combinatorics is an area of mathematics that seeks to establish connections between combinatorics and fields of pure mathematics that involve algebra. These interdisciplinary connections serve to enrich and advance combinatorics and the other fields. Combinatorics is the science of counting, arranging and analyzing discrete configurations. Communications networks and phylogenic trees are examples of a fundamental discrete configuration called a graph. Graphs and other discrete configurations arise in various fields of mathematics, computer science, physics and biology. Combinatorial methods are playing an expanding role in these fields.
