Seminar talk by Dr. A. Dochtermann
Title: Edge ideals, resolutions, and spaces of graph homomorphisms
Abstract: A common theme in combinatorics is to take a discrete object and associate to it some algebraic or topological structure. The idea is that one can then use tools from one discipline(s) to answer questions in the other.
A simple example of the algebra/combinatorics connection is the `edge ideal' I(G) of a graph (or hypergraph) G, where one works in a polynomial ring and constructs an ideal generated by monomials corresponding to the edges of G. The hope is that one can read off algebraic invariants of I(G) such as projective dimension, Betti numbers, etc. in terms of combinatorial properties of G. In the best case scenario we can even obtain a `minimal resolution' of I(G), a powerful description of I(G) analogous to the presentation of a (abelian) group. The study of edge ideals (and more general monomial and binomial ideals) has been an active area of research in recent years.
Next we bring in some topology. Motivated by a construction of Lovasz from the 70s, we describe a way to associate to any labeled graph G a topological space (CW-complex) E(G) whose 0-cells index the directed edges of G. The space E(G) is a directed version of a more general `complex of homomorphisms' Hom(T,G) used to obtain lower bounds on the chromatic number of G. We show how the algebraic complex computing cellular homology of E(G) can be used to obtain a minimal resolution of the ideal I(G), in the case that G is the complement of an `interval graph'. This extends and generalizes constructions of other authors and also leads to other applications of combinatorial topology to commutative algebra.