Geometry/Topology Seminar
Nicholas Wawrykow
University of Chicago
Representation Stability and Disk Configuration Spaces
Abstract: Church-Ellenberg-Farb and Miller-Wilson proved that for a nice enough manifold X and fixed k, the k-th homology group of the ordered configuration space of points in X stabilizes in a representation-theoretic sense as the number of points in the configuration space increases. By fixing a metric on X and replacing points with open unit-diameter disks, we get a new family of configuration spaces where the geometry of X comes to the forefront. One of the simplest of these disk configuration spaces is conf(n,w), the ordered configuration space of unit-diameter disks in the infinite strip of width w. The homology groups of conf(*,w) do not stabilize in the sense of Church-Ellenberg-Farb, Miller-Wilson; however, Alpert proved that when the width is 2 they stabilize in a related sense. Alpert's methods do not extend to larger widths. In this talk I discuss various notions of representation stability, and show that when the width of the strip is at least 2, the rational homology groups of conf(*,w) stabilize in a representation-theoretic sense.
Wednesday October 11, 2023 at 3:00 PM in 636 SEO