# SCIENTIFIC PROGRAMS AND ACTIVITIES

April 23, 2019

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

 Sunday March 29- Thursday April 2, 2015 Forcing and its Applications Retrospective Workshop Fields Institute 222 College St. Toronto Organizing Committee: Justin Moore, Stevo Todorcevic

### Abstracts

Dana Bartosova

Applications of Ramsey theory in topological dynamics

We show that the group of linear isometries of the Gurarij space is extremely amenable and compute the universal minimal flows of the group of affine homeomorphisms of the Poulsen simplex and the group of homeomorphisms of the Lelek fan. This is a joint work with Jordi Lopez-Abad and Brice Mbombo, and Aleksandra Kwiatkowska.

Christina Brech

Subsymmetric sequences in large Banach spaces

We present a method of constructing Banach spaces of large densities without subsymmetric basic sequences, based on the existence of certain sequences of compact, hereditary and large families of finite sets. We also give an idea of how to construct those families for every cardinal smaller than the first inaccessible cardinal, improving results by S. Argyros and P. Motakis. This is a joint work with J. Lopez-Abad and S. Todorcevic.

Sean Cox

Quotients of strongly proper posets, and related topics

I will discuss when quotients of strongly proper posets have the $\omega_1$ approximation property. As an application we prove the conjecture of Viale and Weiss, that $ISP(\omega_2)$ is consistent with arbitrarily large continuum. This is joint work with John Krueger.

Natasha Dobrinen

Higher dimensional Ellentuck spaces

Abstract TBA.

Alan Dow

Long-low iterations and matrix forcing

We prove the existence of models satisfying that the distributivity degree of P(N)/fin is arbitrarily large while still being less than any desired value of the splitting number and the bounding number. This is joint work with S. Shelah.

Michael Hrusak

Parametrized $\diamondsuit$-principles and canonical models

We will review recent results concerning definable versions of the weak diamond.

John Krueger

Mitchell's theorem revisited

Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of $\omega_2 \cap \textrm{cof}(\omega_1)$ in the approachability ideal $I[\omega_2]$. In this talk I will give an overview of a new proof of Mitchell's theorem, which is derived from an abstract framework of side condition methods.

Approximate Ramsey properties of finite dimensional normed spaces.

We present the following result: For every finite dimensional normed spaces $F$ and $G$, every integer $r$ and every numbers $\theta\geq 1$ and $\varepsilon>0$ there exists a finite dimensional space $H$ containing a linear isometric copy of $G$ and such that every $r$-coloring of the set of linear $\theta$-isometric embeddings $\mathrm{Emb}_\theta(F,H)$ has an $\varepsilon$-monochromatic set of the form $\gamma \circ \mathrm{Emb}(F,G)$, for some $\gamma\in \mathrm{Emb}(G,H)$. We will discuss several applications, as the extremely amenability of the group of linear surjective isometries of the Gurarij space or the fact that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is the Poulsen simplex itself. The proof of the main result is of combinatorial nature, as it uses the dual Ramsey theorem.We will also mention other approximate Ramsey results for $\ell_p^n$'s, this time using the concentration of measure.

Maryanthe Malliaris

Open questions on ultrafilters arising from p, t and model theory

The talk will be about a range of open problems, and some related theorems, at the intersection of set theory, model theory, and general topology, mainly around construction of ultrafilters.

Heike Mildenberger

Blass--Shelah forcing revisited

We force with $\sigma$-centred subforcings of forgetful versions of Blass--Shelah forcings. The pure parts of the conditions are taken from suitable centred sets $C$ in a space of sequences of normed subsets of powersets of finite sets. We call the resulting subforcing ${\mathbb BS}(C)$. We sketch a proof of the following: A $P$-point ${\mathcal U}$ in the ground model is preserved by forcing with ${\mathbb BS}(C)$ iff the projection of $C$ to ${\mathcal P}(\omega)$ is not below ${\mathcal U}$ in the Rudin--Blass order.

Miguel Angel Mota

Coherent adequate forcing and preserving CH

In the last years there has been a second boom of the technique of forcing with side conditions (see for instance the recent works of Asper\'{o}-Mota, Krueger and Neeman describing three different perspectives of this technique). The first boom took place in the 1980s when Todorcevic discovered a method of forcing in which elementary substructures are included in the conditions of a forcing poset to ensure that the forcing poset preserves cardinals. More than twenty years later, Friedman and Mitchell independently took the first step in generalizing the method from adding small (of size at most the first uncountable cardinal) generic objects to adding larger objects by defining forcing posets with finite conditions for adding a club subset on the second uncountable cardinal. However, neither of these results show how to force (with side conditions together with another finite set of objects) the existence of such a large object together with the continuum being small. In this talk we will discuss new results in this area. This is joint work with John Krueger.

Yinhe Peng

A Lindelof topolotical group with non-Lindelof square

This is a joint work with Liuzhen. We generalize Moore's construction for an L space to get an L group. We also prove that its square is not Lindelof. This answers a question of Arhangel'skii. We also apply the method to higher finite powers.

Assaf Rinot

A microscopic approach to Souslin trees constructions

We present an approach to construct $\kappa$-Souslin trees that is insensitive to the identity of the cardinal $\kappa$, thereby, allowing to transform constructions from successor of regulars to successor of singulars and to inaccessible. This is obtained by redirecting all constructions through a parametrized proxy principle. The construction is carried as a transfinite sequence of microscopic steps that is indifferent of the "big picture". Indeed, the features of the outcome tree are determined by the parameters of the proxy principle that one starts with. This is joint work with Ari. M. Brodsky.

Marcin Sabok

The topological conjugacy relation of free minimal G-subshifts

During this talk I will discuss the descriptive set-theoretic complexity of the topological conjugacy relation for free minimal G-subshifts for various countable groups G. For residually finite countable groups G we will see that there exists a probability measure on the set of free minimal G-subshifts, which is invariant under a natural action of G and such that the stabilizers of points in this action are a.e. amenable. As a consequence, we will get that if G is a countable residually finite non-amenable group, then the relation of topological conjugacy on free minimal G-subshifts is not amenable. On the other hand, for the group G=Z, we will look at the class of subshifts with separated holes and see that the conjugacy relation is an amenable equivalence relation there. This is joint work (in progress) with Todor Tsankov.

Dima Sinapova

Prikry forcing and square properties

Prikry type forcing is the standard way of constructing models of the failure of SCH. On the other hand not SCH is at odds with the failure of the weaker square principles. I will go over some consistency results about not SCH and failure of squares. Then I will present a dichotomy theorem characterizing what type of Prikry posets add weak square. This is joint work with Spencer Unger.

Juris Steprans

Universal functions and universal graphs

A function of two variables F(x,y) was defined by Sierpinski to be universal if for every other function G(x,y) there exist functions h(x) and k(y) such that G(x,y)=F(h(x),k(y)). Various aspects of this question were examined in a paper LMSW (Larson, Miller, Steprans and Weiss). While the universality of Sierpinski seems similar to model theoretic universality, there is a key difference in the role played by the range of the function in the two cases. This was the motivation for the following question asked in LMSW: Does the existence of a universal graph of cardinality aleph_1 imply the existence of a universal colouring of the complete graph on omega_1 with countably many colours? I will discuss joint work with S. Shelah providing a negative answer to this question.

Konstantinos Tyros

A disjoint union theorem for trees

In this talk we will present an infinitary disjoint union theorem for level products of trees, which can be viewed as a dual form of the Halpern-Läuchli theorem. A consequence of the dual Ramsey theorem due to T.J. Carlson and S.G. Simpson is that for every Suslin measurable finite coloring of the powerset of the natural numbers, there exists a sequence (Xn)n2N of pairwise disjoint non-empty subsets of N such that the set
{ U n2Y Xn : Y non-empty subset of N }
is monochromatic. Our result is similar in spirit. However, the underline structure, that we consider, is the level product of a nite sequence of uniquely rooted and finitely branching trees with no maximal nodes of height ! instead of the natural numbers. The proof required an analogue of the infinite dimensional version of the Hales{Jewett Theorem for maps de ned on a level product of trees that provides additional information about the wildcard sets. This is a joint work with Stevo Todorcevic.

Spencer Unger

The tree property

We survey recent partial progress towards a positive answer to a question of Magidor "Is it consistent that every regular cardinal greater than aleph_1 has the tree property?" We also discuss some obstructions to better results.

Matteo Viale

Universally Baire subsets of 2^\kappa

We generalize the notion of universally Baire set of reals and define the universally Baire subsets of 2^\kappa for \kappa an arbitrary infinite cardinal. We show that the standard theory of universally Baire sets of reals can be naturally generalized to this setting and that the properties of universally Baire subsets of 2^{\omega_1} are inestricably intertwined with forcing axioms. This is joint work with Daisuke Ikegami.

Jindrich Zapletal

Interpreter for Topologists

Given a transitive model M of set theory, we find an interpretation functor between topological spaces of M and V which commutes with very many topological operations. The theory is developed from scratch all the way to computations in functional analysis.