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.

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.

Osvaldo Guzman Gonzalez: There are no P-points in Silver extensions

Place: Fields Institute (Room 210)

Date: September 29 , 2017 (13:30-15:00)

Speaker: Osvaldo Guzman Gonzalez

Title: There are no P-points in Silver extensions

Abstract: We prove that after adding a Silver real no ultrafilter from the ground
model can be extended to a P-point, and this remains to be the case in any
further extension which has the Sacks property. We use this result to show
that there are no P-points in the Silver model or after adding Silver
reals with the side by side product. In particular, we build models with
no P-points where the continuum can be arbitrarily big. This is joint work
with David Chodounský.

Yinhe Peng: PFA implies a class of hereditarily Lindelof spaces are D spaces

Place: Fields Institute (Room 210)

Date: September 22, 2017 (13:30-15:00)

Speaker: Yinhe Peng

Title: PFA implies a class of hereditarily Lindelof spaces are D spaces

Abstract: For a space X, OSM_X asserts that for any open neighbourhood assignment (or open set mapping) N, there is a partition of X into countably many pieces such that for each x, y in the same piece, either x is in N(y) or y is in N(x).

We introduce a property that will force OSM under PFA. We then use OSM to imply D, assuming additional properties (e.g., sub-Sorgenfrey).

Yuan Yuan Zheng: Moderately-abstract parametrized Ellentuck theorem

Place: Fields Institute (Room 210)

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

Speaker: Yuan Yuan Zheng

Title: Moderately-abstract parametrized Ellentuck theorem

Abstract: Mimicking the parametrized Ellentuck theorem in the Ellentuck
space and the parametrized Milliken theorem in the Milliken space, we
present a ‘moderately abstract’ parametrized theorem for ‘moderate’
topological Ramsey spaces. It is a parametrization of the abstract
Ellentuck theorem with infinitely many perfect sets of real numbers,
implying that essentially all infinitely-dimensional Ramsey properties
proven using topological Ramsey space theory can be parametrized by
products of infinitely many perfect sets.

Micheal Pawliuk: The Perfect Expansion Property

Place: Fields Institute (Room 210)

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

Speaker: Micheal Pawliuk

Title: The Perfect Expansion Property

Abstract: The expansion property for classes of finite structures is a well studied Ramsey property for homogeneous structures. Recently, a quantitative version of this property was introduced to answer questions related to amenability and unique ergodicity of automorphism groups of homogeneous structures. A typical way to check this property involves fine estimates and the probabilistic method.

We introduce an even stronger expansion property that is purely combinatorial, while not being so strong as to be impossible. We will then classify which completely n-partite directed graphs have this property. Remarkably, the property is able to isolate the geometry of completely n-partite directed graphs.

This provides a step in the right direction towards the goal of showing that the semigeneric digraph has a uniquely ergodic automorphism group (which is still open).