Gilles Lancien
Szlenk indices. Applications to renorming theory and to the non linear classification of Banach spaces.


In this series of lectures we will introduce the Szlenk index of a Banach space and cover some of the applications of this notion. We will start with a brief description of its historical applications: the non existence of universal spaces and the isomorphic classification of $C(K)$ spaces where $K$ is a countable compact space. Then we will show how it is related with locally uniformly rotund and uniformly Kadec-Klee renormings. Finally we will see how it can be used as an invariant under uniform homeomorphisms between Banach spaces.