Olena Karlova: Extension of Borel maps and Borel retracts of topological spaces

Tuesday, December 19, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Olena Karlova (Chernivtsi National University, Ukraine)

Title: Extension of Borel maps and Borel retracts of topological spaces

Abstract:

We will discuss the problem of extension of (dis)continuous maps between topological spaces. Concepts of Baire and Borel retracts of topological spaces will be introduced. Some open problems will be considered.

Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees

Tuesday, November 28, 2017, 17:15
Wrocław University of Science and Technology, 215 D-1

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

Title: Bernstein, Luzin and Sierpiński meet trees

Abstract:

In [2] we have proven that if $\mathfrak{c}$ is a regular cardinal number, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to Marczewski ideal $s_0$. We will generalize this result for other tree ideals – $m_0$ and $l_0$ – using some lemmas on special kind of fusion sequences for trees of respective type.
Let us introduce a following notion. Let $\mathbb{X}$ be a set of trees.
Definition. We call a set $B$ a $\mathbb{X}$-Bernstein set, if for each $X\in\mathbb{X}$ we have $[X]\cap B\neq\emptyset$.
We shall explore this notion for various set of trees, including Sacks, Miller and Laver trees, with the support of technics developed in [1].

[1] Brendle J., Strolling through paradise, Fundamenta Mathematicae, 148 (1995), pp. 1-25.
[2] Michalski M., Żeberski Sz., Some properties of I-Luzin, Topology and its Applications, 189 (2015), pp. 122-135.

Sakae Fuchino: Downward Löwenheim-Skolem Theorems in stationary logic

Tuesday, November 21, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Sakae Fuchino (Kobe University)

Title: Downward Löwenheim-Skolem Theorems in stationary logic

Tomasz Natkaniec: Perfectly everywhere surjective but not Jones functions

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

Speaker: Tomasz Natkaniec (University of Gdansk)

Title: Perfectly everywhere surjective but not Jones functions

Abstract:

Given a function $f:\mathbb{R}\to\mathbb{R}$ we say that

1. $f$ is perfectly surjective ($f\in \mathrm{PES}$) if $f[P]=\mathbb{R}$ for every perfect set $P$;
2. $f$ is a Jones function ($f\in\mathrm{J}$) if $C\cap f\neq\emptyset$ for every closed $C\subset\mathbb{R}^2$ with $\mathrm{dom}(C)$ of size $\mathfrak{c}$.

M. Fenoy-Munoz, J.L. Gamez-Merino, G.A. Munoz-Fernandez and E. Saez-Maestro in the paper ”A hierarchy in the family of real surjective functions” [Open Math. 15 (2017), 486–501] asked about the lineability of the set $\mathrm{PES}\setminus\mathrm{J}$.
Answering this question we show that the class $\mathrm{PES}\setminus\mathrm{J}$ is $\mathfrak{c}^+$-lineable. Moreover, if
$2^{<\mathfrak{c}}=\mathfrak{c}$ then $\mathrm{PES}\setminus\mathrm{J}$ is $2^\mathfrak{c}$-lineable. We prove also that the additivity number
$A(\mathrm{PES}\setminus\mathrm{J})$ is between $\omega_1$ and $\mathfrak{c}$. Thus $A(\mathrm{PES}\setminus\mathrm{J})=\mathfrak{c}$ under CH,
however this equality can't be proved in ZFC, because the Covering Property Axiom CPA implies $A(\mathrm{PES}\setminus\mathrm{J})=\omega_1<\mathfrak{c}$.

The talk is based on the joint paper:
K.C.Ciesielski, J.L. Gamez-Merino, T. Natkaniec, and J.B. Seoane-Sepulveda, '' On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity'', submitted.

Barnabas Farkas: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration

Tuesday, November 7, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Barnabas Farkas (TU Wien)

Title: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration

Abstract:

I will present two models concerning interactions between the existence of towers in analytic P-ideals and their cardinal invariants. It is trivial to see that if there is no tower in $\mathcal{I}$, then $\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})$. I will prove that this implication cannot be reversed no matter the value of $\mathrm{non}^*(\mathcal{I})$. More precisely, let $\mathcal{I}$ be an arbitrary tall analytic P-ideal, I will construct the following two models:

Model1 of $\mathrm{non}^*(\mathcal{I})=\mathfrak{c}$,
there is a tower in $\mathcal{I}$, and $\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})$. Method: Small filter iteration.

Model2 of $\mathrm{non}^*(\mathcal{I})<\mathfrak{c}$,
there is a tower in $\mathcal{I}$, and $\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})$. Method: Matrix iteration.

This is a joint work with J. Brendle and J. Verner.

Ziemowit Kostana: Non-measurability of algebraic sum

Sunday, September 17, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Ziemowit Kostana (University of Warsaw)

Title: Non-measurability of algebraic sum

Abstract:

Consider following problems:
1. If A is meagre (null) subset of real line, does there necessarily exist set B such that algebraic sum A+B doesn’t have Baire property (is non-measurable)?
2. If A is meagre (null) subset of real line, does there necessarily exist non-meagre (non-null) additive subgroup, disjoint with some translation of A?

It is not hard to prove that positive answer to 2. implies positive answer to 1, both for measure and category.
We answer 2. affirmatively for category, while version for measure turns out to be independent of ZFC. The latter was essentially proved last year
by A. Rosłanowski and S. Shelah. Both results holds for Cantor space with coordinatewise addition mod. 2 as well.

Aleksander Cieślak: Ideals of subsets of plane

Sunday, September 10, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wrocław University of Technology)

Title: Ideals of subsets of plane

Abstract:

For given two ideals $I$ and $J$ of subsets of Polish space $X$ we define a Fubini product $I\times J$ as all these subsets of plane $X^2$ which can be covered by a Borel set $B$ such that $I$-almost all its vertical sections are $J$-small. We will investigate how properties of factors influence properties of product.

Aleksander Cieślak: Cohen-stable families of subsets of integers

Tuesday, June 13, 2017, 17:15
Wrocław University of Technology, 215 D-1

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

Title: Cohen-stable families of subsets of integers

Abstract:

A mad family is Cohen-stable if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-stable. We will find condition necessary and sufficient for mad family to be Cohen-unstabe and investigate when such family exist.

Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jarosław Swaczyna (Lodz University of Technology)

Title: Haar-small sets

Abstract:

In locally compact Polish groups there is a very natural $\sigma$-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.

Joanna Jureczko: Some remarks on Kuratowski partitions, new results

Tuesday, May 9, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Joanna Jureczko (Cardinal Stefan Wyszynski University in Warsaw)

Title: Some remarks on Kuratowski partitions, new results

Abstract:

K. Kuratowski in 1935 posed the problem whether a function $f \colon X \to Y$ from a completely metrizable space $X$ to a metrizable space $Y$ is continuous apart from a meager set.
This question is equivalent to the question about the existence of so called a Kuratowski partition, i. e. a partition $\mathcal{F}$ of a space $X$ into meager sets such that $\bigcup \mathcal{F}’$ for any $\mathcal{F}’ \subset \mathcal{F}$.
With any Kuratowski partition we may associate a $K$-ideal, i.e. an ideal of the form
$$I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A}F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.$$
It would seem that the information about $I_{\mathcal{F}}$ would give us full information about the ideal and the world in which it lives.
My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier.
During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz – Kunen Theorem (1987) on the existence of measurable cardinals.