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.

## 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)
Speaker: Jordi Lopez-Abad
Title: Approximate Ramsey property of normed spaces
Abstract: We will present and discuss the known examples approximate Ramsey
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 $p=1,2,\infty$.

## David Chodounsky: Generic ultrafilters

Place: Fields Institute (Room 210)
Date: November 30, 2018 (12:00-1:00)
Speaker: David Chodounsky
Title: Generic ultrafilters
Abstract: S. Todorcevic proved that given the presence of large
cardinals, selective ultrafilters are precisely ultrafilters
P(omega)/FIN-generic over L(R). We generalize this result. We provide a
characterization of P(omega)/I-generic ultrafilters over L(R) for an
arbitrary F_sigma ideal I.
This is a joint work with Jindra Zapletal.

## Noé de Rancourt: Gowers spaces: unifying standard and strategical Ramsey theory

Place: Fields Institute (Room 210)
Date: November 16 , 2018 (13:30-15:00)
Speaker: Noé de Rancourt
Title: Gowers spaces: unifying standard and strategical Ramsey theory
Abstract:

Strategical Ramsey theory was developed in the nineties by Gowers in the
setting of Banach spaces; in this setting where the natural pigeonhole
principle does not always hold, this theory is an alternative to standard
infinite-dimensional Ramsey theory.

In this talk, I will present the formalism of Gowers spaces, an abstract
formalism unifying both strategical and standard infinite-dimensional
Ramsey theory. In this formalism, we can prove an abstract Ramsey theorem
implying both Gowers’ Ramsey-type theorem in Banach spaces, and more
standard Ramsey results like Galvin-Prikry’s theorem. I will also present
a result unifying infinite-dimensional Ramsey theory and determinacy.

I will then introduce a new family of Gowers spaces that arose from a
recent work in progress with Wilson Cuellar-Carrera and Valentin Ferenczi.
These examples from functional analysis are based on local properties of
subspaces of Banach spaces. We hope that examples of the same kind could
be found in other areas of mathematics.

## William Chen: ANTICHAINS, THE STICK PRINCIPLE, AND A MATCHING NUMBER

Place: Fields Institute (Library)
Date:  September 2, 2018 (13:30-15:00)
Speaker: William Chen
Title: ANTICHAINS, THE STICK PRINCIPLE, AND A MATCHING NUMBER
Abstract: This talk is about some cardinal invariants related to $\omega_1$. The
antichain number is the least cardinal for which there does not exist a
subcollection of that size with pairwise finite intersections, and the
matching number is the least cardinal for which there exists a
subcollection X of that size of order-type $\omega$ subsets of $\omega_1$
so that every uncountable subset of $\omega_1$ has infinite intersection
with a member of X. We explore how these invariants behave in various
forcing extensions. Joint work with Geoff Galgon.

## Stevo Todorcevic: A proof of Galvin’s Conjecture

Place: Fields Institute (Room 210)
Date: October 26, 2018 (13:30-15:00)
Speaker: Stevo Todorcevic
Title: A proof of Galvin’s Conjecture
Abstract: We prove that for every finite colouring of the set of unordered
pairs of real numbers there is a set of reals homeomorphic to the rationals whose pairs use no more than two colours. This solves a problem of F. Galvin from the 1970’s. The proof uses large cardinals. This is a joint work with Dilip Raghavan.

## Ilya Shapirovsky: Modal logics of model-theoretic relations

Place: Fields Institute (Room 210)
Date: October 19 , 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Modal logics of model-theoretic relations
Abstract: Consider a unary operation f on the set of sentences of a model-theoretic language L, and a set T of sentences of L.  Properties of f in T can be studied using  propositional modal language: variables are evaluated as sentences of L, and f interprets the modal operator. The modal theory of f in T is defined as the set of those modal formulas which are in T under every valuation.

An example of this approach is Solovay’s theorem providing a complete modal axiomatization of formal provability in Peano arithmetic. Another example is the theorem of Hamkins and Loewe axiomatizing the modal logic of forcing, where the modal operator expresses satisfiability in forcing extensions. Both these logics  have good semantic and algorithmic properties: in particular, they have the finite model property, are finitely axiomatizable, and hence decidable.

This raises the question of modal theories of other model-theoretic relations R (e.g., the submodel relation or the homomorphic image relation). These theories can be defined in the case when satisfiability in R-images is expressible in L. In this talk we will discuss general properties of such modal systems, and then provide a complete axiomatization for the case of the submodel relation.  This talk is based on a joint work with D.I. Saveliev.

## Vera Fischer: More ZFC inequalities between cardinal invariants

Place: Fields Institute (Room 210)
Date: September 21, 2018 (13:30-15:00)
Speaker: Vera Fischer
Title: More ZFC inequalities between cardinal invariants
Abstract: We will discuss some recent ZFC results concerning the
generalized Baire spaces, and more specifically the generalized
bounding number, relatives of the generalized almost disjointness
number, as well as generalized reaping and domination.