Coloring Complexes

Meeting Time: Nov. 16, 2010, 2:00-2:50pm

Abstract: This talk will serve as an introduction to topological combinatorics and demonstrate connections with commutative algebra. In addition, we present an application of Einar Steingrimsson's to graph theory. Simplicial complexes are built out of vertices, edges, triangles, tetrahedra, and their higher-dimensional counterparts. Associated to a simplicial complex is a vector called the $f$-vector whose coordinates count the number of faces of each dimension. Understanding $f$-vectors, and in particular, which vectors arise as $f$-vectors is an important area of study in topological combinatorics. Also associated to a simplicial complex is a certain quotient of a polynomial ring called the Stanley-Resiner ring. The Hilbert series of the Stanley-Reisner ring is closely related to the $f$-vector of the simplicial complex. We shall begin by discussing these objects and some of the relationships between them. As a graph theoretic application, we present Steingrimsson's construction of a class of simplicial complexes, each of which is associated to a particular graph, and whose faces are in correspondence with certain illegal graph colorings. By studying the combinatorics and algebra of thise complexes, one can give restrictions upon evaluations of chromatic polynomials.