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 $V=L$.

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.

Jacek Tryba: Homogeneity of ideals

Tuesday, March 6, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jacek Tryba (University of Gdansk)

Title: Homogeneity of ideals

Abstract:

The homogeneity family of the ideal $\mathcal{I}$ is a family of subsets such that the restriction of $\mathcal{I}$ to this subset is isomorphic to $\mathcal{I}$.
We say that an ideal $\mathcal{I}$ is homogeneous if all $\mathcal{I}$-positive sets belong to the homogeneity family of $\mathcal{I}$. We investigate basic properties of this notion, give examples of homogeneous ideals and present some applications to ideal
convergence.
Moreover, we present connections between the homogeneity families and the notion of bi-$\mathcal{I}$-invariant functions introduced by Balcerzak, Głąb and Swaczyna and give answers to several questions related to this topic.

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.