There has been considerable interest in the problem of whether every metric space of bounded geometry coarsely embeds into a uniformly convex Banach space due to the work of Kasparov and Yu that established a connection between such embeddings and the Novikov conjecture. Brown and Guentner were able to prove that a metric space with bounded geometry coarsely embeds into a reflexive Banach space. Kalton significantly extended this result to stable metric spaces and asked whether these classes are coarsely equivalent, i.e. whether every reflexive Banach space coarsely embeds into a stable metric space. Baudier introduced the notion of upper stability, a relaxation of stability, for metric spaces as a new invariant to distinguish reflexive spaces from stable metric spaces. In this talk, we show that in fact, every reflexive space is upper stable and also establish a connection of upper stability to the asymptotic structure of Banach spaces. This is joint work with F. Baudier and Th. Schlumprecht.
