# Category Archives: Seminars

## Antonio Aviles: Free Banach lattices

Place: Fields Institute (Room 210)

Date: March 23, 2018 (14:05-15:05)

Speaker: Antonio Aviles

Title: Free Banach lattices

Abstract: A Banach lattice has compatible structures of both Banach space and lattice. In this talk we present free constructions of Banach lattices based on a given Banach space or based on a given lattice, and we discuss some of their properties, like chain conditions ccc and others.

## Slawomir Solecki: Polishable equivalence relations

Place: Fields Institute (Room 210)

Date: March 23, 2018 (13:00-14:00)

Speaker: Slawomir Solecki

Title: Polishable equivalence relations

Abstract: We introduce the notion of Polishable equivalence relations. This class of equivalence relations contains all orbit equivalence relations induced by Polish group actions and is contained in the class of idealistic equivalence relations of Kechris and Louveau. We show that each orbit equivalence relation induced by a Polish group action admits a canonical transfinite sequence of Polishable equivalence relations approximating it. The proof involves establishing a lemma, which may be of independent interest, on stabilization of increasing ω1-sequences of completely metrizable topologies.

## Monroe Eskew: Local saturation of the nonstationary ideals

Talk held by Monroe Eskew (KGRC) at the KGRC seminar on 2018-03-22.

Abstract: It is consistent relative to a huge cardinal that for all successor cardinals $\kappa$, there is a stationary $S \subseteq \kappa$ such that the nonstationary ideal on $\kappa$ restricted to $S$ is $\kappa^+$-saturated. We will describe the construction of the model, focusing how to get this property on all $\aleph_n$ simultaneously. Time permitting, we will also briefly discuss the Prikry-type forcing that extends this up to $\aleph_{\omega+1}$.

## Grzegorz Plebanek: Strictly positive measures on Boolean algebras

Tuesday, March 27, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Grzegorz Plebanek (University of Wroclaw)

Title: Strictly positive measures on Boolean algebras

Abstract:

$SPM$ denotes the class of Boolean algebras possessing strictly positive measure (finitely additive and probabilistic). Together with Menachem Magidor, we consider the following problem: Assume that $B$ belongs to $SPM$ for every subalgebra $B$ of a given algebra $A$ such that $|B|\le\mathfrak c$. Does it imply that the algebra $A$ belongs to $SPM$?

It turns out that the positive answer follows from the existence of some large cardinals, while the counterexample can be found in the model of $V=L$.

## David Chodounsky: How to kill a P-point

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 14 March 2018, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: David Chodounsky

Title: How to kill a P-point

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
The existence of P-points (also called P-ultrafilters) is independent
of the axioms of set theory ZFC. I will present the basic ideas behind
a new and simple proof of the negative direction of this fact; a new
forcing method for destroying P-points.

## Thilo Weinert: Cardinal Characteristics and Partition Properties

Talk held by Thilo Weinert (KGRC) at the KGRC seminar on 2018-03-15.

Abstract: Many a partition relation has been proved assuming the Generalised Continuum Hypothesis. More precisely, many negative partition relations involving ordinals smaller than $\omega_2$ have been proved assuming the Continuum Hypothesis. Some recent results in this vein for polarised partition relations came from Garti and Shelah. The talk will focus on classical partition relations. The relations $\omega_1\omega \not\rightarrow (\omega_1\omega, 3)^2$ and $\omega_1^2 \not\rightarrow (\omega_1\omega, 4)^2$ were both shown to follow from the Continuum Hypothesis, the former in 1971 by Erdős and Hajnal and the latter in 1987 by Baumgartner and Hajnal.

The former relation was shown to follow from both the dominating number and the stick number being $\aleph_1$ in 1987 by Takahashi. In 1998 Jean Larson showed that simply the dominating number being $\aleph_1$ suffices for this. It turns out that the unbounding number and the stick number both being $\aleph_1$ yields the same result. Moreover, also the second relation follows both from the dominating number being  $\aleph_1$ and from both the unbounding number and the stick number being $\aleph_1$ thus answering a question of Jean Larson.

This is both joint work with Chris Lambie-Hanson and with both William Chen and Shimon Garti.

## David J. Fernández Bretón: Models of set theory with union ultrafilters and small covering of meagre, II

Thursday, March 15, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Models of set theory with union ultrafilters and small covering of meagre, II

Abstract:

Union ultrafilters are ultrafilters that arise naturally from Hindman’s finite unions theorem, in much the same way that selective ultrafilters arise from Ramsey’s theorem, and they are very important objects from the perspective of algebra in the Cech–Stone compactification. The existence of union ultrafilters is known to be independent from the ZFC axioms (due to Hindman and Blass–Hindman), and is known to follow from a number of set-theoretic hypothesis, of which the weakest one is that the covering of meagre equals the continuum (this is due to Eisworth). In the first part of this two-talk series I exhibited a model of ZFC with union ultrafilters whose covering of meagre is strictly less than the continuum, obtained by means of a short countable support iteration. In this second talk, I will exhibit two more such models, one obtained by means of a countable support iteration of proper forcings, and the other by means of a single-step forcing (modulo being able to obtain an appropriate ground model).

## Yasser Fermán Ortiz Castillo: Crowded pseudocompact spaces of cellularity at most the continuum are resolvable

Place: Fields Institute (Room 210)

Date: March 9 , 2018 (13:30-15:00)

Speaker: Yasser Fermán Ortiz Castillo

Title: Crowded pseudocompact spaces of cellularity at most the continuum are resolvable

Abstract: It is an open question from W. Comfort and S. Garcia-Ferreira if it is true that every crowded pseudocompact space is resolvable. In this talk will be present a partial positive answer for spaces of cellularity at most the continuum.

## Grzegorz Plebanek: On almost disjoint families with property (R)

Tuesday, March 13, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Grzegorz Plebanek (University of Wroclaw)

Title: On almost disjoint families with property (R)

Abstract:

We consider (with A.Aviles and W. Marciszewski) almost disjoint families with some combinatorial property that has applications in functional analysis. We are looking for the minimal cardinality of m.a.d. family with property (R). It turns out that this cardinal is not greater than $non(\mathcal{N})$ the uniformity of null sets.

## Šárka Stejskalová – The tree property and the continuum function

Talk held by Šárka Stejskalová (KGRC) at the KGRC seminar on 2018-03-08.

Abstract: We will discuss the tree property, a compactness principle which can hold at successor cardinals such as $\aleph_2$ or $\aleph_3$. For a regular cardinal $\kappa$, we say that $\kappa$ has the tree property if there are no $\kappa$-Aronszajn trees. It is known that the tree property has the following non-trivial effect on
the continuum function:

(*) If the tree property holds at $\kappa^{++}$, then $2^\kappa> \kappa^+$.

After defining the key notions, we will review some basic constructions related to the tree property and state some original results regarding the tree property which suggest that (*) is the only restriction which the tree property puts on the continuum function in addition to the usual restrictions provable in ZFC.