Winter School

Invited lectures at 48th Winter School

Richard M. Aron, Department of Mathematical Sciences, Kent State University, Kent, Ohio, USA

    3 lectures

Norm attaining operators
We describe work with D. Garca, D. Pellegrino, and E. Teixeira studying the so-called weak maximizing property (wmp) of operators between Banach spaces. A pair of Banach spaces $(E,F)$ has the wmp means that if $T\in\mathcal L(E,F)$ is such that it has a non-weakly null norm-one maximizing sequence, then $T$ attains its norm. We investigate the relation of this property to norm attainment of linear and polynomial operators.

Analytic structure in fibers of $\mathcal H^\infty(B_{c_0})$
We describe recent work with V. Dimant, S. Lassalle, M. Maestre, and others about the fiber structure of the set of homomorphisms (i.e. maximal ideal space) for the algebras $A_u(B)$, resp. $H^\infty(B)$, of uniformly continuous holomorphic functions, resp. bounded holomorphic functions, on the open unit ball $B$ of a complex Banach space $X$. We pay particular attention to the case $X=c_0$, providing (reasonably) good, complete information about cluster sets, the structure of the fibers, and Gleason parts for these algebras.

Olga Maleva, School of Mathematics, University of Birmingham, Birmingham, UK

    3 lectures

A dichotomy of sets via typical differentiability
The classical Rademacher Theorem guarantees that every set of positive Lebesgue measure in a finite-dimensional Euclidean space contains points of differentiability of every Lipschitz (real-valued) function defined on the whole space. A major direction in geometric measure theory research of the last two decades was to explore to what extent this is true for Lebesgue null subsets of finite-dimensional spaces. First, Zahorski showed in the 1940s that for any null subset N of R there is a Lipschitz function defined on R nowhere differentiable in N. In contrast, Preiss proved in 1990 that every finite-dimensional space of dimension at least 2 has Lebesgue null subsets S such that every Lipschitz function on the whole space has points of differentiability in S. Sets with the latter property are called universal differentiability sets (UDS), and examples with additional important features (such as closed of Minkowski dimension 1) of such sets were constructed in joint works with Dore and with Dymond.
But if there exists a Lipschitz function nowhere differentiable on a given set N, one naturally wonders what happens with a typical (in the sense of Baire category) Lipschitz function on N. In these lectures, I will present a recent joint work with Dymond, where we show how a question of differentiability of a typical Lipschitz function inside a given analytic subset of a finite-dimensional space is settled. Namely, we give a complete characterisation of the sets in which a typical 1-Lipschitz function has points of differentiability: these are the sets which cannot be covered by an F-sigma purely unrectifiable set. We also show that for all remaining sets a typical 1-Lipschitz function is nowhere differentiable, even directionally, at each point. The proof involves a topological Banach-Mazur game, an introduction to which will also be given in these lectures.

Jos Orihuela, Departamento de Matemticas, Universidad de Murcia, Murcia, Spain

    3 lectures

Compactness, Optimality and Risk
One of the most important achievements in optimisation in Banach space theory is the James's 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's 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's 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. Prez. (2017), finishing with new versions of the classical James's theorem (preprint 2019).
In the second lecture we shall provide techniques for a proof of Theorem 1 in arbitrary Banach spaces. Our approach comes from Ruz Galn and Simons and it goes back to the Pryce's undetermined function technique. We will show the strong connection of James theorem with variational principles and optimisation 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 Ruz Galn, Simons, Calvert and Fitzpatrick we have collected in a joint work with M. Ruz Galn (2012). Moors deserves special mention here since he has recently obtained a closely related variational principle too. For a coercive function $\alpha$ with non-empty Mackey-interior for $\partial \alpha(E)$ the level sets $\{\alpha \leq c\}$ are weakly compact too (preprint 2019).
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. Ruz Galn. We shall study the Mackey topology $\tau(\LL^\infty, \LL^1)$ and a new characterisation 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 2019). 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 necessarily bounded) subsets of a dual Banach space $E^*$. One-sided versions of classical Godefroy's results will be presented and new applications considered. Some of them comes from the same joint work with B. Cascales and A. Prez (2017).

Antonn Slavk, Katedra didaktiky matematiky, Matematicko-fyzikln fakulta, Univerzita Karlova, Praha, Czech Republic

    2 lectures

Kurzweil-Stieltjes integral
We provide a basic overview of the Kurzweil-Stieltjes integral on the real line. This nonabsolutely convergent integral generalises the well-known Henstock-Kurzweil integral, and is useful in the study of ordinary differential equations with discontinuous solutions. We focus on existence theorems, relations to other Stieltjes-type integrals, etc.

Regulated functions
Regulated functions are ubiquitous in the Kurzweil-Stieltjes integration theory. We present some equivalent definitions of regulated functions, and provide a characterisation of relatively compact sets in the Banach space of regulated functions (a counterpart to the Arzela-Ascoli theorem). Finally, we describe continuous linear functionals on the space of regulated functions (analogue of the Riesz representation theorem).