In the last fifteen years, the question of embedding metric spaces into “nice” Banach spaces has gathered researchers coming from very diverse origins, in particular from theoretical computer science, geometry of Banach spaces and geometric group theory. A very natural motivation is to aim at a better understanding of complicated metric spaces, such as graphs with a high number of vertices and edges, by embedding them into well understood Banach spaces, like Hilbert spaces.
In the so-called non linear geometry of Banach spaces, which is the subject of this talk, the approach is somewhat reversed. The starting point is to try to exhibit the linear properties of Banach spaces that are stable under some particular non linear maps. These non linear maps can be of very different nature: Lipschitz isomorphisms or embeddings, uniform homeomorphisms, uniform or coarse embeddings. Then, the next goal is to characterize these linear properties in purely metric terms. Usually, these characterizations are given by the (non)embeddability of special metric spaces, which are very often fundamental metric trees or graphs.
Finally, this can motivate the study of these properties, whose linear character can now be forgotten, in the larger setting of metric spaces. We will try to illustrate these ideas with a few results and open questions. We will explain what is known as the “Ribe Program”, which aims at characterizing local properties of Banach spaces (properties of their finite dimensional subspaces) in purely metric terms. This will be illustrated with famous results by J. Bourgain or A. Naor. If time allows, we will describe a recent line of research: looking for metric characterizations of “asymptotic properties” of Banach spaces. This subject could rightfully be named the “Kalton program”.