## Andrzej Starosolski: The Rudin-Keisler ordering of P-points under b=c

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

Speaker: Andrzej Starosolski (Silesian University of Technology)

Title: The Rudin-Keisler ordering of P-points under $\mathfrak b=\mathfrak c$

Abstract:

M. E. Rudin proved under CH that for each P-point there exists another P-point strictly RK-greater . Assuming $\mathfrak p=\mathfrak c$, A. Blass showed the same; moreover, he proved that each RK-increasing $\omega$-sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-preordering. He also asked what ordinals can be embedded in the set of P-points.
In my talk the results cited above are proved and the mentioned question is answered under a (weaker) assumption $\mathfrak b =\mathfrak c$.

## Marek Bienias: About universal structures and Fraisse theorem

Tuesday, April 24, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marek Bienias (Łódź University of Technology)

Title: About universal structures and Fraisse theorem

Abstract:

For a given structure D of language L we can consider age of D, i.e. the family of all finitely generated L-substructures od D. It turns out that age has property (HP) and (JEP). Fraisse theorem let us revers the procedure: if K is nonempty countable family of finitely generated L-structures having properties (HP), (JEP) and (AP), then there exists exactly one (up to isomorphism) L-structure D (so called Fraisse limit) which is countable ultrahomogenous and has age K.
The aim of the talk is to define basic notions from Fraisse theory, proof the main theorem and show some alternative way of looking at the construction of Fraisse limit.

## Piotr Borodulin-Nadzieja: Tunnels through topological spaces

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

Speaker: Piotr Borodulin-Nadzieja (University of Wroclaw)

Title: Tunnels through topological spaces

Abstract:

I will show a ZFC example of a compact space (without
isolated points) through which one cannot drill a tunnel. I will discuss
when and when not $\omega^*$ has a tunnel.

## 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.