Tutorial Track

S. Müller: Determinacy, Large Cardinals, and Inner Models

The study of inner models was initiated by Gödel’s analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy. This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy. This tutorial outlines the main concepts and results connecting determinacy hypotheses with the existence of mice with large cardinals as well as recent progress in the area, assuming only basic set theoretic background.

slides I, slides II, slides III

J. Steprans: An introduction to some aspects of P-points and related ultrafilters

P-points were originally studied in the context of maximal ideals of rings of continuous functions on various topological spaces, but they have proven to be useful in various applications. Very early results showed that the P-points in the specific space $\beta \mathbb N\setminus \mathbb N$ have a very nice combinatorial characterization as a certain class of ultrafilters on $\mathbb N$. Further study has revealed various alternate characterizations, some of which will be examined in the first lecture. This introductory lecture will also discuss various methods of constructing P-points and related ultrafilters.

The second lecture will examine the forcing arguments needed to create models of set theory without any P-points, while the final lecture will discuss methods for killing some P-points while preserving others. Many questions in this are remain open and I will try to present some of them along with the context for asking them.

The first lecture will not assume any knowledge beyond elementary mathematics. The last two lectures will assume some more sophisticated knowledge of set theory, of the type that is usually covered in an introductory graduate course on set theory.

slides I, slides II

Z. Vydnyánszky: Complexity and Borel Combinatorics

The first lecture will be an introduction to Borel combinatorics, with defining the basic concepts and showing some of the most important examples of Borel graphs. Then, I will discuss how can one use results about projective complexity to refute conjectures, and I will outline proofs of the theorems which describe the complexity of deciding the existence of Borel colorings and homomorphism problems.

The tutorial will be accessible without prior familiarity with Borel combinatorics or projective sets.

lecture notes

A. Zucker: Big Ramsey degrees, structures, and related dynamical phenomena

In the past decade, rapid progress has been made in the understanding of big Ramsey degrees, namely, how much infinite Ramsey theory do various countable, ultrahomogeneous structures satisfy? In particular, by work of Balko, Chodounský, Dobrinen, Hubička, Konečný, Vena, and myself, we now have a complete understanding of the situation for Fraïssé limits of finitely constrained binary free amalgamation classes, i.e. graph-like structures which forbid a specified finite list of clique-like substructures.

In the setting of small Ramsey degrees, works of Kechris--Pestov--Todorčević, Nguyen Van Thé, and myself connect this combinatorial property of a Fraïssé class to dynamical properties of the automorphism group of the Fraïssé limit. This tutorial will survey this connection, discuss recent progress in big Ramsey degrees, and explore the possible connections of big Ramsey degrees to dynamical phenomena.

slides I, slides II, slides III

Research Track

Szymon ŻeberskiEggleston meets Mycielski, measure caseabstract slides
Tomasz ŻuchowskiThe Nikodym property and filters on $\omega$abstract slides
Serhii BardylaCountably compact extensions and cardinal characteristics of the continuumabstract slides
Adam BartošFraïssé-like constructions of compacta slides
Jonathan Cancino ManríquezA model with p-measures and no p-point slides
David ChodounskyImportant closing remarks slides
Aleksander CieślakAntichain numbersabstract slides
Cesar CorralHigh dimensional sequential compactnessabstract slides
Jorge Antonio Cruz ChapitalOn gaps, almost disjoint families and a Ramsey ultrafilter.abstract slides
Matheus Duzi Ferreira CostaGeneralized Krom spaces and the Menger gameabstract slides
Curial Gallart RodríguezStrong chains of subsets of $\omega_1$ of length $\omega_3$abstract slides
Damian GłodkowskiEpic math battle of history: Grothendieck vs Nikodym - round 2abstract slides
Tatsuya GotoKeisler's theorem and cardinal invariants at uncoutable cardinalsabstract slides
Jan HubickaRamsey theorem for trees with sucessor operation slides
Klára KarasováTopological fractals slides
Tamás KátayElusive properties of countably infinite graphsabstract
Jerzy Kąkol$\Delta_1$-spaces and Asplund spaces $C_k(X)$ over $\Delta_1$-spaces $X$abstract slides
Maciej KorpalskiStraightening almost chains in P($\omega$)abstract slides
Wiesław KubiśUncountable ultrahomogeneous structures slides
Borisa KuzeljevicOrderings on P-point ultrafilters slides
Chris Lambie-HansonGeneralized almost disjoint families and injective Banach spaces slides
Arturo Martínez-CelisMarczewski-like ideals related to superperfect trees.abstract slides
Łukasz MazurkiewiczIdeal Analytic Setsabstract slides
Marcin MichalskiEggleston meets Mycielski - category caseabstract slides
Adam MorawskiFew words on P-measures - almost sigma-additive measuresabstract slides
Aleksandar PavlovićCan an ideal give you a maximal space?abstract slides
Daria PerkowskaNon-meager filtersabstract slides
Beatrice PittonThe SLO principle for Borel subsets of the generalized Cantor spaceabstract slides
Robert RalowskiThe Baire theorem, an analogue of the Banach fixed point theoremabstract slides
Calliope Ryan-SmithUpwards homogeneity of symmetric extensions slides
Kamil RyduchowskiLusin sets and uncountable Auerbach systemsabstract slides
Šárka StejskalováForcing over a free Suslin tree slides
Corey SwitzerOn Generic Independent Familiesabstract slides
Tristan van der VlugtCombinatorial κ-reals in the higher Baire spaceabstract slides
Thilo WeinertOn the binary linear orderingabstract slides
Agnieszka WidzEpic math battle of history: Grothendieck vs Nikodym - round 1abstract slides
Wolfgang WohofskyDualizing the distributivity number h? slides
Takashi YamazoeCichoń's maximum with evasion numberabstract slides
Krzysztof ZakrzewskiFunction spaces on Corson-like compactaabstract slides

Scientific committee

David Chodounský, Institute of Mathematics, Czech Academy of Sciences
Jan Grebík, Mathematics Institute, University of Warwick
Chris Lambie-Hanson, Institute of Mathematics, Czech Academy of Sciences