Winter School

Invited speakers

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


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 Rn 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.