Descriptive Set Theory and the Geometry of Banach spacesEquipe d'Analyse Fonctionnelle, Université Pierre et Marie Curie, Paris 6
We shall discuss some recent advances on the interaction between Banach Space Theory and Descriptive Set Theory. We will focus on classical problems in the Geometry of Banach spaces (problems going back to the beginnings of the theory) which are of the following type:
Suppose that (P) is a certain property of Banach spaces. Suppose further that we are given a class C of separable Banach spaces such that every space in the class C has property (P). When can we find a separable Banach space Y which has property (P) and contains an isomorphic copy of every member of the given class C?
As it turns out, for most natural properties, an affirmative answer to the above problem is possible if and only if the class C is "simple". The "simplicity" of the class is measured in set theoretic terms. In particular, if the class C is analytic in a natural coding of separable Banach spaces, then we can indeed find a space Y as required above.
We will present the historical background and overview related results, starting from the seminal work of Jean Bourgain in the 80s which led, eventually, in the solution the last few years.