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: Set theoretical dichotomies in the theory of continuous quotients

Place: Fields Institute (Room 210)

Date: April 28, 2017 (13:30-15:00)

Speaker: Alessandro Vignati, York University

Title: Set theoretical dichotomies in the theory of continuous quotients

Abstract: We state and (depending on time) prove some dichotomies of set theoretical nature arising in the theory of continuous quotients. In particular we show that the assumption of CH on one side, and of Forcing Axioms on the other, affects the nature of possible embeddings of certain corona algebras, as well as the behavior of their automorphisms group. This is partly joint work with P. McKenney.

## Ari Brodsky: Distributive Aronszajn trees

Place: Fields Institute (Room 210)

Date: April 21, 2017 (13:30-15:00)

Speaker: Ari Brodsky, Bar-Ilan University

Title: Distributive Aronszajn trees

Abstract: We address a conjecture asserting that, assuming GCH, for every singular cardinal $\lambda$, if there exists a $\lambda^+$-Aronszajn tree, then there exists one which is moreover $\lambda$-distributive.A major component of this work is the study of postprocessing functions and their effect on square sequences. This is joint work with Assaf Rinot

## Francisco Guevarra Parra: An application of the the Ultra-Ellentuck theorem

Place: Fields Institute (Room 210)

Date: April 7, 2017 (14:15-15:00)

Speaker: Francisco Guevara Parra

Title: An application of the the Ultra-Ellentuck theorem

Abstract: We will use the Ultra-Ellentuck theorem to construct countable
local $\pi$-bases in a given sequential-definable topology on $\omega$
that is $p^+$ (or $\alpha_4$ if we replace sequential by Frechet).

## Yuan Yuan Zheng: The Ehrenfeucht Game

Place: Fields Institute (Room 210)

Date: April 7, 2017 (13:30-14:15)

Speaker: Yuan Yuan Zheng, University of Toronto

Title: The Ehrenfeucht Game

Abstract: The Ehrenfeucht Game is interesting in its own right as a game.
It was originally a method given by Roland Fraïssé to verify elementarily
equivalence. It was reformulated as a game by Andrzej Ehrenfeucht. We will
define the game, see how it plays a role in deciding whether a property is
first order expressible, and give a vague idea of how it relates to the
Zero-One Law.

## Timothy Trujillo: Parametrizing by the Ellentuck space

Place: Fields Institute (Room 210)

Date: March 31, 2017 (13:30-15:00)

Speaker: Timothy Trujillo, Colorado School of Mines

Title: Parametrizing by the Ellentuck space

Abstract: We introduce a new construct that can be used to parametrize some
topological Ramsey spaces by the collection of inﬁnite subsets of the natural
numbers. We show that these parametrized spaces are also topological Ramsey spaces.Then we use these spaces to give new proofs of some known parametrized perfect set theorems. We conclude with a discussion of how to extend the results to the abstract setting and open questions related to applying the results to the Tukey theory of ultraﬁlters.

## Zoltán Vidnyánszky: Anti-basis results for graphs of infinite Borel chromatic number

Place: Fields Institute (Room 210)

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

Speaker: Zoltán Vidnyánszky, York University

Title: Anti-basis results for graphs of infinite Borel chromatic number

Abstract: One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that a Borel graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs with infinite Borel chromatic number. Using descriptive set theoretic methods we answer this question and a couple of similar questions negatively, showing that one cannot hope for the existence of a Borel graph whose embeddability would characterize Borel (or even closed) graphs with infinite Borel chromatic number.

## Marcin Sabok: Hyperfiniteness of boundary actions of cubulated hyperbolic groups

Place: Fields Institute (Room 210)

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

Speaker: Marcin Sabok

Title: Hyperfiniteness of boundary actions of cubulated hyperbolic groups

Abstract:  An old result of Dougherty, Jackson and Kechris implies that the
boundary action of the free group F2 induces a hyperfinite equivalence
relation. During the talk, I will discuss generalizations of this theorem
to the class of hyperbolic groups. The examples discussed will include
groups acting properly and cocompactly on CAT(0) cube complexes. This is
joint work with Jingyin Huang and Forte Shinko.

## Sergio Garcia-Balan: On star selection principles

Place: Fields Institute (Room 210)

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

Speaker: Segio Garcia-Balan

Title: On star selection principles

Abstract: In the theory of selection principles, an important result (due to L. Aurichi), states that every Menger space is a D-space. Motivated by this result, we will discuss the star versions of the Menger property and some other selection principles in specific topological spaces. We will also talk about the game version of some of these principles. This is joint work with Javier Casas de la Rosa and Paul Szeptycki.

## Yinhe Peng: Product of countable Frechet spaces

Place: Fields Institute (Room 210)

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

Speaker: Yinhe Peng, University of Toronto

Title: Product of countable Frechet spaces

Abstract: I will discuss the preservation of Frechet property in the
product, mainly in the class of countable spaces. Some result in the
higher powers will also be mentioned.

## Zoltán Vidnyánszky: Random elements of large groups

Place: Fields Institute (Room 210)

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

Speaker: Zoltán Vidnyánszky

Title: Random elements of large groups

Abstract: The automorphism groups of countable homogeneous structures are usually interesting objects from group theoretic and set theoretic perspective. The description of typical (with respect to category) elements of such groups is a flourishing topic with a wide range of applications. A natural question is whether there exist measure theoretic analogues of these results. An obvious obstacle in this direction is that such automorphism groups are often non-locally compact, hence there is no natural translation invariant measure on them. Christensen introduced the notion of Haar null sets in non-locally compact Polish groups which is a well-behaved generalisation of the null ideal to such groups. Using Christensen’s Haar null ideal it makes sense to consider the properties of a random element of the group. We investigate these properties, giving a full description of random elements in the case of the automorphism group of the random graph and the rational numbers (as an ordered set).