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.

## Alessandro Vignati: Homeomorphisms of Cech-Stone remainders

Place: Bahen Centre Information T (Room BA 2165)
Date: May 18, 2018 (13:30-15:00)
Speaker: Alessandro Vignati
Title: Homeomorphisms of Cech-Stone remainders
Abstract: From a locally compact space X one construct its Cech-Stone remainder X*=beta X minus X. We analyze the problem on whether X* and Y* can be homeomorphic for different spaces X and Y. In the 0-dimensional case, a solution to this problem has been proved to be independent of ZFC, by the work of Parovicenko, Farah, Dow-Hart and Farah-McKenney among others.
We prove, under PFA, the strongest possible rigidity result: for metrizable X and Y, we prove that X* is homeomorphic to Y* only if X and Y are homeomorphic modulo compact subsets. We also show that every homeomorphism X* –> Y* lifts to an homeomorphism between cocompact subsets of X and Y.

## Osvaldo Guzman: hm and the ultrafilter number

Place: Bahen Centre Information T .   BA 2165
Date: May 11, 2018 (13:30-15:00)
Speaker: Osvaldo Guzman
Title: hm and the ultrafilter number
Abstract: The cardinal invariant $\mathfrak{hm}$ is defined as the minimum size of a family of $\mathsf{c}_{\mathsf{min}}$-monochromatic sets that cover $2^{\omega}$ (where $\mathsf{c}_{\mathsf{min}}\left( x,y\right)$ is the parity of the biggest initial segment both $x$ and $y$ have in common). We prove that $\mathfrak{hm}=\omega_{1}$ holds in the Shelah’s model of $\mathfrak{i<u}$ so the inequality $\mathfrak{hm<u}$ is consistent with the axioms of $\mathsf{ZFC.}$ This answers a question of Thilo Weinert.

## David Fernandez Breton: Partition theorems on uncountable abelian groups

Place: Fields Institute (Room 210)

Date: April 27, 2018 (13:30-15:00)

Speaker: David Fernandez Breton

Title: Partition theorems on uncountable abelian groups

Abstract:

In the past two years, a number of Ramsey-theoretic results concerning the additive structure of uncountable abelian groups have been investigated by diverse subsets of the set {Komjáth, Rinot, D. Soukup, W. Weiss, myself} (among others). From Ramsey results generalizing Hindman’s theorem for certain groups and colourings, to anti-Ramsey statements asserting the existence of ”rainbow colourings” for sets of finite sums, I plan to provide a panoramic overview of this exciting line of research, and point towards possible future lines of enquiry.

## Anthony Bonato: The new world of infinite random geometric graphs

Place: Fields Institute (Room 210)

Date: April 20, 2018 (13:30-15:00)

Speaker: Anthony Bonato

Title: The new world of infinite random geometric graphs

Abstract:

The \emph{infinite random} or \emph{Rado graph} $R$ has been of interest to graph theorists, probabilists, and logicians for the last half-century. The graph $R$ has many peculiar properties, such as its \emph{categoricity}: $R$ is the unique countable graph satisfying certain adjacency properties. Erd\H{o}s and R\'{e}nyi proved in 1963 that a countably infinite binomial random graph is isomorphic to $R$.

Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space $\ell _{\infty }^{n}$, which consists of $\mathbb{R}^{n}$ equipped with the $L_{\infty }$-norm. Balister, Bollob\'{a}s, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the $L_{\infty }$-norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric $\ell _{\infty}^{n}$ gives non-isomorphic limits with probability $1$.

With Janssen and Anthony Quas, we have discovered unique limits in infinite dimensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.

## Menachem Magidor: Compactness and Incompactness for being Corson and Eberline compacta

Place: Fields Institute (Room 210)

Date: April 6, 2018 (13:30-15:00)

Speaker: Menachem Magidor

Title: Compactness and Incompactness for being Corson and Eberline compacta

Abstract:

The problems we shall discuss are examples of compactness (or dually a
reflection) problems. A typical compactness property is a statement that a
given structure has a certain property, provided smaller some structures have
this property. Reflection property is a dual statement, namely if a structure
has the property then there is a smaller substructure having the property. The
notion of “smaller substructure ” may depend on the domain we talk about. Thus
for an algebraic structure “smaller substructure” typically means a subalgebra
having a smaller cardinality. For topological spaces “smaller substructure ”
may mean a continuous image of the space of smaller weight.

Compactness problems tend to form clusters, which share the same pattern. For
instance sharing the same cardinals which are compact for the given property.
In this talk we shall survey some of these patterns. But we shall concentrate
on the problem of compactness for a compact space being Corson.

A compact space is a Corson compact if it can be embedded into $\Sigma\left( R^{k}\right)$ where $\Sigma\left( R^{k}\right)$ is the the subspace of $R^{k}$ (with the product topology) of those sequences which are non zero only
on countably many coordinates. The compactness problem for Corson compacta is whether a space is Corson compact if compact given that all its continuous images of small weight are Corson. We shall report on some ongoing work about this problem.

## Carlos Di Prisco: Chromatic numbers of Borel graphs

Place: Fields Institute (Room 332)

Date: March 28, 2018 (13:30-15:00)

Speaker: Carlos Di Prisco

Title: Chromatic numbers of Borel graphs

Abstract: A graph G=(X, R) defined on a Polish space X is Borel if the binary relation R is a Borel subset of the cartesian product of X with itself.
The Borel chromatic number of such a graph is the least cardinal k such that there is a Borel measurable function c from X to k coloring of the graph, that is, connected elements of X get different images under c.
The study of Borel chromatic numbers was initiated by Kechris, Solecki and Todorcevic (Advances in Mathematics 141 (1999) 1-44) and has received considerable attention since then.
We will survey some of the basic results regarding this concept and mention some open questions.

## Antonio Aviles: Free Banach lattices

Place: Fields Institute (Room 210)

Date: March 23, 2018 (14:05-15:05)

Speaker: Antonio Aviles

Title: Free Banach lattices

Abstract: A Banach lattice has compatible structures of both Banach space and lattice. In this talk we present free constructions of Banach lattices based on a given Banach space or based on a given lattice, and we discuss some of their properties, like chain conditions ccc and others.

## Slawomir Solecki: Polishable equivalence relations

Place: Fields Institute (Room 210)

Date: March 23, 2018 (13:00-14:00)

Speaker: Slawomir Solecki

Title: Polishable equivalence relations

Abstract: We introduce the notion of Polishable equivalence relations. This class of equivalence relations contains all orbit equivalence relations induced by Polish group actions and is contained in the class of idealistic equivalence relations of Kechris and Louveau. We show that each orbit equivalence relation induced by a Polish group action admits a canonical transfinite sequence of Polishable equivalence relations approximating it. The proof involves establishing a lemma, which may be of independent interest, on stabilization of increasing ω1-sequences of completely metrizable topologies.

## Yasser Fermán Ortiz Castillo: Crowded pseudocompact spaces of cellularity at most the continuum are resolvable

Place: Fields Institute (Room 210)

Date: March 9 , 2018 (13:30-15:00)

Speaker: Yasser Fermán Ortiz Castillo

Title: Crowded pseudocompact spaces of cellularity at most the continuum are resolvable

Abstract: It is an open question from W. Comfort and S. Garcia-Ferreira if it is true that every crowded pseudocompact space is resolvable. In this talk will be present a partial positive answer for spaces of cellularity at most the continuum.

## Bruno Braga: On the rigidity of uniform Roe algebras of coarse spaces

Place: Fields Institute (Room 210)

Date: March 2 , 2018 (13:30-15:00)

Speaker: Bruno Braga

Title: On the rigidity of uniform Roe algebras of coarse spaces.

Abstract: (joint with Ilijas Farah) Given a coarse space (X,E), one can define a $C^*$-algebra $C^∗_u(X)$ called the uniform Roe algebra of (X,E). It has been proved by J. \v{S}pakula and R. Willet that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this talk, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.