Abstract
The tutorial lectures (each tutorial comprising of several 50 minute lectures), delivered by leading experts, were devoted to advanced topics of Combinatorial Set Theory. Prof.
Antonio Avilés (Murcia, Spain) presented recent results in collaboration with Stevo Todorcevic. The results deal with a finite number of incompatible hereditary families of infinite subsets of the natural numbers.
When these families are definable, for instance analytic, there are finite combinatorial structures that rule this interaction, imposing surprising constrains. Prof.
Jaroslav Nešetřil (Charles University, Prague) surveyed recent development showing fine interplay between Homomorphisms, Structural Ramsey Theory and Limits. Some of the key words mentioned in his lectures were: density and universality of homomorphism order, forbidden homomorphisms and dualities, Ramsey classes and ultrahomogeneous structures, limits and FO limits.
Prof. Dilip Raghavan (National University of Singapore) presented techniques for constructing different types of MAD families (Laflamme, tight families and completely separable families).
These techniques were recently devised by Shelah and further developed by the speaker and his collaborators. Prof.
Matteo Viale (University of Torino, Italy) presented the theory of iterated forcing by means of commutative systems of complete homomorphisms of atomless cBAs. This approach has already been pursued in the 90s by Donder and Fuchs, however few of their results have been published.
In the tutorial the direct and inverse limit of iterated systems of cBAs was defined followed by definitions of proper and semiproper iterations and a proof of the preservation of properness under cs iterations and of semiproperness under rcs iterations. The research presentations were given mainly by participants, topics included General Topology, Set Theory, Boolean Algebras and Forcing, Descriptive Set Theory, Functional Analysis and the Theory of ultrafilters.