DAN

Abstract

The linear properties of Banach spaces considered in this talk will be the ex- istence of an equivalent asymptotically uniformly smooth (or convex) equivalent norm. We shall study the stability of these properties under various non linear transformations, but we will concentrate on the coarse Lipschitz embeddings (i.e. maps that are bi-Lipschitz for very large distances). These questions in relation with uniform asymptotic smoothness are now quite well understood. We will try to present the progress made last year by N.J. Kalton on the stability of uniform asymptotic convexity under coarse embeddings. We will focus on the use of some fundamental metric graphs or trees in the subject, and present a few open questions that we find interesting.