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.

## Frank Tall: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Place: Fields Institute (Room 210)

Date: February 23 , 2018 (13:30-15:00)

Speaker: Frank Tall

Title: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Abstract: We continue the study of K-analytic and related spaces started last time, especially the connections between descriptive set theory as presented by Rogers and Jayne, and generalized metric spaces. We shall mention a number of unsolved problems and also give applications to productively Lindelof spaces and to topological groups.

## Will Brian: Autohomeomorphisms of ω∗ : the quotient relation

Place: Fields Institute (Room 210)

Date: February 16, 2018 (13:30-15:00)

Speaker: Will Brian

Title: Autohomeomorphisms of ω∗ : the quotient relation

Abstract: Given two autohomeomorphisms f and g of N*, we say that f is a quotient of g when there is a continuous surjection Q from N* to N* such that Qg = fQ. In other words, f is a quotient of g if it is the “continuous image” of g, in the appropriate sense.

I have been investigating this relation, and will present some of the results of that investigation in my talk. For example, under CH: there are many universal autohomeomorphisms (an autohomeomorphism is universal if everything else is a quotient of it); the quotient relation has uncountable chains and antichains; there is an exact description of the quotients of a given trivial map. Under OCA+MA the picture is still murky: for example, there is a jointly universal pair of autohomeomorphisms (meaning everything else is a quotient of one or the other), but I do not know if there is a single universal automorphism. I will sketch some of these results and include several open questions.

## Frank Tall: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Place: Fields Institute (Room 210)

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

Speaker: Frank Tall

Title: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Abstract: I will not assume knowledge from my previous talks on this subject. We define co-K-analytic spaces and provide evidence that this is the “correct generalization” of ‘co-analytic’ to non-metrizable spaces. As before, we view the classic work of Rogers and Jayne on analytic sets through the lens of

Arhangel’skii’s work on generalized metric spaces, while we investigate the question of whether definable Menger spaces are sigma-compact.

## Stevo Todorcevic: P-ideal dichotomy and versions of Souslin Hypothesis, continued

Place: Fields Institute (Room 210)

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

Speaker: Stevo Todorcevic

Title: P-ideal dichotomy and versions of Souslin Hypothesis, continued

Abstract: This is a joint work with B. kuzeljevic. This talk will be about the relationship of PID with various forms of SH such as, for example, the statement that all Aronszajn trees are Q-embeddable.

## Stevo Todorcevic: P-ideal dichotomy and versions of Souslin Hypothesis

Place: Fields Institute (Room 210)

Date: January 26, 2018 (13:30-15:00)

Speaker: Stevo Todorcevic

Title: P-ideal dichotomy and versions of Souslin Hypothesis

Abstract: This is a joint work with B. kuzeljevic. This talk will be about the relationship of PID with various forms of SH such as, for example, the statement that all Aronszajn trees are $\mathbb{Q}$-embeddable.

## Fulgencio Lopez: Adding Cohen reals also adds a capturing Construction Scheme

Place: Fields Institute (Room 210)

Date: January 19, 2018 (13:30-15:00)

Speaker: Fulgencio Lopez

Abstract: We show that adding $\kappa\geq \omega_1$ Cohen reals adds a capturing construction scheme.

## 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.