Series of 3 lectures: Non-linear geometry of Banach spaces is a relatively new but rapidly expanding field of research. One of its goals is to determine how much of the linear structure of a Banach spaces can be recovered from a knowledge of its metric or its uniform structure. In other words, if two Banach spaces are Lipschitz-isomorphic, or merely uniformly isomorphic, are they linearly isomorphic? Or at least, do they share some significant isomorphic properties? We will devote three lectures to this field, which will be organized as follows. 1. The free space of a metric space.A simple and canonical construction permits to associate to every metric space $M$ a Banach space $\mathcal{F}(M)$ in such a way that Lipschitz maps between metric spaces become linear maps between their corresponding free spaces. We will display this construction and see how to use the free spaces. In particular, they will provide canonical examples of Lipschitz isomorphic (non separable!) Banach spaces which are not linearly isomorphic. 2. The Gorelik principle.The Gorelik principle is a topological result which roughly asserts that a Lipschitz isomorphism from $X$ to $Y$ maps a large enough ball of any subspace of finite codimension of $X$ to a subset of $Y$ of «compact codimension». When properly used, this principle provides a substitute to the fact that Lipschitz isomorphisms cannot be transposed to isomorphisms between the dual spaces. We will see how to use it for showing that a space which is Lipschitz-isomorphic to $c_0$ is actually linearly isomorphic to $c_0$. 3. Approximation properties.A proper use of free spaces show that Grothendieck’s bounded approximation property (BAP) is stable under Lipschitz-Isomorphisms. We will further study the interplay between BAP and extension of Lispchitz maps. For instance, a canonical approach permits to relate the existence of a Banach spaces failing BAP and the existence of Banach-space valued Lipschitz functions on a subset of a metric space $M$ which do not admit Lipschitz extensions to $M$. In all these lectures, related open questions will be recalled and commented. An effort will be made to avoid technicalities.
Series of 3 lectures: One of the most important achievements in optimization in Banach space theory is the James' weak compactness theorem. It says that a weakly closed subset $A$ of a Banach space $E$ is weakly compact if, and only if, every linear form $x^*\in E^*$ attains its supremum over $A$ at some point of $A$. We propose a tour around it with three lectures in the natural framework of variational analysis. Of course, we shall bring related open questions in every one of them. We will concentrate on recent extensions of James' theorem. Among them we shall study the following one: Theorem 1 Let $A$ be a closed, convex, bounded and not weakly compact subset of a Banach space $E$. Let us fix a convex and weakly compact subset $D$ of $E$, a functional $z^*_0 \in E^*$ and $\epsilon>0$. Then there is a linear form $x^*_0\in B_{p_W}(z^*_0, \epsilon)$, i.e. $$|x^*_0(d)-z^*_0(d)|<\epsilon$$ for all $d\in D$, which does not attain its supremum on $A$. Moreover, if $z^*_0(A)<0$ the same can be provided for the former non attaining linear form : $x^*_0(A)<0$ (one sided James' theorem). In the first lecture we shall concentrate in the case of Banach
spaces with $w^*$-sequentially compact dual unit ball. We shall present
one-sided versions of the well known results by Bishop and Phelps, Simons,
Fonf and Lindenstrauss, which play their job and go back to ideas of a joint
work with B. Cascales and A. P'erez. (2017)In the second lecture we shall provide techniques for a proof of
Theorem 1 in arbitrary Banach spaces. Our approach comes from Ru'iz
Gal'an and Simons and it goes back to the Pryce' undetermined function
technique. We will show the strong connection of James theorem with
variational principles and optimization theory. In order to do it, we will
study unbounded versions of the former results. The first case should be the
epigraph of a weakly lower semicontinuous function $$\alpha: E\longrightarrow
(-\infty, +\infty],$$ where we shall see that $\partial \alpha(E)=E^*$ if, and
only if, the level sets $\{\alpha \leq c\}$ are weakly compact (the Fenchel
conjugate $\alpha^*$ should be finite for the "if" implication), which goes
back to ideas of Ru'iz Gal'an, Simons, Calvert and Fitzpatrick we have
collected in a joint work with M. Ru'iz Gal'an (2012). Moors deserves
special mention here since he has recently obtained a closely related
variational principle too.In the third lecture we shall present some new applications. We will see
that reflexive spaces are the natural frame to develop variational analysis
and we will show a robust representation theorem for a {\it risk measure}
$\rho: \LL^\infty\longrightarrow \R$ in natural dual pairs appearing in
financial mathematics, both applications are based in a joint work with M.
Ru'iz Gal'an. We shall study the Mackey topology $\tau(\LL^\infty, \LL^1)$
and a new characterization for risk measures $\rho$ verifying the Lebesgue
dominated convergence theorem, as the expectated value does. The proof of
Theorem \ref{theo} together with these applications have been obtained in
joint work with F. Delbaen and T. Pennanen ( preprint 2018). We shall finish
our programme with $\sigma(E^*,E)$ versions of the discussed results. Indeed,
we shall look for conditions that provide $\sigma(E^*,E)$-closedness of norm
closed convex (not necessarely bounded) subsets of a dual Banach space $E^*$.
One-sided versions of classical Godefroy' results will be presented and new
applications considered. Some of them comes from the same joint work with B.
Cascales and A. P'erez (2017).
Series of 3 lectures: The classical theory of harmonic analysis and dynamical systems is usually restricted to actions of locally compact groups but in recent years, some of the theory has been extended to more general Polish groups. In this series of talks, I will concentrate on closed subgroups of S_infty, which is perhaps the best understood class of non-locally compact groups in this context, and in the study of which model-theoretic tools play an important role. I will explain some aspects of the theory concerning the unitary representations of some of those groups and their measure-preserving actions.
Series of 2-3 lectures: An equivalence relation $E$ on a Polish space $X$ (e.g., $X := \mathbb{R}$, $L^1(\mathbb{R})$, $\mathbb{N}^\mathbb{N}$) is \emph{Borel} if it is a Borel subset of $X^2$, and it is \emph{countable} if each $E$-class is countable. Countable Borel equivalence relations (CBERs) naturally arise as orbit equivalence relations of Borel actions of countable groups (e.g., $\mathbb{Q} \curvearrowright \mathbb{R}$ by translation). From another, rather combinatorial, angle, a CBER $E$ can always be viewed as the connectedness relation of a locally countable Borel graph (e.g., take $E$ as the set of edges). These connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability, descriptive graph combinatorics, and geometric group theory. These lectures will feature this interplay. On one hand, we will learn some tools from each of the aforementioned subjects to analyze the structure of CBERs. On the other hand, we will utilize the basic theory of CBERs to prove some well-known results in those subjects |