Winter School 2013

Michael Hrušák, Jan van Mill, Slawek Solecki, Stevo Todorčević, Jindra Zapletal


This year the tutorial lectures (each tutorial comprising of several 50 minute lectures) were devoted to advanced topics of Set Theory & Descriptive Set Theory, Forcing and Set-Theoretic topology. They were delivered by leading experts.

Professor Michael Hrušák (UNAM, Morelia) reviewed basic forcing and preservation properties of natural partial orders associated with filters and ideals on countable sets. He paid special attention to destructibility of ideals by forcing. As an application he sketched the proof of the solution to Malykhin's problem: Consistently every separable Frechet topological group is metrizable.

Professor Jan van Mill (Vrije Universiteit, Amsterdam) gave a tutorial on the topic of uniquely homogeneous spaces. A (nontrivial) space $X$ is called uniquely homogeneous provided that for all $x,y\in X$ there is a unique homeomorphism of $X$ that takes $x$ onto $y$. This concept is due to Burgess who asked in 1955 whether there exists a uniquely homogeneous metrizable continuum. Barit and Renaud showed in 1978 that there are no locally compact uniquely homogeneous metrizable spaces. There is a nontrivial uniquely homogeneous Baire space of countable weight and it is unknown whether there is a Polish uniquely homogeneous space. He discussed these results and others and presented many fundamental open problems which remain unsolved. For example, whether there exists a nonmetrizable uniquely homogeneous compact space. Such an example, if it exists, would definitely be completely different from all the known examples of homogeneous compacta.

Professor Slawek Solecki (University of Illinois at Urbana-Champaign, Urbana IL) had a series of lectures on the Tukey reduction between directed orders. The notion originated in the theory of convergence in topological spaces. More recently, it proved useful in classification problems and in the study of cardinal invariants of the continuum. He presented an outline of recent research in which Tukey reduction is applied to investigate the structure of the classes of analytic P-ideals of subsets of $\omega$ and of analytic $\sigma$-ideals of closed subsets of compact spaces.

Professor Stevo Todorčević (University of Toronto, Toronto) talked about how forcing with a coherent Souslin tree can be analyzed using a sufficient amount of PFA that allow the existence of such a tree. He also presented a list of few applications to set-theoretic topology.

Professor Jindřich Zapletal (University of Florida, Gainesville) gave an introduction to Idealized forcing. His tutorial was based on his book Forcing idealized. The topics he talked about included: The quotient poset PI of Borel I-positive sets for a sigma-ideal I on a Polish space; Translating forcing properties of PI into analytic properties of I and back; Operations on sigma-ideals vs. operations on forcings; The countable support iteration and product;

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 Filters.

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 Aviles, Bohuslav Balcar, Taras Banakh, Maria Bas, Wojciech Bielas, Marek Bienias, Piotr Borodulin-Nadzieja, David Chodounsky, Michal Dečo, Michal Doucha, Piotr Drygier, Ahmad Farhat, Barnabás Farkas, Jana Flašková, Kevin Fournier, Jocer Franquiz, Joanna Garbulińska, Szymon Glab, Magdalena Grzech, Edward Grzegorek, Osvaldo Guzmán González, K. P. Hart, Michael Hrušák, Jakub Jasinski, Joanna Jureczko, Witold Kisielewski, Zdzislaw Kosztolowicz, Mikolaj Krupski, Alicja Krzeszowiec, Michal Krzeszowiec, Wieslaw Kubis, Andrzej Kucharski, Adam Kwela, Roman Lubczyk, Leon Luo, Arturo Antonio Martínez Celis Rodríguez, Marta Martynenko, Nenad Moraca, Magdalena Nowak, Aleksandar Pavlović, Tomáš Pazák, Yann Pequignot, Claribet Piña, Grzegorz Plebanek, Dušan Radičanin, Robert Ralowski, Cristobal Rodriguez Porras, Růžena Roháčková, Omar Selim, Jeffrey Serbus, Oleg Sidorov, Petr Simon, Sergii Slobodianiuk, Damian Sobota, Slawomir Solecki, Wojciech Stadnicki, Jan Starý, Michal Stas, Šárka Stejskalová, Campo Elías Suárez Villagrán, Jaroslav Supina, Jaroslaw Swaczyna, Slawomir Szczepaniak, Anda Ramona Tanasie, Stevo Todorčević, Daniela Torrealba, Slawomir Turek, Jan van Mill, Jonathan Verner, Zoltan Vidnyanszky, Dimitris Vlitas, Luisa Vogel, Thilo Weinert, Wolfgang Wohofsky, Rachel Alisha Zachariah, Piotr Zakrzewski, Jindřich Zapletal, Lyubomyr Zdomskyy, Szymon Zeberski


Talks, slides, abstracts

Antonio AvilesA new C(K) space with few operators slides
Taras BanakhSolecki submeasures and densities on groups slides
Wojciech BielasGenerating Borel Measurable Functions with Continuous Functions slides
Piotr Borodulin-NadziejaHausdorff towers and gaps slides
Michal DečoStrongly Dominating Sets of Reals slides
Michal DouchaUniversal structures on the Urysohn universal space slides
Ahmad FarhatGenerating compact spaces from trees slides
Barnabás FarkasRepresentations of ideals in Polish groups and in Banach spaces slides
Jana FlaškováSome remarks concerning the van der Waerden ideal slides
Kevin FournierWadge Hierarchy of Difference of Analytic sets slides
Joanna GarbulińskaIsometric uniqueness of a complementably universal Banach space for Schauder decompositions slides
Edward GrzegorekOuter measure and hull property slides
Osvaldo Guzmán GonzálezNamba forcing and set theory of the reals
K. P. HartDimension(s) of compact F-spaces slides
Michael HrušákForcing with filters and ideals I. slides
Michael HrušákForcing with filters and ideals II. slides
Michael HrušákForcing with filters and ideals III. slides
Joanna JureczkoOn existence of independent sets in partially ordered sets slides
Zdzisław KosztołowiczGeneralizations of Strok-Szymański theorem slides
Mikolaj KrupskiSome new observations on the t-equivalence relation slides
Wieslaw KubisUniversal homomorphisms slides
Andrzej KucharskiA normally supercompact Parovicenko space slides
Adam KwelaIdeals represented on Polish spaces slides
Arturo Antonio Martínez Celis RodríguezCardinal invariants of porous sets. slides
Marta MartynenkoOn skeletally factorizable spaces slides
Magdalena NowakTopological classification of countable IFS-attractors slides
Yann PequignotBetween Bqo and Wqo: the space of ideals slides
Grzegorz PlebanekOn isomorphisms and embeddings of C(K) spaces slides
Omar SelimSubmeasures and signed measures slides
Jeffrey SerbusConfigurations of Cardinal Invariants Consistent with the P-ideal Dichotomy slides
Oleg SidorovThe innovative structure of point sets in terms of the next point principle slides
Slawomir SoleckiTukey reduction among analytic P-ideals and $\sigma$-ideals I. slides
Slawomir SoleckiTukey reduction among analytic P-ideals and $\sigma$-ideals II. slides
Slawomir SoleckiTukey reduction among analytic P-ideals and $\sigma$-ideals III. slides
Wojciech StadnickiWallman representations of hyperspaces slides
Jan StarýCompactness of the order-sequential topology slides
Jaroslav SupinaSequence selection principles for functions slides
Sławomir SzczepaniakSome remarks on splittings slides
Stevo TodorčevićForcing With a Coherent Souslin Tree I.
Stevo TodorčevićForcing With a Coherent Souslin Tree II.
Stevo TodorčevićForcing With a Coherent Souslin Tree III.
Sławomir TurekHereditarily supercompact spaces slides
Jan van MillUniquely homogeneous spaces I. slides
Jan van MillUniquely homogeneous spaces II. slides
Jan van MillUniquely homogeneous spaces III. slides
Zoltan VidnyanszkyBorel hulls of shy sets slides
Wolfgang WohofskyCharacterizing Strong Measure Zero Sets in Polish groups slides
Rachel Alisha ZachariahRandomness and Universality in Topological Spaces. slides
Jindřich ZapletalForcing Idealized I. slides
Jindřich ZapletalForcing Idealized II. slides
Jindřich ZapletalForcing Idealized III. slides
Lyubomyr ZdomskyyCovering properties and CDH filters slides
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