Departmental Colloquium

Date: March 28, 2024

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Persi Diaconis, Stanford University

Title: Hyperplane walks

Abstract: Picture a collection of hyperplanes in d-dimensional Euclidean space. These divided space into chambers (points not on any of the hyperplanes) and faces (points on some hyperplanes). The geometry and combinatorics of such arrangements is a world of its own, with applications in topology,algebraic geometry and every kind of algebra. I'll supplement this by introducing a simple family of random walks on the chambers. These include classical walks (Ehrenfest urn, card shuffling, dynamic storage allocation) but also lots of fresh examples(walks on parking functions!). Strangely, in more or less complete generality, there is a complete theory (all eigenvalues of the associated transition operators and sharp rates of convergence to stationarity--known). Naturally, there are open problems-- Understanding the stationary distribution of these walks involves the classical problem of sampling from an urn without replacement in various guises and there is a lot we don't know. I'll try to explain all this to a non-specialist audience.