### Zvaní přednášející

**Jesús María Fernández Castillo**, Universidad de Extremadura, Badajoz, Spain

**Marián Fabian**, Czech Academy of Sciences, Praha, Czech Republic

**Tamás Keleti**, Eötvös Loránd University, Budapest, Hungary

**Stevo Todorčević**, University of Toronto, Toronto, Canada

### Abstrakty

**Jesús M. F. Castillo:** Homological methods in Banach space theory far and beyond

**Marián Fabian:** Asplund story

We present the theory of Asplund spaces from their beginning in 1968 up to now.
We mention properties equivalent to the Asplund property - in particular dentability and the Radon-Nikodym property.
We show examples/counterexamples, open questions, applications in renormings, and in variational analysis.
We also recall adjacent concepts - weak Asplund spaces, Gâteaux differentiability spaces, Asplund-generated spaces, and topological counterparts: scattered and Radon-Nikodym compact spaces.
Related techniques are also presented; in particular separable reductions, rich families, and projectional skeletons.

**Tamás Keleti:** Lipschitz images, measure and dimensions

We study the following general question: given two compact metric spaces X and Y, can we find a Lipschitz map from X onto Y?
An important special case is the following, more than 30 year old open problem of Miklós Laczkovich:
Can every measurable set of positive Lebesgue measure in R

^{n} be mapped onto a ball by a Lipschitz map?
We discuss some partial results, including a proof by Jiří Matoušek for the affirmative answer for n=2.
Since most notions of dimension cannot be increased by Lipschitz maps,
in order to have a Lipschitz map from X onto Y, it is necessary that the (Hausdorff/box/packing) dimension of X is at least the dimension of Y.
It turns out that although the converse is clearly false, a bit stronger condition is already sufficient.
We also study the special case when X and Y are self-similar sets,
and as a spin-off, we find the smallest and the largest "reasonable" dimensions that behave nicely with respect to Lipschitz or bi-Lipschitz maps.

**Stevo Todorčević:** Finite-dimensional amalgamation phenomena in non-separable Banach spaces

Study of possible uncountable structures present in a given class of Banach spaces frequently reduces to solving finite-dimensional amalgamation problems in normed linear spaces.
The series of lectures will try to shed some light on this.
While this is a study that involves multiple mathematical subjects, very basic mathematical background will be assumed.