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.

## David Fernández: Every strongly summable ultrafilter is sparse

24/May/2013, 13:30–15:00
Fields institute,Room 210

Speaker: David J. Fernández Bretón

Title: “Every strongly summable ultrafilter is sparse!”

Abstract: The concept of a Strongly Summable Ultrafilter was born from Hindman’s efforts for proving the theorem that now bears his name (which at the time was known as Graham-Rothschild’s conjecture), although later on it got a life of its own and started to be studied for its own sake, mostly because of its nice algebraic properties. At the time the focus was on ultrafilters over the semigroup $(\mathbb N,+)$, but eventually Hindman, Protasov and Strauss generalized much of this theory to abelian groups in general in a 1998 paper. In that same paper, they introduced the notion of a sparse ultrafilter, one which subsumes that of strongly summable as a particular case but that has even nicer algebraic properties. In a 2012 paper, Hindman, Steprans and Strauss found a large class of abelian groups (which included $(\mathbb N,+)$) over which every strongly summable ultrafilter must be sparse.
In this talk I extend this result to all abelian groups. Moreover we show that in most cases the strong summability of these ultrafilters is due to their being additively isomorphic to a union ultrafilter (I will explain what this means). However, this does not happen in all cases: I will also construct (assuming $\mathfrak p=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.

## Arnold W. Miller: Countable subgroups of Euclidean Space

17/May/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Arnold W. Miller

Title: Countable subgroups of Euclidean Space

Abstract:

In his PhD Thesis Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is K-sigma – the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is K-sigma (since it is countable) but not compactly generated.

Beros showed that for any Polish group G, every K-sigma subgroup of G is compactly generated iff every countable subgroup of G is compactly generated. Beros showed that any K-sigma subgroup of Z^omega (infinite product of the integers) is compactly generated and more generally, for any Polish group G, if every countable subgroup of G is finitely generated, then every countable subgroup of G^omega is compactly generated.

In unpublished work Beros asked whether finitely generated may be replaced by compactly generated in his theorem. He conjectured that the reals R under addition might be an example such that every countable subgroup of R is compactly generated but not every countable subgroup of R^omega is compactly generated. We prove that this is not true. The general question remains open.

In the course of our proof we came up with some interesting countable subgroups. We show that there is a dense subgroup of the plane which meets every line in a discrete set. Furthermore, for each n there is a dense subgroup of Euclidean space R^n which meets every (n-1)-dimensional subspace in a discrete set. Similarly there is a dense subgroup of R^omega which meets every finite dimensional subspace of R^omega in a discrete set.

## Martino Lupini: Borel complexity of equivalence relations from operator algebras

03/May/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Martino Lupini

Title: Borel complexity of equivalence relations from operator algebras

Abstract: I will give an overview of the study from the point of view of descriptive set theory of the Borel complexity of equivalence relatons arising within the theory of C*-algebras. No previous knowledge of operator algebras will be assumed.

## Dana Bartosova: Filter dynamical systems II

26/April/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Dana Bartosova

Title: Filter dynamical systems II

Abstract: This time, we will focus on general topological groups and show that many dynamical notions naturally translate into the language of filters. We will construct disjoint large subsets of non-precompact topological groups showing that the greatest ambit has infinitely many disjoint minimal left ideals. This result was inspired by a problem of Ellis asking for which groups homomorphisms from the greatest ambit into the universal minimal flow separate points. We will finish up with a sketch of a simplified proof that groups of isometries of generalized Urysohn spaces are extremely amenable.

## Konstantinos Tyros: A discussion on Density Ramsey Theory

19/April/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Konstantinos Tyros

Abstract: We will present some recent results in Density Ramsey Theory.  In particular, we will present a density version of a result due to Carlson and Simpson concerning left variable words, which consists a common extension of the Density Hales-Jewett Theorem and the Density Halpern-Läuchli Theorem. If time permits we will also present some applications

## Daniel Soukup: Partitioning bases of topological spaces

12/April/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Daniel Soukup

Title: Partitioning bases of topological spaces

Abstract: The purpose of this talk is to investigate whether an arbitrary base for a dense in itself topological space can be partitioned into two bases; these spaces will be called base resolvable. First, we review positive results, i.e. that several classes of spaces are base resolvable: metric spaces and left-or right separated spaces. Furthermore, every T_3 (locally) Lindelöf space is base resolvable. Second, we aim to outline the construction of a non base resolvable space; this is done by isolating a new partition property of partially ordered sets. Our strongest result in this direction is that, consistently, there is a 0-dimensional, 1st countable Hausdorff space of weight $\omega_1$ and size continuum which is non base resolvable.

joint work with L. Soukup.

## David Chodounsky: Gaps and Towers in $P(\omega)/fin$

05/April/2013, 13:30–15:00
Fields institute,Room 210

Speaker: David Chodounsky

Title: Gaps and Towers in $P(\omega)/fin$

Abstract: We study the structure of $\subset$ relation on towers ($\subset^*$-chains) and gaps in $P(\omega)/fin$. We define Suslin towers and Hausdorff towers and discuss their existence in various models of set theory. Then some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap.

## Dana Bartosova: Filter dynamical systems

22/March/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Dana Bartosova

Title: Filter dynamical systems

Abstract: We show how we can view the universal minimal flow of a topological group $G$ as a space of filters on $G$ with the structure of right topological semigroup. This approach allows us to translate a variety of dynamical properties into the language of filters and use set theoretic and combinatorial methods to understand dynamics of $G.$

## Miguel Angel Mota: On a question of Abraham and Cummings

15/March/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Miguel Angel Mota

Title: On a question of Abraham and Cummings

Abstract: The technique of ensuring properness of a given forcing notion by incorporating elementary substructures of some large enough model into its definition as side conditions may be traced back to Todorcevic. The more specific approach of considering symmetric systems of countable structures as side conditions in the context in which one starts with a model of CH and wants to obtain a forcing notion which is proper and does not collapse cardinals is quite natural. In fact, this approach (also created by Todorcevic) has already shown up in several places in the literature. The main novelty of the method created by Asperó and Mota is that it incorporates the use of symmetric systems of structures as side conditions affecting all iterands of a given forcing iteration rather than a single forcing as in the above references. As an interesting application of this method, we answer a question of Abraham and Cummings by showing that a negative polychromatic Ramsey relation is consistent together with MA and a large continuum. This is joint work with Asperó

## Chris Eagle: Omitting types in infinitary $[0, 1]$-valued logic

8/March/2013, 13:30–15:00
Fields institute,Room 210

Speaker: Chris Eagle

Title: Omitting types in infinitary $[0, 1]$-valued logic.

Abstract: In first-order logic many interesting non-elementary classes of
mathematical structures can be classified by the types that they realize
or omit. The classical Omitting Types Theorem characterizes those types
which can be omitted in models of a fixed theory $T$ as the ones which are
not generated over $T$ by a single formula. The Omitting Types Theorem
has close connections to the Baire Category Theorem, which we will use to
give a topological proof of an Omitting Types Theorem for a logic for
metric structures which is analogous to $\mathcal{L}_{\omega_1, \omega}$.