## Piotr Borodulin-Nadzieja : Measures and slaloms

Tuesday, December 15, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Borodulin-Nadzieja (University of Wrocław)

Title: Measures and slaloms

Abstract:

We examine combinatorial properties of certain families of slaloms. In particular, we investigate destructibility by random forcing. Then, using these results and a technique due to Todorcevic we construct several examples of compact spaces supporting measures. E.g. we show that a non-trivial piece of measure algebra always embed in $P(\omega)/fin$. Joint work with Tanmay Inamdar.

## David Chodounsky: Y-cc and Y-proper forcing notions

Tuesday, December 8, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: David Chodounsky (Czech Academy of Sciences)

Title: Y-cc and Y-proper forcing notions

Abstract:

In our recent joint work with J. Zapletal, we explored a new type of properties of forcing notions, of which the Y-cc and Y-proper are the most prolific examples. These two notions can be seen as a generalization of the notion of $\sigma$-centered. While keeping similar consequences a larger and better behaved class of posets is cover by these notions.
I will give an introduction to this topic with the focus on understating the basic techniques and ideas.

## Piotr Szewczak: Products of Menger spaces

Tuesday, November 24, 2015, 17:15
Wrocław University of Technology, 215 D-1

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

Coauthor: Boaz Tsaban (Bar-Ilan University)

Title: Products of Menger spaces

Abstract:

A topological space $X$ is Menger if for every sequence of open covers $O_1, O_2,\ldots$ there are finite subfamilies $F_1$ of $O_1$, $F_2$ of $O_2$, . . . such that their union is a cover of $X$. The above property generalizes sigma-compactness.

One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.

## Marcin Michalski: A generalized version of the Rothberger theorem

Tuesday, November 17, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marcin Michalski (Wrocław University of Technology)

Title: A generalized version of the Rothberger theorem

Abstract:

We call a set $X$ a generalized Luzin set if $|L\cap M|<|L|$ for every meager set $M$. Dually, if we replace meager set with a null set, we obtain a definition of a generalized Sierpiński set.

We will show that if $2^\omega$ is a regular cardinal then for every generalized Luzin set $L$ and every generalized Sierpiński set $S$ an algebraic sum $L+S$ belongs to the Marczewski ideal $s_0$ (i.e. for every perfect set $P$ there exists a perfect set $Q$ such that $Q\subseteq P$ and $Q\cap (L+S)=\emptyset$). To prove the theorem we shall prove and use a generalized version of the Rothberger theorem.

We will also formulate a series of results involving algebraic, topological and measure structure of the real line, that emerged during searching for a proof of the above theorem.

## Julia Wódka: Comparison of some families of real functions in sense of porosity

Thursday, January 1, 1970, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Julia Wódka (Łódź University of Technology)

Title: Comparison of some families of real functions in sense of porosity

Abstract:

We consider set $\mathbb{R}^\mathbb{R}$ with uniform convergence metric, i.e:
$\rho(f,g)=\min{1,\sup\limits_{x\in\mathbb{R}}|f(x)-g(x)|}\quad \text{for f,g\in\mathbb{R}^\mathbb{R}}$
and the following subsets of $\mathbb{R}^\mathbb{R}$:

Darboux functions ($f\in\mathscr{D}$ if whenever $a <b$ there exists a nontrivial interval $J\subset I$ such that ${\rm diam} (f[J\cup \{x\}]) <\varepsilon$.

Świątkowski functions ($f\in\mathscr{\acute S}$ if for all $a<b$ with $f(a)\ne f(b)$, there is a $y$ between $f(a)$ and $f(b)$ and an $x\in (a,b) \cap \mathcal{C}(f)$ such that $f(x)=y$, where $\mathcal{C}(f)$ denotes the set of all continuity points of function $f$).

strongly Świątkowski functions ($f\in\mathscr{\acute S}_s$ if for all $a<b$ and each $y$ between $f(a)$ and $f(b)$ there is an $x\in (a,b) \cap \mathcal{C}(f)$ such that $f(x)=y$).
\end{itemize}

The aim of this is to compare this sets in terms of porosity.

Let $(X,d)$ be a metric space, $x\in A\subset M$, and $r\in\mathbb{R}_+$. We define
$\gamma(x,r,M)=\sup\{{t\geq 0}:\ \exists_{z\in M} B(z,t)\subset B(x,r)\setminus M\}$
and $p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.$
$\Bigl(p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\Bigr)$
Quantity $p^u(M,x)$ is called upper porosity of $M$ at the point $x$.
We say that $M$ is upper $p-$porous if $p=\inf\{p^u(M,x):\ x\in M\}>0$.

Analogously we define lower porosity.

## Aleksander Cieślak: On nonmeasurable subsets of $\mathbb{R}$ and $\mathbb{R}^2$

Tuesday, October 27, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wrocław University of Technology)

Title: On nonmeasurable subsets of $\mathbb{R}$ and $\mathbb{R}^2$

Abstract:

I would like to present some results connected with the existence of a subset $X$ of the square $[0,1]^2$ with the property that for any line $L$ outside $[0,1]^2$ the projection $\pi_L[X]$ is completely nonmeasurable in some interval with respect to selected $sigma$-ideal with Borel base on the line $L$.

Moreover, I will discuss the existence of large midpoint-free subsets of arbitrary subset of the real line.

## Antonio Aviles: Boolean algebras obtained by push-out iteration

Tuesday, October 20, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Antonio Aviles (University of Murcia)

Title: Boolean algebras obtained by push-out iteration

Abstract:

We discuss the notion of push-out in the category of Boolean algebras, and we describe a method of constructing Boolean algebras by transfinite iterative push-outs. Under $CH$ and in a model obtained by adding $\aleph_2$ Cohen reals to a model of $CH$, $P(\omega)/fin$ is such an algebra.

## Damian Sobota: The Nikodym property and cardinal invariants of the continuum

Tuesday, October 13, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Damian Sobota (Polish Academy of Sciences)

Title: The Nikodym property and cardinal invariants of the continuum

Abstract:

A Boolean algebra $\mathcal{A}$ is said to have the Nikodym property if every sequence $(\mu_n)$ of measures on $\mathcal{A}$ which is elementwise bounded (i.e. $\sup_n|\mu_n(a)|<\infty$ for every $a\in\mathcal{A}$) is uniformly bounded (i.e. $\sup_n|\mu_n|<\infty$). The property is closely related to the classical Banach-Steinhaus theorem for Banach spaces.

My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside $\wp(\omega)$ or $\omega^\omega$, e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of $\wp(\omega)$ having some special measure-theoretic properties.

## Magdalena Nowak: Zero-dimensional spaces as topological and Banach fractals

Tuesday, June 16, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Magdalena Nowak (Jan Kochanowski University in Kielce)

Title: Zero-dimensional spaces as topological and Banach fractals

Abstract:

A topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\mathcal{F}}f(X)$ for a finite system $\mathcal{F}$ of continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal{U}$ of $X$ there is a number $n\in\mathbb{N}$ such that for any functions $f_1,\dots,f_n\in \mathcal{F}$, the set $f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal{U}$. If, in addition, all functions $f\in\mathcal{F}$ have Lipschitz constant $<1$ with respect to some metric generating the topology of $X$, then the space $X$ is called a Banach fractal. It is known that each topological fractal is compact and metrizable. We prove that a zero-dimensional compact metrizable space $X$ is a topological fractal if and only if $X$ is a Banach fractal if and only if $X$ is either uncountable or $X$ is countable and its scattered height $\hbar(X)$ is a successor ordinal.

## Szymon Żeberski: Applications of Shoenfield Absoluteness Lemma

Tuesday, June 2, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Szymon Żeberski (Wrocław University of Technology)

Title: Applications of Shoenfield Absoluteness Lemma

Abstract:

We will recall Shoenfield Absoluteness Lemma about $\Sigma^1_2$ sentences. We will show applications of this theorem connected to topological and algebraic structure of Polish spaces in publications co-authored by the speaker.