Invited lectures at the 51st Winter School
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.