# Tutorial Track

## L. Aurichi: Some games and their topological consequences

Alice and Bob are always playing games. Sometimes Alice plays points and Bob answers with open neighborhoods of the points (Alice wants to force Bob to cover the whole space). Some other times Alice and Bob play things related to density or tightness. Some games are short ($\omega$ innings), some others are longer ($\omega_1$ innings).

We will present some games and discuss the consequences of existence of winning strategies for some of the players.

## J. D. Hamkins: Set-theoretic potentialism

I shall introduce and develop the theory of set-theoretic potentialism. A potentialist system is a collection of first-order structures, all in the same language $\mathcal{L}$, equipped with an accessibility relation refining the inclusion relation. Any such system, viewed as an inflationary-domain Kripke model, provides a natural interpretation for the modal extension of the underlying language $\mathcal{L}$ to include the modal operators. We seek to understand a given potentialist system by analyzing which modal assertions are valid in it.

Set theory exhibits an enormous variety of natural potentialist systems. For example, with forcing potentialism, one considers the models of set theory, each accessing its forcing extensions; with rank potentialism, one considers the collection of of rank-initial segments $V_\alpha$ of a given set-theoretic universe; with Grothendieck–Zermelo potentialism, one has the collection of $V_\kappa$ for (a proper class of) inaccessible cardinals $\kappa$; with top-extensional potentialism, one considers the collection of countable models of ZFC under the top-extension relation; and so on with many other natural examples.

In this tutorial, we shall settle the precise potentialist validities of each of these potentialist systems and others, and we shall develop the general tools that enable one to determine the modal theory of a given potentialist system. Many of these arguments proceed by building connections between certain sweeping general features of the models in the potentialist system and certain finite combinatorial objects such as trees or lattices. A key step involves finding certain kinds of independent control statements — buttons, switches, ratchets and rail-switches — in the collection of models.

## J. Lopez-Abad: Extremely amenable automorphism groups

The goal of this mini course is to present the main points in the study of the topological dynamics of topological groups when represented as automorphism groups of approximate ultrahomogeneous metric structures. More particularly, we will discuss:

• Fraïssé and metric Fraïssé correspondence for (approximate) ultrahomogeneous structures (AuH).

• Representation of polish groups as automorphism groups $\mathrm{Aut}(\mathcal M)$ of (AuH) structures $\mathcal M$.

• The Kechris–Pestov–Todorcevic (KPT) correspondence between extremely amenable groups of automorphisms of (AuH) structures and the approximate Ramsey property.

• The computation of universal minimal flows.

• Examples, including

1. $\mathrm{Aut}(\mathbb Q,<)$,

2. $\mathrm{Aut}(\mathbb B)$, where $\mathbb B$ is the countable atomless boolean algebra,

3. $\mathrm{Aut}(\mathbb G)$, where $\mathbb G$ is the Gurarij space,

4. $\mathrm{Aut}(L_p[0,1])$.

# Research Track

SpeakerTitleAbstract/Slides
Viera ŠottováTBA
Szymon Żeberski
Taras BanakhA parallel metrization theorem
Serhii BardylaOn locally compact semitopological graph inverse semigroupsabstract
Adam BartošCompactifiability and Borel complexity up to equivalenceabstract
Wojciech BielasSeparation axiom for regular closed sets
Andrew Brooke-TaylorProducts of CW complexesabstract
Noé de RancourtRamsey theory with and without the pigeonhole principle
Stamatis DimopoulosCardinal characteristics and strong compactnessabstract
Monroe EskewRigid collapseabstract
Saeed GhasemiAlmost disjoint families and C*-algebrasabstract
Michał Tomasz GodziszewskiSatisfaction classes over models of set theory
Martin GoldsternCichoń's Maximum
Miha HabičSurgery and nonamalgability for Cohen reals
Eliza JabłońskaOn Steinhaus properties and families of "small" sets
Olena KarlovaAlmost strongly zero-dimensional spacesabstract
Michał KorchSpecial subsets of the generalized Cantor space $2^\kappa$abstract
Ziemowit KostanaNon-measurability of the algebraic sums of sets of real numbersabstract
Wieslaw Kubis
Ondřej KurkaLarge separated sets of unit vectors in Banach spaces of continuous functionsabstract
Marta Kwela
Maxwell LevineForcing Square Sequencesabstract
Marcin Michalski
Heike MildenbergerBlock Sequences with Projections into a Sequence of Happy Familiesabstract
Kaethe MindenSubcomplete Forcing, Trees, and Generic Absoluteness
Volodymyr MykhaylyukUpper Namioka property of multi-valued mappingsabstract
Sonia NavarroBorel ideals associated to Ramsey spaces
José de Jesús Pelayo GómezSome combinatorial properties and a cut and choose game
Grzegorz PlebanekMardesic problem on products of compact lines
Robert Rałowski
Olga SipachevaBoolean topological groups and ultrafiltersabstract
Damian SobotaConvergence of measures in forcing extensions
Wolfgang WohofskyNon-stationary topology on $2^\kappa$abstract