Groups and Dynamics Seminar

Given a discrete subset P of a metric space X, the Voronoi tessellation decomposes X into tiles according to the nearest point of P. This construction appears in several guises. When P is an orbit of a discrete group Γ, the tiles are Dirichlet domains, and good Dirichlet domains can in turn certify that Γ is discrete. When P is a random point set, such as a Poisson point process, one obtains a random tessellation. When P is a labeled data set, the tiles are the decision regions of the 1-nearest-neighbor classifier from machine learning. We study these tessellations when X = G/K is a nonpositively curved symmetric space of higher rank, with P arising from any of these motivations. Besides the invariant Riemannian metric, such spaces carry other natural G-invariant Finsler metrics, which turn out to be well suited to transferring ideas from rank one (negative curvature) to higher rank. This talk will feature joint work with Colin Davalo and Anna Wienhard.