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.

## Christopher Eagle: Baire Category and the Omitting Types Theorem

Place: BA6183

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

Speaker: Christopher Eagle

Title: Baire Category and the Omitting Types Theorem

Abstract: It is well-known that the Omitting Types Theorem from model theory can be proved by topological means, and the central ingredient of that proof is the Baire Category Theorem.  The goal of this talk is to consider the extent to which the Omitting Types Theorem is equivalent to the Baire Category Theorem.  To do so, we will describe a topological framework (based on work of Robin Knight) that generalizes the classical type spaces from model theory.   Many classical logics (including first-order, infinitary, and continuous logics) fit into this general setting, and conversely we will show that each instance of the general framework yields a model-theoretic logic.  We then distinguish several version of the Omitting Types Theorem these logics may have, based on Baire Category properties of the underlying topological spaces.  All of these properties are equivalent for first-order logic, but are distinct in the general setting.  This is joint work with Frank Tall.

## Saeed Ghasemi: Scattered C*-algebras

Place: Bahen Centre (Room 6183)

Date: July 7th, 2017 (13:30-15:00)

Speaker: Saeed Ghasemi, Polish Academy of Sciences

Title: Scattered C*-algebras

Abstract: By the Gelfand duality, the theory of C*-algebras can be
regarded as “non-commutative topology”. In a joint work with Piotr
Koszmider at IMPAN, we investigated the non-commutative analogues of
the scattered spaces, parallel to the classical research in
set-theoretic topology. The so called scattered C*-algebras, despite
being around in the literature, have not been subject to the tools
from set-theoretic topology. The techniques and constructions of
compact, Hausdorff scattered spaces, or equivalently (by the Stone
duality) superatomic Boolean algebras, have already led to many
fundamental results in the theory of Banach spaces of the form C(K),
or more generally Asplund spaces. In fact scattered C*-algebras were
introduced as C*-algebras which are Asplund as Banach spaces. I will
introduce the notion of the Cantor-Bendixson derivatives for these
C*-algebras, and present some of the basic properties of such
algebras. I will also show how it can be used to construct C*-algebras
with exotic properties, which are non-commutative versions of
well-known scattered spaces. In particular, the constructions of
non-commutative Psi-spaces and thin tall spaces lead to new
discoveries about the preservation of the “stability” for
non-separable C*-algebras.

## Dana Bartosova: When can we act freely?

Place: Fields Institute (Stewart Library)

Date: May 19, 2017 (13:30-15:00)

Speaker: Dana Bartosova, CMU

Title: When can we act freely?

Abstract:  A topological group admits a free action if there is a compact
Hausdorff space on which the group acts without fixed points. I will
discuss this notion and explain how to translate it into colourings of
graphs and Ramsey type properties.

## Fulgencio Lopez: A result of Dzamonja with construction schemes

Place: Fields Institute (Room 210)

Date: May 5, 2017 (14:20-15:00)

Speaker: Fulgencio Lopez, University of Toronto

Title: A result of Dzamonja with construction schemes

Abstract: We present a proof with construction schemes of the following result of Mirna Dzamonja: Forcing a Cohen real adds a Banach space of density $\omega_1$ which does not isomorphically embed in any ground model Banach space of density $\omega_1$.

## Sergio Garcia-Balan: The star-Menger property on $\Psi$-spaces

Place: Fields Institute (Room 210)

Date: May 5, 2017 (13:30-14: 10)

Speaker: Sergio Garcia- Balan, York University

Title: The star-Menger property on $\Psi$-spaces

Abstract: Bonanzinga and Matveev showed that the space $\Psi(\mathcal{A})$ is strongly star-Menger if and only if $|\mathcal{A}|<\mathfrak{d}$. We will discuss what can be said about the star-Menger property on $\Psi(\mathcal{A})$.

## 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.