## Shashi Srivastava: Some Applications of Descriptive Set Theory to Transition Probabilities

Tuesday, September 20, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Shashi Srivastava (Kalkuta)

Title: Some Applications of Descriptive Set Theory to Transition Probabilities

Abstract:

We use measurable selection theorems and prove several results on extensions and existence of transition probabilities with prescribed domain. This is part of joint work with E. E. Doberkat. The remaining part of the work will be presented at Mathematical Institute, University of Wroclaw on 21 September 2016.

## Wiesław Kubiś: Abstract Banach-Mazur game

Tuesday, May 31, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Wiesław Kubiś (Czech Academy of Sciences, KSW University)

Title: Abstract Banach-Mazur game

Abstract:

We will discuss an infinite game in which two players alternately choose some objects (structures) from a given class. The only rule is that at each move the structure chosen by the player should extend the one chosen in the previous move by the opponent. One of the players wins if the limit of the chain of structures resulting from the play is isomorphic to some concrete (fixed in advance) object. We will show some basic results and relevant examples concerning the existence of winning strategies.

## Andrzej Kucharski: $\kappa$-metrizable spaces

Tuesday, May 17, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Andrzej Kucharski (Silesian University in Katowice)

Title: $\kappa$-metrizable spaces

Abstract:

We introduce a new supclass of $\kappa$-metrizable spaces, namely $\omega$ $\kappa$-metrizable spaces.
We show that $\kappa$-metrizable spaces form a proper subclass of $\omega$ $\kappa$-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with $\kappa$-metrizable spaces.

## Barnabas Farkas: Towers in filters and related problems

Tuesday, May 10, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Barnabas Farkas (University of Vienna)

Title: Towers in filters and related problems

Abstract:

I am going to present a survey on my recently finished joint work with J. Brendle and J. Verner. In this paper we investigated which filters can contain towers, that is, a $\subseteq^*$-decreasing sequence in the filter without any pseudointersection (in $[\omega]^\omega$). I will present Borel examples which contain no towers in $\mathrm{ZFC}$, and also examples for which it is independent of $\mathrm{ZFC}$. I will prove that consistently every tower generates a non-meager filter, in particular (consistently) Borel filters cannot contain towers. And finally, I will present the “map” of logical implications and non-implications between (a) the existence of a tower in a filter $\mathcal{F}$, (b) inequalities between cardinal invariants of $\mathcal{F}$, and (c) the existence of a peculiar object, an $\mathcal{F}$-Luzin set of size $\geq\omega_2$.

## Aleksandra Kwiatkowska: Universal flows and Ramsey theory

Tuesday, March 22, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksandra Kwiatkowska (University of Bonn)

Title: Universal flows and Ramsey theory

Abstract:

The subject lies on the crossroad of topological dynamics, topology, topological groups and Ramsey theory.
We will present Kechris-Pestov-Todorcevic theorem about connections between structural Ramsey theory, extremely amenable groups and universal minimal flows.
We will show some examples. Next, we will focus on groups of homeomorphisms (Cantor set, Lelek fan, pseudoarc, Hilbert cube). We will recall known results and ask some questions.

## Tomasz Żuchowski: Nonseparable growth of omega supporting a strictly positive measure

Tuesday, March 15, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Tomasz Żuchowski (University of Wroclaw)

Title: Nonseparable growth of $\omega$ supporting a strictly positive measure

Abstract:

We will construct in ZFC a compactification $\gamma\omega$ of
$omega$ such that its remainder $\gamma\omega\backslash\omega$ is not
separable and carries a strictly positive measure, i.e. measure positive
on nonempty open subsets. Moreover, the measure on our space is defined by
the asymptotic density of subsets of $\omega$.

Our remainder is a Stone space of a Boolean subalgebra of Lebesgue
measurable subsets of $2^{\omega}$ containing all clopen sets.

## Piotr Borodulin-Nadzieja: Mathias forcings for slaloms

Tuesday, March 8, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Borodulin-Nadzieja (University of Wroclaw)

Title: Mathias forcings for slaloms

Abstract:

We will show an example of a Boolean algebra which is not sigma-centered but sigma-n-linked. Moreover, it has property (*) of Fremlin. Such examples were known before. We will construct our algebra using Mathias forcing for something resembling the density filter.

## Robert Rałowski: Bernstein set and continuous functions

Tuesday, March 1, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Robert Rałowski (Wrocław University of Technology)

Title: Bernstein set and continuous functions

Abstract:

Alexander V. Osipov asked “It is true that for any Bernstein subset $B\subset \mathbb{R}$ there are countable many continous functions from $B$ to $\mathbb{R}$ such that the union of images of $B$ is a whole real line $\mathbb{R}$”. We give the positive answer for this question, but we show that this result is not true for a $T_2$ class of functions.

We show some consistency results for completely nonmeasurable sets with respect to $\sigma$-ideals of null sets and meager sets on the real line.

These results was obtained commonly with Jacek Cichoń, Michał Morayne and me.

## Aleksander Cieślak: Filters and sets of Vitali’s type

Thursday, January 1, 1970, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wrocław University of Technology)

Title: Filters and sets of Vitali’s type

Abstract:

In construction of classical Vitali set on $\{0,1\}^{\omega}$ we use filter of cofinite sets to define rational numbers. We replece cofinite filter by any nonprincipal filter on $\omega$ and ask some questions about measurability and cardinality of selectors and equevalence classes.

## Jan Stary: Coherent ultrafilters

Tuesday, January 12, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jan Stary (Czech Technical University, Prague)

Title: Coherent ultrafilters

Abstract:

The notion of a P-ultrafilter on omega can be straightened in a natural way to the notion of a coherent P-ultrafilter on a complete ccc Boolean algebra. These ultrafilters exist generically under the condition isolated by Ketonen, namely c = d.
Similarly, under the Canjar condition c = cov(Meager), coherently Ramsey ultrafilters can be shown to exist.
Existence of “coherent” versions of other traditional objects is an ongoing programme.
The coherent ultrafilters are relevant in an old topological question: a coherent P-ultrafilter on an algebra B is an untouchable point in the Stone space of B, witnessing its nonhomogeneity.