Higher medians for higher rank spaces

The notion of a "coarse median" on a metric space, introduced by Bowditch, is a non-positive curvature condition that generalizes both metric trees and Euclidean spaces. Answering a question of Bowditch, Haettel showed that the only higher-rank symmetric spaces that admit a "coarse median" are products of rank-one, e.g. H^2 times H^2. In this talk, we will introduce the notion of a "coarse higher median" and show that many irreducible higher-rank symmetric spaces admit such higher medians. This is a work in progress with Grazia Rago.