Speaker: John Holbrook (University of Guelph, Canada)

Title of the talk: Matrix means and statistics in spaces of negative curvature

Abstract: Some years ago (with Rajendra Bhatia) we studied the problem of geometric mean for several positive definite matrices, focusing on the Cartan centroid interpretation. The question of monotonicity for this notion of geometric mean remained puzzling. In a striking recent development, it turns out that monotonicity follows from an elegant probabilistic argument (Lawson-Lim, Sturm, Bhatia-Karandikar). We'll describe this development, giving the necessary background concerning laws of large numbers in a space of non-positive curvature (defined by a semi-parallelogram law). As time permits, we'll mention some applications and survey related binary procedures that converge to various matrix geometric means (Ando-Li-Mathias, Bini et al., Palfia). In particular we suggest that “God does not place dice” in this matter. Instead, there appear to be simple deterministic procedures that converge to the Cartan centroid.