## Marcin Michalski: Luzin’s theorem

Tuesday, April 25, 2017, 17:15
Wrocław University of Technology, 215 D-1

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

Title: Luzin’s theorem

Abstract:

In 1934 Nicolai Luzin proved that each subset of the real line can be decomposed into two full subsets with respect to ideal of measure or category. We shall present the proof of this result partially decoding his work and we will also briefly discuss possible generalizations.

## Aleksander Cieślak: Indestructible tower

Tuesday, April 11, 2017, 17:15
Wrocław University of Technology, 215 D-1

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

Title: Indestructible tower

Abstract:

Following the Kunen’s construction of m.a.d. family which is indestructible over adding $\omega_2$ Cohen reals we provide analogous construction for indestructibe tower.

## Judyta Bąk: Domain theory and topological games

Tuesday, March 28, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Judyta Bąk (University of Silesia)

Title: Domain theory and topological games

Abstract:

Domain is a partially ordered set, in which there was introduced some specific relation. We say that a space is domain representable if it is homeomorphic to a space of maximal elements of some domain. In 2015 W. Fleissner and L. Yengulalp introduced a notion of $\pi$–domain representable space, which is analogous of domain representable. We prove that a player $\alpha$ has a winning strategy in the Banach–Mazur game on a space $X$ if and only if $X$ is countably $\pi$–domain representable. We give an example of countably $\pi$–domain representable space, which is not $\pi$–domain representable.

## Piotr Szewczak: The Scheepers property and products of Menger spaces

Tuesday, March 14, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Szewczak (Cardinal Stefan Wyszynski University in Warsaw)

Title: The Scheepers property and products of Menger spaces

Abstract:

A topological space $X$ is Menger if for every sequence of open covers $\mathcal{O}_1, \mathcal{O}_2,\ldots$ of the space $X$, there are finite subfamilies $\mathcal{F}_1\subseteq \mathcal{O}_1, \mathcal{F}_2\subseteq\mathcal{O}_2,\ldots$ such that their union is a cover of $X$. If, in addition, for every finite subset $F$ of $X$ there is a natural number $n$ with $F\subseteq\bigcup\mathcal{F}_n$, then the space $X$ is Scheepers. The above properties generalize $\sigma$-compactness, and Scheepers’ property is formally stronger than Menger’s property. It is consistent with ZFC that these properties are equal.

One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely
combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces

This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).

## Aleksander Cieślak: Strongly meager sets and subsets of the plane

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

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

Title: Strongly meager sets and subsets of the plane

Abstract:

We will show some results proved by J. Pawlikowski in “Strongly meager sets and subsets of the plane”.

## Marcin Michalski: Decomposing the real line into Borel sets closed under addition

Tuesday, December 13, 2016, 17:15
Wrocław University of Technology, 215 D-1

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

Title: Decomposing the real line into Borel sets closed under addition

Abstract:

We will show some results proved by M. Elekes and T.Keleti in “Decomposing the real line into Borel sets closed under addition”.

## Artur Bartoszewicz: On the sets of subsums of series

Tuesday, November 29, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Artur Bartoszewicz (Łódź University of Technology)

Title: On the sets of subsums of series

Abstract:

The first observations connected with sets of subsums of series (s.c. achievement sets) belong to Kakeya and are over 100 years old. In my lecture I want to present the story of the studies of the problem and the results obtained by my cooperators and me quite recently. These results concern the series generating Cantorvals, connections between the achievement sets of series and the atractors of affine IFS’s and achievement sets of conditionally convergent series in the plane.

## 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.