## Daria Michalik: Degree of homogeneity of connes over locally connected curves

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

Speaker: Daria Michalik (Cardinal Stefan Wyszyński University)

Title: Degree of homogeneity of connes over locally connected curves

Abstract:
$\mathcal{H}(X)$ denotes the group of self-homeomorphisms of $X$.
An orbit of a point $x_0$ in $X$ is the set:
$$\mathcal{O}_X(x_0) = \{h(x_0) : h\in\mathcal{H}(X)\}.$$
$X$ is $1/n$-homogeneous if $X$ has exactly $n$ orbits. In such a case we say that the degree of homogeneity of $X$ equals $n$.
P. Pellicer-Covarrubias, A. Santiago-Santos calculated the degree of homogeneity of connes over local dendrites depending on the degree of homogeneity of their bases.
We will generalize above result on connes over locally connected curves.

## Szymon Głąb: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

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

Speaker: Szymon Głąb (Łódź University of Technology)

Title: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Abstract:

Let $1 \le n \le\omega$. Let $A_n$ be a set of natural numbers less than $n$. Define $<$ on $A_n$ so that for no
$x, y \in A_n$ is $x<y$. Let $B_n = A_n \times\mathbb{Q}$ where $\mathbb{Q}$ is the set of rational numbers. Define $<$ on $B_n$ so
that $(k, p) < (m, q)$ iff $k = m$ and $p < q$. Let $C_n = B_n$ and define $<$ on $C_n$ so that $(k, p) < (m, q)$
iff $p < q$. Finally, let $(D, <)$ be the universal countable homogeneous partially ordered set, that is a
Fraisse limit of all finite partial orders.
A structure is called ultrahomogeneous, if every embedding of its finitely generated substructure
can be extended to an automorphism. Schmerl showed that there are only countably many,
up to isomorphism, ultrahomogeneous countable partially ordered sets. More precisely he proved the
following characterization:

Let $(H, <)$ be a countable partially ordered set. Then $(H, <)$ is ultrahomogeneous iff it
is isomorphic to one of the following:
$(A_n, <)$ for $1 \le n \le\omega$;
$(B_n, <)$ for $1 \le n \le\omega$;
$(C_n, <)$ for $1 \le n \le\omega$;
$(D, <)$.

Moreover, no two of the partially ordered sets listed above are isomorphic.
Consider automorphisms groups $Aut(A_\omega) = S_\infty$, $Aut(B_n)$, $Aut(C_n)$ and $Aut(D)$. We prove that
each of these groups contains two elements f, g such that the subgroup generated by f and g is free
and dense. By Schmerl’s Theorem the automorphism group of a countable infinite partially ordered
set is freely topologically 2-generated.

## Marcin Michalski: Universal sets for bases of $\sigma$-ideals

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

Speaker: Marcin Michalski (Wrocław University of Science and Technology)

Title: Universal sets for bases of $\sigma$-ideals

Abstract:

We shall construct universal sets of possibly low Borel rank for classic $\sigma$-ideals of sets: $\mathcal{N}$-family of measure zero sets, $\mathcal{M}$-family of meager sets, $\mathcal{M}\cap\mathcal{N}$ and $\mathcal{E}$. We will also discuss briefly cases of other ideals.

## David Chodounsky: Combinatorial properties of the Mathias-Prikry forcing

Tuesday, October 25, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: David Chodounsky (Czech Academy of Sciences)

Title: Combinatorial properties of the Mathias-Prikry forcing

Abstract:

I will review basic fact and results about the Mathis-Prikry forcing and I will present and prove sufficient condition for genericity of reals with respect to this poset. Time permitting, further connections of parameters of the forcing with its properties will be explored.

## Marcin Michalski: On some properties of sigma-ideals

Tuesday, October 18, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marcin Michalski (Wrocław University of Science and Technology)

Title: On some properties of sigma-ideals

Abstract:

We shall consider a couple of properties of $\sigma$-ideals and relations between them. Namely we will prove that $\mathfrak c$-cc $\sigma$-ideals are tall, Weaker Smital Property implies that every Borel $\mathcal I$-positive set contains a witness for non($\mathcal I$) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of $\mathcal I$-Luzin sets and prove that the ideal $[{\mathbb R}]^{\leq\omega}$ does not posses the Fubini Property using some interesting lemma about perfect sets.

## Aleksander Cieślak: Nonmeasurable images in Polish space with respect to selected sigma ideals

Tuesday, October 11, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wrocław University of Science and Technology)

Title: Nonmeasurable images in Polish space with respect to selected sigma ideals

Abstract:

We present results on nonmeasurability (with respect to a selected
sigma-ideal on a Polish space) of images of functions defined on Poilish spaces. In particular, we give a positive answer to the following question: Is there a subset of the unit disc in the real plane such that continuum many projections onto lines are Lebesgue measurable and continuum many projections are not?
Results were obtained together with Robert Rałowski.

## 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 $\geq\omega_2$.