Winter School 2014

Antonio Avilés, Jaroslav Nešetřil, Dilip Raghavan, Matteo Viale


The program with titles and slides is available here.

This year the tutorial lectures (each tutorial comprising of several 50 minute lectures) were devoted to advanced topics of Combinatorial Set Theory. They were delivered by leading experts.

Professor Antonio Avilés (Murcia, Spain) presented some recent results in collaboration with Stevo Todorcevic. The results are about understanding how a finite number of incompatible hereditary families of infinite subsets of the natural numbers interact. When these families are definable, for instance analytic, there are finite combinatorial structures that rule this interaction, imposing surprising constrains.

Professor 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, graphons, graphings and modelings. For further reading see An unified approach to structural limits and limits of graphs of bounded tree depth, or the monographs Nesetril, Ossona de Mendez: Sparsity, Springer 2013 and Hell, Nesetril: Graphs and Homomorphisms, OUP

Professor 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. (see also a pdf version of his abstract with citations).

Professor Matteo Viale (University of Torino, Italy) developed in his tutorial the theory of iterated forcing by means of commutative systems of complete homomorphisms of complete and atomless boolean algebras. This approach is equivalent to the standard development of iterated forcings but he aimed to show that it is much simpler to handle. It has already been pursued in the nineties by Donder and Fuchs, however few of their results have been published and their work remained largely unaccomplished. In the tutorial the direct and inverse limit of iterated systems of complete boolean algebras was defined. That was then followed by the definition of proper and semiproper iterations and sketch the proof of the preservation of properness under countable support iterations and of semiproperness under RCS iterations.

The research presentations were given mainly by participants who presented their newest results. The topics included General Topology, Set Theory, Boolean Algebras and Forcing, Descriptive Set Theory, Functional Analysis and the Theory of ultrafilters.

Each day participants had time to discuss their work with colleagues. The topics of the discussions were diverse and included recent developments in set theory.


Giorgio Audrito, Antonio Avilés, Jaroslav Šupina, Bohuslav Balcar, Taras Banakh, Adam Bartoš, Wojciech Bielas, Marek Bienias, Christina Brech, Filippo Calderoni, Jonathan Cancino-Manríquez, Gemma Carotenuto, Raphaël Carroy, David Chodounsky, Katarzyna Chrząszcz, Scott Cramer, Michal Doucha, Ohad Drucker, Piotr Drygier, Peter Eliaš, Barnabas Farkas, Rafal Filipow, Jana Flašková, Kevin Fournier, Miguel Angel Gaspar Arreola, Szymon Glab, Jan Grebík, Fiorella Guichardaz, Gabriele Gullà, Oleg Gutik, Osvaldo Guzman, KP Hart, Clayton Suguio Hida, Peter Holy, Armand Idarraga Lopez, Joanna Jureczko, Piotr Kalemba, Paweł Klinga, Tomasz Kochanek, Michał Korch, Zdzisław Kosztołowicz, Wieslaw Kubis, Adam Kwela, Robert Leek, Richard Lupton, Arturo Antonio Martínez Celis Rodríguez, Gregory McKay, Andrea Medini, Diana Carolina Montoya Amaya, Frederic Morneau-Guerin, Jaroslav Nešetřil, Ana Njegomir, Iryna Pastukhova, Honza Pequignot, Claribet Piña, Max Pitz, Grzegorz Plebanek, Szymon Plewik, Oles Potiatynyk, Dilip Raghavan, Ulises Ariet Ramos-García, Cristobal Rodriguez Porras, Růžena Roháčková, Héctor Gabriel Salazar Pedroza, Lynn Scow, Omar Selim, Petr Simon, Damian Sobota, Wojciech Stadnicki, Marcin Staniszewski, Jan Starý, Silvia Steila, Filip Strobin, Jarosław Swaczyna, Anda - Ramona Tanasie, Jonathan Verner, Matteo Viale, Marta Walczyńska, Julia Wódka, Thilo Weinert, Wolfgang Wohofsky


Talks, slides, abstracts

Antonio AvilésBasis problem for analytic multiple gaps I slides
Antonio AvilésBasis problem for analytic multiple gaps II slides
Antonio AvilésBasis problem for analytic multiple gaps III slides
Jaroslav ŠupinaIdeal versions of wQN-space and QN-spaceabstract slides
Taras BanakhPytkeev $\aleph_0$-spaces slides
Wojciech BielasAn example of a rigid superuniversal metric space slides
Marek BieniasSome properties of compact preserving functions slides
Christina BrechOn PID and biorthogonal systemsabstract slides
Jonathan Cancino-ManríquezAnother cardinal invariant for ideals on $\omega$abstract slides
Raphaël CarroyOrdering functionsabstract slides
Scott CramerMeasures and combinatorics on $\lambda^+$abstract slides
Michal DouchaUniversal groups and group structures on the Urysohn spaceabstract slides
Ohad DruckerHjorth Analysis of General Polish Group Actionsabstract slides
Barnabas FarkasAlmost disjoint refinementsabstract slides
Rafal FilipowThe reaping and splitting numbers of nice idealsabstract
Miguel Angel Gaspar ArreolaMinimally generated Boolean algebras and its Stone space slides
Jan GrebíkAsymptotic Density in Generic Extensions slides
Oleg GutikPseudocompact inverse primitive (semi)topological semigroupsabstract slides
Osvaldo GuzmanMAD families and idealsabstract
KP HartSubcontinua of Hstarabstract slides
Peter HolySimplest Possible Locally Definable Wellordersabstract slides
Joanna JureczkoOn some problems concerning strong sequences slides
Paweł KlingaRearranging series of vectors on a small set slides
Wieslaw KubisUltrafilter selection properties slides
Adam KwelaOn a new $F_\sigma$ idealabstract slides
Robert LeekRadiality and compactificationsabstract slides
Richard LuptonThe Filter Dichotomy slides
Arturo Antonio Martínez Celis RodríguezPorous sets and Martin numbers. slides
Gregory McKayAbstract well and better quasi-orders.abstract slides
Andrea MediniSeven characterizations of non-meager P-filtersabstract slides
Jaroslav NešetřilHomomorphisms, Structural Ramsey Theory and Limits I slides
Jaroslav NešetřilHomomorphisms, Structural Ramsey Theory and Limits II slides
Jaroslav NešetřilHomomorphisms, Structural Ramsey Theory and Limits III slides
Honza PequignotWadge hierarchy on second countable spacesabstract slides
Max PitzCompactifications of $\omega^*\setminus\{x\}$abstract slides
Grzegorz PlebanekOn small measures on Boolean algebras and compact spacesabstract slides
Dilip RaghavanConstructing special almost disjoint families I slides
Dilip RaghavanConstructing special almost disjoint families II slides
Dilip RaghavanConstructing special almost disjoint families III slides
Ulises Ariet Ramos-GarcíaExtremal disconnectedness and ultrafiltersabstract slides
Héctor Gabriel Salazar PedrozaShelah's Easy Black Boxabstract slides
Lynn ScowRamsey classes of treesabstract slides
Damian SobotaOn tightness in the space of measures on Boolean algebras and compact spaces slides
Wojciech StadnickiGuessing principles in the Mathias model
Marcin StaniszewskiOn ideal equal convergenceabstract
Jan StarýUltraproduct of measures
Filip StrobinLarge free subgroups of automorphisms groups of ultrahomogeneous spacesabstract slides
Jonathan VernerThe last talk slides
Matteo VialeIterated forcing defined by means of Boolean algebras I slides
Matteo VialeIterated forcing defined by means of Boolean algebras II slides
Matteo VialeIterated forcing defined by means of Boolean algebras III slides
Marta WalczyńskaEmbeddability properties of $\sigma$-discrete metrizable spaces slides
Thilo WeinertPartition Relations for Linear Orders in a non-Choice-Context slides
Wolfgang WohofskyStrong measure zero and meager-shiftable in topological groups
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