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.

## Antonio Aviles: Twisted sums of spaces of continuous functions

Place: Fields Institute (Room 210)
Date: March 22, 2019 (13:30-15:00)
Speaker: Antonio Aviles
Title: Twisted sums of spaces of continuous functions
Abstract: Given two Banach spaces $Z$ and $X$, can we find a Banach space $Y$ that contains $X$ as an uncomplemented subspace and $Y/X = Z$? We will mention two instances of this problem connected to set theoretic questions. When $X = c_0$ and $Z=C(K)$ is a space of continuous functions on a nonmetric compactum, the answer may be negative under $MA_{\omega_1}$ but it is always positive under CH (joint work with W. Marciszewski and G. Plebanek). When $X = \ell_\infty/c_0$ and $Z=c_0(\mathfrak{c})$, the answer is positive provided splitting chains exist in $\mathcal{P}(\omega)/fin$ (joint work with P. Borodulin-Nadzieja, F. Cabello, D. Chodounsk\'{y} and O. Guzm\'{a}n)

## Francisco Guevara Parra: Analytic spaces and their Tukey types

Place: Fields Institute (Room 210)
Date: March 15, 2018 (13:30-15:00)
Speaker: Francisco Guevara Parra
Title: Analytic spaces and their Tukey types
Abstract: In this Thesis we study topologies on countable sets from the
perspective of Tukey reductions of their neighbourhood
filters. In particular we will study $k$-analytic group topologies
on $\omega$. This will allow us to obtain a metrization theorem
for analytic sequential group topologies on $\omega$, as well
as a classification of such groups in terms of the Tukey type of
the filter of neighbourhoods of the identity. Recall that
a countable topological space is analytic if the topology is analytic
as a subset of the Cantor space.

## Yann Pequignot: Finite versus infinite: an insufficient shift

Place: Fields Institute (Room 210)
Date: , 2018 (13:30-15:00)
Speaker: Yann Pequignot
Title:    Finite versus infinite: an insufficient shift
Abstract: The Borel chromatic number – introduced by Kechris, Solecki, and Todorcevic (1999) – generalizes the chromatic number to Borel graphs. While the G_0 dichotomy states that there exists a minimal graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, the situation is quite complicated. The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. In 1999, Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. I will sketch a proof that the answer is negative using descriptive complexity considerations and a representation theorem for Sigma^1_2 sets due to Marcone (1994). This result has recently been considerably strengthened by Todorcevic and Vidnyanszky who proved that the set of closed subsets of the Shift Graph that have infinite Borel Chromatic number is Pi^1_2 complete, therefore ruling out most interesting basis results for this class of Borel graphs.

## David Schrittesser: The Ramsey property, MAD families, and their multidimensional relatives

Place: Fields Institute (Room 210)
Date: February 15, 2018 (13:30-15:00)
Speaker: David Schrittesser
Title: The Ramsey property, MAD families, and their multidimensional relatives
Abstract: Suppose every set of real numbers has the Ramsey property and “uniformization on Ellentuck-comeager sets” as well as Dependent Choice hold (as is the case under the Axiom of Determinacy, but also in Solovay’s model). Then there are no MAD families. As it turns out, there are also no (Fin x Fin)-MAD families, where Fin x Fin is the two-dimensional Fubini product of the ideal of finite sets. We also comment on higher dimensional products. Results are joint work with Asger Törnquist.

## Damjan Kalajdzievski: How to show Con(ZFC + omega_1=u<a) from Con(ZFC)

Place: Fields Institute (Room 210)
Date: February 8 , 2019 (13:30-15:00)
Speaker: Damjan Kalajdzievski
Title: How to show Con(ZFC + omega_1=u<a) from Con(ZFC)
Abstract: I will outline how to prove the result in the title by joint work with Osvaldo Guzman

## Hossein Lamei Ramandi: $\Sigma^2_2$-absoluteness

Place: Fields Institute (Room 210)
Date: January 25 , 2018 (13:30-15:00)
Speaker: Hossein Lamei Ramandi
Title: $\Sigma^2_2$-absoluteness
Abstract:  We will show there is a $\Sigma^2_2$ sentence $\Phi$ such that both $\Phi$ and $\neg \Phi$ are consistent with $\diamondsuit$. This answers a question due to Woodin.

## Marcin Sabok: Measurable Hall’s theorem for actions of Z^n

Place: Fields Institute (Room 210)
Date: January 18, 2018 (13:30-15:00)
Speaker: Marcin Sabok
Title: Measurable Hall’s theorem for actions of Z^n
Abstract: In the 1920’s Tarski asked if it is possible to divide the unit square into finitely many pieces, rearrange them by translations and get a disc of area 1. It turns out that this is possible and proved by Laczkovich in the 1990’s. His decomposition, however, used non-measurable pieces and seemed paradoxical. Recently, Grabowski, Mathe and Pikhurko and Marks and Unger showed that such decompositions can be obtained using nice measurable pieces. During the talk, I will discuss a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. This result implies that for measurable actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposalble using measurable pieces. The latter generalizes the measurable circle squaring result by Grabowski, Mathe and Pikhurko. This is joint work with Tomasz Ciesla.

## Clovis Hamel: Stability and Definability in Continuous Logics, Cp-theory and the Tsirelson space.

Place: Fields Institute Library
Date: Decembver 7, 2018 (13:30-15:00)
Speaker: Clovis Hamel
Title: Stability and Definability in Continuous Logics, Cp-theory and the Tsirelson space.
Abstract: An old question in Functional Analysis inquired whether there is a Banach space that
does not contain a copy of either lp or c0. Tsirelson defined such a space through
an infinitary process that inspired many other constructions of pathological spaces.
Then Gowers popularized the problem: does every explicitly definable infinite
dimensional Banach space contain a copy of lp or c0?
In first-order logic, the notion of explicit definability is closely related to that
of stability, which has been a driving force of Model Theory in the last decades
since Shelah introduced the idea. We will discuss how these concepts extend to
continuous logics in order to present Casazza and Iovino’s positive answer to
Gowers’s question in finitary continuous logic and our current work around this
result in infinitary continuous logics using Cp-theoretic tools.

## Miloš Kurilić: Vaught’s Conjecture for Monomorphic Theories

Place: Fields Institute (Room 210)
Date: November 30, 2018 (1:00-2:00)
Speaker: Miloš Kurilić
Title: Vaught’s Conjecture for Monomorphic Theories
Abstract: A complete first order theory of a relational signature is called monomorphic iff
all its models are monomorphic (i.e.\ have all the $n$-element substructures
isomorphic, for each positive integer $n$).
We show that a complete theory ${\mathcal T}$ having infinite models is monomorphic
iff it has a countable monomorphic model
and confirm the Vaught conjecture for monomorphic theories.
More precisely, we prove that if ${\mathcal T}$ is a complete monomorphic theory
having infinite models, then the number of its non-isomorphic countable models,
$I({\mathcal T} ,\omega)$, is either equal to $1$ or to ${\mathfrak c}$. In addition,
$I({\mathcal T},\omega )= 1$ iff some countable model of ${\mathcal T}$ is simply
definable by an $\omega$-categorical linear order on its domain.

## Jordi Lopez-Abad: Approximate Ramsey property of normed spaces

Place: Fields Institute (Room 210)
Date: November 30, 2018 (2:00-3:00)
properties of normed spaces. In some cases, this can be seen as a multidimensional Borsuk-Ulam Theorem, and as a reformulation of the extreme amenability of the automorphism group of an appropriate space (a Fraisse limit). Concerning the proofs, we will sketch them for the 3 main families of spaces: $\{\ell_p^n\}_$ for $p=1,2,\infty$.