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.

## Osvaldo Guzman Gonzalez: On (1,w_1)-weakly universal functions

Place: Fields Institute (Room 210)

Date: January 12, 2018 (13:30-15:00)

Speaker: Osvaldo Guzman Gonzalez

Title: On (1,w_1)-weakly universal functions

Abstract: We will study a very weak notion of universality of functions in Sacks models. We will answer a question of Shelah and Steprans by showing that there are no (1,w_1)-weakly universal functions after adding uncountably many Sacks reals side by side.

## Piotr Koszmider: Dimension drop phenomena and compact supports in noncommutative topology

Place: Fields Institute (Room 210)

Date: December 15, 2017 (13:30-15:00)

Speaker: Piotr Koszmider

Title: Dimension drop phenomena and compact supports in noncommutative topology

Abstract: “When X is a locally compact Hausdorff space, continuous functions on X with compact support can approximate every continuous function in C_0(X). There is a natural notion of elements with compact supports for general, not necessarily commutative, C*-algebras and a  result of Blackadar saying that  in every separable C*-algebra one can choose from such elements  an approximate unit (Blackadar calls it an almost idempotent approximate unit).

We address the issue of the existence of such an approximate unit for general, not necessarily separable C*-algebra and show that such  approximate units exist in every C*-algebra of density omega_1, that they do not exist in some C*-algebras of density min{2^k: 2^k>continuum} and that their existence in all operator algebras acting on the separable Hilbert space is  independent from ZFC. The infinitary combinatorics used involves CH, Canadian trees and Q-sets.

No knowledge of noncommutative mathematics beyond multiplication of 2×2 matrices will be assumed. These are the results of a joint research project with Tristan Bice available at arxiv.org/pdf/1707.09287.pdf

## Jose Iovino: Definability in linear functional analysis

Place: Fields Institute (Room 210)

Date:  December 8 , 2017 (13:30-15:00)

Speaker: Jose Iovino

Title: Definability in linear functional analysis

Abstract: I will discuss some recent results in the theory of second-order definability and applications of these results in Banach space theory.

## Jan Pachl: Topological centres for group actions

Place: Fields Institute (Room 210)

Date: December 1, 2017 (13:30-15:00)

Speaker: Jan Pachl

Title: Topological centres for group actions

Abstract: Based on joint work with Matthias Neufang and Juris Steprans. By a variant of Foreman’s 1994 construction, every tower ultrafilter on $\omega$ is the unique invariant mean for an amenable subgroup of $S_\infty$, the infinite symmetric group. From this we prove that in any model of ZFC with tower ultrafilters there is an element of $\ell_1(S_\infty)^{\ast\ast} \setminus \ell_1(S_\infty)$ whose action on $\ell_1(\omega)^{\ast\ast}$ is w* continuous. On the other hand, in ZFC there are always such elements whose action is not w* continuous.

## Paul Szeptycki: Ladder systems after forcing with a Suslin tree

Place: Fields Institute (Room 210)

Date: November 24, 2017 (13:30-15:00)

Speaker: Paul Szeptycki

Title: Ladder systems after forcing with a Suslin tree

Abstract: Uniformization properties of ladder systems in models obtained by forcing with a Suslin tree S over a model of MA(S) are considered.

## Osvaldo Guzman Gonzalez: The Shelah-Steprans property of ideals

Place: Fields Institute (Room 210)

Date: November 17, 2017 (13:30-15:00)

Speaker: Osvaldo Guzman Gonzalez

Title: The Shelah-Steprans property of ideals

Abstract: An ideal I has the Shelah-Steprans property if for every set X of finite sets, there is an element of I that either intersects every element of X or contains infinitely many elements of X. We will give a characterization of the Borel Shelah-Steprans ideals in terms of the Katetov order and we will see some applications in the destructibility of MAD families.

## Francisco Guevara Parra: Finite products of M-separeble spaces

Place: Fields Institute (Room 210)

Date: November 3, 2017 (13:30-15:00)

Speaker: Francisco Guevara Parra

Title: Finite products of M-separeble spaces

Abstract: A topological space is called M-separable if for all sequence of dense sets, we can select a finite subset from each dense set so that the union of those finite sets is dense. We will study the finite productivity of this property when we assume the spaces are countable and sequential.

## Haim Horowitz: Martin’s Maximum and the saturation of the nonstationary ideal

Place: Fields Institute (Room 210)

Date: October 27, 2017 (13:30-15:00)

Speaker: Haim Horowitz

Title: Martin’s Maximum and the saturation of the nonstationary ideal

Abstract:

By a classical result of Foreman, Magidor and Shelah, MM implies that the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated. We shall prove that MM actually implies a stronger saturation property for the nonstationary ideal. As a corollary, we obtain a new proof of the fact that the continuum is $\aleph_2$ under MM.

This is joint work with Shimon Garti and Menachem Magidor

## Bruno Braga: Coarse embeddings into superstable spaces

Place: Fields Institute (Room 210)

Date: October 20 , 2017 (13:30-15:00)

Speaker: Bruno Braga

Title:Coarse embeddings into superstable spaces

Abstract: In 1981, J. Krivine and B. Maurey introduced the definition of stable  Banach spaces, and, in 1983, Y. Raynaud introduced the notion of superstability and studied uniform embeddings of Banach spaces into superstable Banach spaces. In this talk, we will talk about coarse embeddings into superstable spaces. This is a joint work with Andrew Swift.

## Frank Tall: Completely Baire spaces, Menger spaces, projective sets, Hurewicz’ theorems, and an application to Model Theory

Place: Fields Institute (Room 210)

Date: October 13, 2017 (13:30-15:00)

Speaker: Frank Tall

Title:Completely Baire spaces, Menger spaces, projective sets, Hurewicz’ theorems,  and an application to Model Theory

Abstract: We prove the following are equiconsistent:
(1) There is an inaccessible cardinal.
(2) Every projective Menger set of reals is σ-compact.
(2’) Every co-analytic Menger set of reals is σ-compact.
(3) Every projective set of reals with every closed subset Baire is Polish.
(3’) Every analytic set of reals with every closed subset Baire is Polish.
(1), (2), (2’) are from Tall-Todorcevic-Tokg ̈z 2017; (1), (3), (3’) are from Tall-Zdomskyy, in preparation.

Researchers previously derived (2), (3) from the Axiom of Projective Determinacy, and negations of (2’) and (3’) from V = L. We substitute a perfect set version of Todorcevic’s Open Graph Axiom for PD and the L[a] existence of an a ⊆ ω such that $\omega_1=\omega_1$ for V = L.

We (Tall-Zdomskyy) also construct in ZFC a separable metric space X such that every closed subset of X ω is Baire, but X includes no dense completely metrizable subspace. Such a space was previously constructed by Eagle-Tall (2017) from a non-meager P-filter, which is not known to exist in ZFC. Such a space can be used to construct an abstract logic in which the Omitting Types Theorem holds but a stronger, game-theoretic version of the OTT does not.