# Tutorial Track

## D. Asperó: Side conditions and (iterated) forcing

I am planning to give a slowly paced introduction to the construction of forcing notions incorporating suitable systems of models as side conditions. Some of these forcing notions will be naturally built as limits of forcing iterations and some will not. The main focus will be on the applications of the method.

slides I, slides II, slides III

## J. Bagaria: An Introduction to Hyperstationary Sets

We shall give an introduction to the theory of hyperstationary sets. Hyperstationarity generalizes the notion of stationary set (i.e., 1-stationarity) and stationary reflection (i.e., 2-stationarity). It turns out that $\xi$-simultaneous-stationary ordinals are precisely the non-isolated points in the $\xi$-th topology on the ordinal numbers, which is obtained by refining the usual interval topology using the Cantor's derivative operator and iterating the refining process $\xi$-many steps. Thus, we will discuss the connections between $\xi$-stationarity, $\xi$-simultaneous-stationarity, derived topologies on the ordinals, and the ideals of non-$\xi$-stationary sets and their dual filters. We will show that in the constructible universe $L$ a regular cardinal is $(\xi +1)$-stationary if and only if it is $\Pi^1_\xi$-indescribable, and we will present some very recent results about the consistency strength of $\xi$-stationarity.

slides I, slides II, slides III

## C. Brech: Generalizing Schreier families to large index sets

Products of the Schreier family have been used in Banach space theory to built important objects such as the Tsirelson space. In order to generalize these constructions to the nonseparable setting, the families had to be generalized to uncountable index sets. The Schreier family is hereditary, compact and spreading but when passing to the uncountable level, we cannot expect them to be spreading if we want to keep compactness, for example.

We will briefly present the link between the families and their corresponding Banach spaces and will focus on how to define and construct those families. The right definition of a product of a family on a large index set by the Schreier family is crucial and there is a weaker and a stronger version of it in terms of the corresponding Cantor-Bendixson index. After introducing these notions, we will present the method (appearing in our recent joint work with Jordi Lopez-Abad and Stevo Todorcevic) to construct such families below the first Mahlo cardinal. It involves defining a family on a tree out of a family on its chains and a family on its antichains, and analyzing the combinatorial structure of it using Ramsey methods.

slides I, slides II, slides III

## A. Marks: Geometrical paradoxes and descriptive set theory

The well-known Banach–Tarski paradox states that the unit ball in $\mathbf R^3$ can be partitioned into finitely many pieces that can be rearranged by rotations and translations to form two unit balls. The study of descriptive-set-theoretic aspects of this paradox has a long history. For example, in 1930, Marczewski asked whether the pieces used in this equidecomposition can have the Baire property. Dougherty and Foreman gave a positive answer to this question in 1994.

Recent progress on matching theorems in the field of descriptive graph combinatorics has shed new light on the general question of when the pieces used in equidecompositions such as the Banach–Tarski paradox may possess various regularity properties such as Lebesgue measurability or the Baire property. We will discuss some of these developments, and applications to the existence of finitely additive invariant measures.

slides I, slides II, slides III

# Research Track

SpeakerTitleAbstract/Slides
Szymon ŻeberskiInvariant sigma-ideals with analytic base on good Cantor measure spacesabstract slides
Taras BanakhThe Interplay between two generalizations of first-countabilityabstract slides
Serhii BardylaOn the semitopological locally compact $\alpha$-bicyclic monoidabstract slides
Adam BartošOn maximal connected I-spacesabstract slides
Judyta BąkDomain representable spaces and topological games slides
Mariam BeriashviliOn some paradoxical point sets in different models of set theoryabstract slides
Jana BlobnerUltrafilters and divergent series slides
Tomasz CieślaA filter on a collection of finite sets. A non-bisequential Eberlein compactum. slides
Sakae FuchinoReflection cardinals of coloring of graphsabstract slides
Szymon GłąbDense free subgroups of automorphism groups of homogeneous partially ordered setsabstract slides
Jan GrebíkFraïssé-like structuresabstract slides
Osvaldo GuzmanLinearly ordered splitting families slides
K P HartBrouwer and cardinalitiesabstract slides
Radek HonzikLaver indestructibility for cardinals smaller than supercompacts slides
Olena KarlovaExtension of Baire-one functionsabstract slides
Adam KwelaIdeal equal Baire classesabstract slides
Marcin MichalskiUniversal sets for $\sigma$-idealsabstract slides
Sonia NavarroPseudointersection numbers for topological Ramsey spaces slides
Magdalena NowakAre you self-similar?abstract slides
José de Jesús Pelayo GómezRamsey porperties and the Katetov order slides
Márk PoórAnswer to a question of Roslanowski and Shelahabstract slides
Damian SobotaThe Nikodym property of Boolean algebras and cardinal characteristics of the continuumabstract slides
Sarka StejskalovaThe tree property at $\aleph_{\omega+2}$ with a finite gapabstract slides
Jarosław SwaczynaIntroduction to Haar-small sets slides
Dorottya SzirákiGames and perfect independent subsets of the generalized Baire spaceabstract slides
Jacek TrybaHomogeneity of idealsabstract slides
Jonathan VernerThe Last Talk slides
Thilo WeinertCardinal Characteristics, Partition Properties and Generalised Scattered Ordersabstract slides
Wolfgang WohofskyCofinalities of Marczewski-type idealsabstract slides