The Toronto Set Theory Seminar is normally held on Fridays, from 1:30 to 3pm, in room 210 of the Fields Institute. For listing of talks from years earlier than 2006, see this page.

## David Fernandez: A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number

Place: Fields Institute, Room 210
Date and time: Friday 23 January 2015 (13:30-15:00)
Speaker: David Fernandez
Title: A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number.

Abstract: Strongly summable ultrafilters are a variety of ultrafilters that relate with Hindman’s finite sums theorem in a way that is somewhat analogous to that in which Ramsey ultrafilters relate to Ramsey’s theorem. It is known that the existence of these ultrafilters cannot be proved in ZFC, however such an existencial statement follows from having the covering of meagre to equal the continuum. Furthermore, using ultraLaver forcing in a short finite support iteration, it is possible to get models with strongly summable ultrafilters and a small covering of meagre, and these models will also have small dominating number. Using this ultraLaver forcing in a countable support iteration to get a model with small covering meagre and strongly summable ultrafilters is considerably harder, but it can be done and in this talk I will explain how (it involves a characterisation of a certain kind of strongly summable ultrafilter in terms of games). Interesingly, this way we also get the dominating number equal to the continuum, unlike the previously described model.

## Marcin Sabok: Automatic continuity for isometry groups

Place: Fields Institute, Room 210
Date and time: Friday 16 January 2015 (13:30-15:00)
Speaker:  Marcin Sabok
Title: Automatic continuity for isometry groups

Abstract: We present a general framework for automatic continuity  results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the  groups of isometries of the Urysohn space and the Urysohn  sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\mathrm{Aut}([0,1],\lambda)$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov. The results and proofs are stated in the language of model theory for metric structures.

## Double Session at Fields Institute (Antonio Aviles and Istvan Juhasz)

Place: Fields Institute (Stewart Library)

Date and time:  28-11-2014 from 12:30 to 15:00

Speaker 1: Antonio Aviles  (12:30-13:30)
Title: A combinatorial lemma about cardinals $\aleph_n$ and its applications on Banach spaces

Abstract: The lemma mentioned in the title was used by Enflo and Rosenthal to show that the Banach space $L_p[0,1]^\Gamma$ does not have an unconditional basis when $|\Gamma|\geq \aleph_\omega$. In a joint work with Witold Marciszewski, we used some variation of it to show that there are no extension operators between balls of different radii in nonseparable Hilbert spaces.

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Speaker 2:  Istvan Juhasz  (13:40 to 15:00)
Title: Lindelof spaces of small extent are $\omega$-resolvable
Abstract: I intend to present the proof of the following result, joint with L. Soukup and Z. Szentmiklossy: Every regular space $X$ that satisfies $\Delta(X) > e(X)$ is $\omega$-resolvable, i.e. contains infinitely many pairwise disjoint dense subsets. Here $\Delta(X)$, the dispersion character of $X$, is the smallest size of a nonempty open set in $X$ and $e(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelof spaces of uncountable dispersion character are $\omega$-resolvable.
This improves some results of Pavlov and of Filatova, respectively, concerning Malychin’s problem if regular Lindelof spaces of uncountable dispersion character are resolvable at all. The question if regular Lindelof spaces of uncountable dispersion character are maximally resolvable, i.e. $\Delta(X)$-resolvable, remains wide open.

## Miodrag Sokic: Functional classes

Place: Fields Institute, Room 210

Date: 21-11-2014

Time: 13:30-15:00

Speaker: Miodrag Sokic

Title: Functional classes
Abstract: We consider the class of finite structures with functional symbols with respect to the Ramsey property.

## Martino Lupini: Fraisse limits of operator spaces and the noncommutative Gurarij space

Place: Fields Institute, Room 210

Date: 14-11-2014

Time: 13:30-15:00

Speaker: Martino Lupini

Title: Fraisse limits of operator spaces and the noncommutative Gurarij space
Abstract: We realize the noncommutative Gurarij space introduced by Oikhberg as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. As a consequence we deduce that such a space is unique, homogeneous, universal among separable 1-exact operator spaces, and linearly isometric to the Gurarij Banach space.

## Juris Steprans: The descriptive set theoretic complexity of the weakly almost periodic functions in the dual of the group algebra

Place: Fields Institute,  Steart Library

Date and time : 7 November 2014,  13:30-15:00

Speaker:  Juris Steprans
Title: The descriptive set theoretic complexity of the weakly almost periodic functions in the dual of the group algebra

Abstract: The almost periodic functions on a group G are those functions F from G to the complex number such that the uniform norm closure of all shifts of F is compact in the uniform norm. The weakly almost periodic functions are those for which the analogous statement holds for the weak topology. The family of sets whose characteristic functions are weakly almost periodic forms a Boolean algebra. The question of when this family is a complete $\Pi^1_1$ set will be examined.

## Double session at Fields Institute (Menachem Magidor and Vera Fischer)

Place: Fields Institute, Room 210

Date: 31-October- 2014

Time: 12:30-15:00

Speaker 1 (from 12:30 to 13:30): Vera Fischer

Title: Definable Maximal Cofinitary Groups and Large Continuum

Abstract: A cofinitary group is a subgroup of the group of all permutations of the natural numbers, all non-identity elements of which have only finitely many fixed points. A cofinitary group is maximal if it is not properly contained in any other cofinitary group. We will discuss the existence of nicely definable maximal cofinitary groups in the presence of large continuum and in particular, we will see the generic construction of a maximal cofinitary group with a $\Pi^1_2$ definable set of generators in the presence of $2^\omega=\aleph_2$.

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Speaker 2 (from 13:30 to 15:00): Menachem Magidor
Title: On compactness for being $\lambda$ collectionwise hausdorff
Abstract: A compactness property is the statement for a structure in a given class, if every smaller cardinality substructure has a certain property then the whole structure has this property. In this talk we shall deal with the compactness for the property of a topological space being collection wise Hausdorff. The space is X is said to be $\lambda$–collection wise Hausdorff ($\lambda$–cwH) if every closed discrete subset of X of cardinality less than $\lambda$ can be separated by a family of open sets. X is cwH if it is $\lambda$–cwH for every cardinal $\lambda$.
We shall deal with the problem of when $\lambda$–cwH implies cwH, or just when does $\lambda$–cwH implies $\lambda^+$–cwH. A classical example of Bing provides for every cardinal $\lambda$ a space $X_\lambda$ which is $\lambda$-cwH but not $\lambda^+$–cwH. So if we hope to get any level of compactness for the the property of being cwH, we have to restrict the class of spaces we consider. A fruitful case is the case where we restrict the local cardinality of the space. A motivating result is the construction by Shelah (using supercompact cardinal) of a model of Set Theory in which a space which is locally countable and which is $\omega_2$–cwH is cwH.
Can the Shelah result be generalized to larger cardinals , e.g. can you get a model in which for spaces which are locally of cardinality $\leq \omega_1$ and which are $\omega_3$-cwH are cwH? In general for which pair of cardinals $(\lambda, \mu)$ we can have models in which a space which is locally of cardinality $< \mu$ and which is $\lambda$–cwH are $\lambda^+$–cwH? In this lecture we shall give few examples where we get some ZFC theorems showing that for some pairs $(\lambda, \mu)$ compactness necessarily fails, and cases of pairs for which one can consistently have compactness for the property of being cwH.

## Jordi Lopez Abad: Ramsey properties of embeddings between finite dimensional normed spaces

Date:  24-October-2014

Time: 13:30- 15:00

Place:  Fields Institute, Room 210

Title: Ramsey properties of embeddings between finite dimensional normed  spaces.

Abstract:  Given $d\le m$, let $E_{m,n}$ be the set of all $m\times d$ matrices $(a_{i,j})$ such that

(a) $\sum_{j=1}^d |a_{i,j}|\le 1$ for every $1\le i\le m$.

(b) $\max_{i=1}^m |a_{i,j}|=1$ for every $1\le j\le d$.

These matrices  correspond to the linear isometric embeddings from the normed space $\ell_\infty^d:=(\mathbb{R}^d,\| \cdot \|_\infty)$ into $\ell_\infty^d$, in their unit bases.

We will discuss and give (hints of) a proof of the following new approximate Ramsey result:

For every integers $d$, $m$ and $r$ and every $\varepsilon>0$ there exists $n$ such that for every coloring of $E_{d,n}$ into $r$-many colors there is $A\in E_{m,n}$ and a color $i<r$ such that
\begin{equation*}
A \cdot  E_{d,m}\subseteq (c^{-1}(i))_\varepsilon.
\end{equation*}
Its proof uses the Graham-Rothschild Theorem on partitions of finite sets.
We extend this result, first for embeddings between \emph{polyhedral} normed spaces, and finally for arbitrary finite dimensional normed spaces to get the following:

For every finite dimensional normed spaces $E$ and $F$, every $\theta>1$ and $\varepsilon>0$, and every integer $r$, there is some $n$ such that for every coloring of $\mathrm{Emb}_{\theta^2}(F,\ell_\infty^n)$ into $r$-many colors there is $T\in \mathrm{Emb}_{\theta}(G,\ell_\infty^n)$ and some color $i<r$ such that $T\circ \mathrm{Emb}_\theta(F,G)\subseteq (c^{-1}(i))_{\theta^2-1+\varepsilon}$.

As a consequence, we obtain that the group of linear isometries of the Gurarij space is extremely amenable. A similar result for positive isometric embeddings gives that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is the Poulsen simplex itself.

This a joint work (in progress) with Dana Bartosova (University of Sao Paulo) and Brice Mbombo (University of Sao Paulo)

## Double session at Fields Institute (D. Bartosova and A. Rinot)

Place: Fields Institute, Room 210

Date: 17-October- 2014

Time: 12:30-15:00

Speaker 1 (from 12:30 to 13:30): Dana Bartosova
Title: Finite Gowers’ Theorem and the Lelek fan
Abstract: The Lelek fan is a unique non-degenrate subcontinuum of the Cantor fan with a dense set of endpoints. We denote by $G$ the group of homeomorphisms of the Lelek fan with the compact-open topology. Studying the dynamics of $G$, we generalize finite Gowers’ Theorem to a variety of operations and show how it applies to our original problem. This is joint work with Aleksandra Kwiatkowska.
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Speaker 2 (from 13:30 to 15:00): Assaf Rinot
Title: Productivity of higher chain condition
Abstract: We shall survey the history of the study of the productivity of the k-cc in partial orders, topological spaces, and Boolean algebras. We shall address a conjecture that tries to characterize such a productivity in Ramsey-type language. For this, a new oscillation function for successor cardinals, and a new characteristic function for walks on ordinals will be proposed and investigated.

## Sheila Miller: Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras

Time: Friday 10 October, 13:30-15:00

Place: Fields Institute, Room 210

Speaker: Sheila Miller

Title: Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras

Abstract: In the early 1990’s Richard Laver discovered a deep and striking correspondence between critical sequences of rank-to-rank embeddings and finite left distributive algebras on integers. Each $A_n$ in the tower of finite algebras can be defined purely algebraically, with no reference to the elementary embeddings, and yet there are facts about the Laver tables that have only been proven from a large cardinal assumption. We present here some of Laver’s foundational work on the algebra of critical sequences of rank-to-rank embeddings and some work of the author’s, describe how the finite algebras arise from the large cardinal embeddings, and mention several related open problems.