Isometry groups on Banach spaces

Abstract:

The following question will be addressed:

How different can a general separable Banach space be from the Hilbert space? After listing some classical properties of the Hilbert space and discussing in which form they may or may not extend to general separable Banach spaces, I shall concentrate on Mazur's rotation problem: is the Hilbert space the only separable infinite dimensional Banach space for which the isometry group acts transitively on the sphere?

Some basic ideas from renorming theory and from descriptive set theory of groups will be presented and applied to the study of isometry groups on Banach spaces. I shall then show how these may be used, together with classical notions of Banach space theory as well as more recent results of Gowers and Maurey, to solve questions posed by Wood and by Deville-Godefroy-Zizler about different forms of transitive actions of isometry groups.