# Tutorial Track

## C. Laflamme: Topics in Combinatorial Set Theory

#### Symmetries of some Homogeneous Structures

slides
Automorphisms may not describe all the possible "symmetries" of the structure. As an example and exercise can you describe all functions on the rationals which map all copies to copies? Homogeneous structures are "highly" symmetric in the sense that any partial finite automorphisms extends to an automorphisms, and we investigate other symmetries for the Rado graph and other homogeneous structures.

#### How many siblings do you have?

slides
Two structures A and B are siblings if each embeds in the other. As an example and exercise can you count / describe all siblings of the rationals (up to isomorphism)? We will discuss the case of all countable linear orders, and conclude with a discussion of the more general case of trees and graphs.

#### Is converging bounded sequences really harder than converging binary sequences?

slides
This is a purely set theoretical problem originating from the work of Vojtas and others in the late 80's; in his notation the question is whether $r=r_\sigma$. As an exercise can you show that these cardinals are indeed equal? I will review some of the history of this problem, exhibiting the frustratingly low level of understanding we still have on this basic combinatorial problem, but will try to conclude with an optimistic outlook.

## D. Milovich: Topological applications of long $\omega_1$-approximation sequences

slides I, slides II, slides III

Every ordinal has a uniformly definable cardinal normal form analogous to its Cantor normal form. As a consequence, given any transfinite sequence of countable elementary substructures of (a large fragment of) the set-theoretic universe, the union of the range of the sequence is a finite union of (possibly uncountable) elementary substructures, provided the sequence satisfies the very weak requirement that every structure in the sequence contains as an element (but not necessarily as a subset) the subsequence of all preceding structures. Such sequences can be loosely thought of as extremely coarse morasses available in ZFC.

The original, implicit application of cardinal normal form is a 1964 ZFC proof by R. O. Davies that the Euclidean plane is a countable union of rotated graphs of functions. (A much earlier construction by Sierpinski required the Continuum Hypothesis.) A 2008 topological application is that every compact group has an open-in-finite base, that is, a topological base such that every nonempty open subset of the group has at most finitely many supersets in the base.

The tutorial will review basic facts about elementary substructures, introduce the cardinal normal form and long $\omega_1$-approximation sequences, prove the fundamental finite union property of long $\omega_1$-approximation sequences, prove the two mentioned applications, and conclude with a brief discussion of more recent and potential future applications.

## J. T. Moore: Iterated forcing and the Continuum Hypothesis

slides I, slides II, slides III

One of the most fundamental questions one can ask about an iterated forcing construction is whether it introduces new real numbers. Clearly it is necessary that the iterands themselves do not introduce new reals, but it is well known that this is not sufficient. This tutorial will develop the theory of iterated proper forcing in the presence of CH through a specific construction: we will show how to construct a model of ZFC in which $\omega_1$ and $-\omega_1$ are the only minimal uncountable order types.

## A. Roslanowski: Properness for iterations with uncountable supports

slides I, slides II, slides III

While there are still open problems in the theory of forcing iterated with finite and/or countable support and we still need to expand our preservation theorems, there is a sense that we understand these iterations pretty well. We use them frequently, even if we tend to skip iterated forcing in favour of the respective forcing axioms (MA or PFA).

It is natural to look at iterations with uncountable supports and ask for parallel theorems. However, it has been known since the beginning of 1980s that there is no straightforward theory of iterations with uncountable supports and cardinals are frequently collapsed in such iterations. Still, there are in literature also positive results concerning not collapsing cardinals in iterations with uncountable supports. For instance, in 1980, Kanamori considered $\lambda$-support iterations of $\lambda$-Sacks forcing notion and he proved that under some circumstances these iterations preserve $\lambda^+$. In 2003, Eisworth discussed iterated forcing for successors of regular cardinals introducing a variant of Shelah's properness over semi-dimonds. Fusion properties of iterations of tree-like forcing notions were used in by Friedman and Zdomskyy in 2010, and then by Friedman, Honzik and Zdomskyy in 2013.

In a series of articles (1,2,3,4,5,6,7) by Shelah and myself we introduced and studied several strong versions of $\lambda$-properness appropriate for iterations with supports of size $\leq\lambda$ of natural forcing notions adding a new member of ${}^\lambda\lambda$ without adding new elements of ${}^{ {<}\lambda} \lambda$. The tutorial will present the motivation, highlights and dependencies among those properties. We will also look at some of the relevant iteration theorems.

# Research Track

SpeakerTitleAbstract/Slides
Giorgio AudritoResurrection axioms and generic absolutenessabstract slides
Jaroslav ŠupinaSequences of continuous and semicontinuous functions slides
Tomasz ŻuchowskiTukey types of orthogonal ideals slides
Adam BartošClosure schemes on topological spacesabstract slides
Will BrianUltrafilters on semifiltersabstract slides
Jonathan Cancino-ManríquezAbout the reaping number of dense subsets of the rationalsabstract slides
Michal DečoIdeals related to Laver and Miller trees slides
Piotr DrygierThe growth of $\omega$ with positive measure slides
Saeed GhasemiIsomorphisms between corona algebras of the form $\prod_{n} M_{k(n)}(\mathbb{C})/\bigoplus_{n} M_{k(n)}(\mathbb{C})$abstract slides
Artur GiżyckiHindman theorem for semigroups slides
Osvaldo GuzmanRemarks on Parametrized diamonds slides
K P HartTBAabstract slides
Michał KorchThe class of perfectly null sets and its transitive versionabstract slides
Konrad KrólickiSome new coanalytic complete collections of continua in cubesabstract slides
Wieslaw KubiśBanach-Mazur games played with arrows slides
Ashutosh KumarAvoiding rational distancesabstract slides
Chris Lambie-HansonPatterns of stationary reflectionabstract slides
Robert LeekThe structure of Fréchet-Urysohn and radial spacesabstract slides
Arturo Antonio Martinez Celis RodriguezUltrafilters and Michael spaces slides
Diego MejíaSplitting, bounding and almost disjointness numberabstract slides
Marcin MichalskiSome properties of I-Luzin setsabstract slides
Magdalena NowakZero-dimensional spaces as topological and Banach fractals.abstract slides
Stuart NygardA Survey of the Density Topology on the Realsabstract slides
José de Jesús Pelayo GómezBorel chromatic numbersabstract slides
Grzegorz PlebanekCompacti fications of $\omega$ and the Banach space $c_0$abstract slides
Szymon PlewikGame theoretic approach to skeletally Dugundji and Dugundji spacesabstract slides
Robert RalowskiSome aspects of nonmeasurability with respect to selected $\sigma$-ideals slides
Damian SobotaOn Nikodym's Uniform Boundedness Principleabstract slides
Silvia SteilaAvoidable polynomials and $\mathbb{R} \subseteq L$abstract slides
Zoltan VidnyanszkyOrdered sets of Baire class 1 functionsabstract slides
Julia WódkaLooking for differencesabstract slides
Wolfgang WohofskySacks dense ideals and Marczewski type null ideals slides
Teruyuki YoriokaFragments of Martin's Axiom and the existence of a non-special Aronszajn treeabstract slides
Lyubomyr ZdomskyyRemainders of topological groups and ultrafilters slides
Szymon ZeberskiMid-point free sets slides