Program

I will present some results and problems pertaining to classification and complexity of separable Banach spaces, two closely related topics.

• In the first course, I will focus on Gowers' classification program, a research project initiated in the late 1990's, aiming at building a classification of separable Banach spaces "up to subspaces".

• In the second course, I will explain how descriptive set theory can help us to express how difficult it is to classify a given class of separable Banach spaces up to isomorphism, and I will present some related results and questions. One of those questions, introduced by Ferenczi and Rosendal in the early 2000s, asks what a separable Banach space having only few pairwise non-isomorphic subspaces can look like.

• In the third course, I'll present some recent results obtained by Cuellar Carrera, Ferenczi and myself about Gowers-like classifications of some specific classes of separable Banach spaces, and I will show how they can be applied to Ferenczi--Rosendal's forementioned problem.

Let $X$ be an infinite-dimensional, separable and reflexive Banach space, and denote by $\mathcal B_1(X)$ the set of all contraction operators on $X$, i.e. the closed unit ball of the space of all bounded operators on $X$. On $\mathcal B_1(X)$, there are several natural Polish topologies weaker than the operator norm topology (e.g. the Strong Operator Topology, or the Weak Operator Topology). For any of these topologies, it makes sense to ask whether a given interesting operator-theoretic property is "generic'' in the sense of Baire category. For example: does a generic contraction operator on $X$ admit a non-trivial invariant subspace? One may also wonder to which extent the "generic picture'' is modified when one considers different topologies. In this mini-course, I will first focus on some very nice Hilbert space results of Tanja Eisner and Tamas Mátrai, and then try to describe a few results obtained with Sophie Grivaux and Quentin Menet for $\ell_p$ spaces.

To every metric space $M$, it is possible to assign a Banach space $\mathcal{F}(M)$ generated by $M$ in such a way that Lipschitz maps between metric spaces are converted into bounded linear operators between the corresponding Banach spaces. We call Banach space $\mathcal{F}(M)$ the \emph{Lipschitz-free space} over $M$ (also known as the Arens-Eells space or Transportation Cost Space). The above universal property makes Lipschitz-free spaces an important tool in functional analysis because it allows the application of linear methods to nonlinear problems.

In this tutorial, we will provide a brief introduction to Lipschitz-free spaces, their basic properties, and their role in the study of the metric classification of Banach space geometry. Then we will present some recent results on the structure of Lipschitz-free spaces and the representation of their elements, and some open questions.

The structural analysis will take place in a more general framework of all continuous functionals on spaces of Lipschitz functions. We will introduce a notion of supports for such functionals and discuss two types of integral representation. The first representation -- via integration of Lipschitz functions with respect to Borel measures on a compactification of the metric space -- enjoys uniqueness, but is not always possible. We will characterize the measure-induced functionals as those that admit a Jordan-like decomposition into a positive and a negative part, and describe metric spaces for which all functionals are of this form. On the other hand, the second representation -- by integrals over elementary molecules using the de Leeuw's map -- is universal, but not unique. We will therefore look for representing measures that are optimal in both the norm and domain.

The second part of the tutorial will be based on joint work with Ramón J. Aliaga and Richard J. Smith.