Seminar

A complete geodesic metric space of non-positive curvature in the sense of Alexandrov is called a Hadamard space. We will explain why Hadamard spaces have sharp metric cotype 2, thus producing the first large class of nonlinear metric spaces for which the optimal metric cotype has been evaluated. As a consequence, we will conclude that there exist metric spaces (even reflexive Banach spaces) which do not admit a coarse embedding into any Hadamard space, in sharp contrast to the situation for Alexandrov spaces of non-negative curvature. Time permitting, we will discuss an extension of this result to a more general class of metric spaces which includes $p$-convex Banach spaces, and provide an application on the bi-Lipschitz distortion of $\ell_\infty$-grids in $L_p$ for $p\in(2,\infty)$. The talk is based on joint work with M. Mendel and A. Naor.