Title:  Independence Complexes, Matching Trees, and Discrete Morse Theory

Abstract:  An independent set in a combinatorial graph is a subset of vertices that are pairwise non-adjacent.  Since set independence is preserved under deletion of elements, we can construct a simplicial complex from the independent sets of a graph of interest.  In this talk, we will survey some of the tools from discrete Morse theory that are useful for analyzing these so-called independence complexes.  We will also discuss some recent results pertaining to the independence complex of small grid graphs including, but not limited to, a particular algorithm giving cell-counting recursions that connect back to some interesting combinatorial sequences.