Georgia Institute of Technology College of Sciences

A matroid is a structure that captures the notion of "independence". For example, given a set of vectors in a vector space, one can define a matroid. Graphs also naturally give rise to matroids. I will talk about various simplicial complexes associated to matroids. These include the "matroid complex", the "broken circuit complex", and the "order complex" of the associated geometric lattice. They carry some of the most important invariants of matroids and graphs. I will also show how the Bergman fan and its refinement (which arise in tropical geometry) relate to the classical theory. If time permits, I will give an outline of a recent breakthrough result of Huh and Katz on log-concavity of characteristic (chromatic) polynomials of matroids. No prior knowledge of the subject will be assumed. Most of the talk should be accessible to advanced undergraduate students.