Topology Seminar: Computation of the homology groups of graph configuration spaces through quantitative representation stability

Abstract: Configuration spaces of graphs have seen an explosion of interest in recent years due to their intriguing theoretical properties, as well as their applicability to problems in physics and robotics. For instance, work of Bibby, Chan, Gaddish, and Yun has shown that the compactly supported cohomology groups of these spaces naturally appears in the study of the tropical moduli spaces of curves. Despite all of this interest, explicit computations of the homologies of these spaces are rare outside of certain special families of graphs. In this talk, we will present an approach to computing the homology groups of graph configuration spaces in certain infinite families of graphs. Our approach involves applying the principals of representation stability to reduce the computation of each family to a finite computation, which we accomplish using discrete morse theory and computer algebra. Through these computations, we come away with a number of interesting conjectures on the interplay between the combinatorics of the graph, and the behaviors of the higher homology groups of its configuration space.

This talk will take place on Zoom and will be broadcast in EH 3866.