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

3 lectures

We describe work with D. García, 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.

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

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.

**José Orihuela**, Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

3 lectures

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:

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**Antonín Slavík**, Katedra didaktiky matematiky, Matematicko-fyzikální fakulta, Univerzita Karlova, Praha, Czech Republic

2 lectures

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 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).