# 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.