Seminarium 22.05.2025
Szymon Żeberski
Politechnika Wrocławska
We will discuss the classical Galvin-Mycielski-Solovay theorem in the Cantor space. It connects strongly null sets with meager sets via *-operation. Furthermore, we will explore an attempt to generalize this result to encompass microscopic sets and porous sets. Additionally, we will discuss some properties of the *-operation. The presented results were obtained in collaboration with my PhD student Daria Perkowska.
Seminarium 23.05.2025 godz 11:30
Marcin Michalski
Politechnika Wrocławska
Let $\mathbb{T}$ be some family of trees in $2^\omega$ or $\mathbb{Z}^\omega$ and let $\mathcal{I}$ be a $\sigma-$ideal. In this talk, we explore the following question: Given any set $A\in\mathcal{I}$ and any tree $T\in\mathbb{T}$ there is a subtree $T'\subseteq T$ of the same type such that $$ A+\underbrace{[T']+[T']+\dots +[T']}_{n-times}\in \mathcal{I} $$ for every natural $n$? We will examine this question for standard $\sigma-$ideals in the Cantor space, such as those associated with Lebesgue measure, Baire category, and the $\sigma-$ideal generated by closed null sets. Additionally, we will consider the $\mathcal{M}_-$ ideal and "fake null" sets in the Baire space, which emerge from adapting combinatorial characterizations of their analogues (meager sets and null sets) in the Cantor space.
These results were obtained together with Łukasz Mazurkiewicz, Robert Rałowski and Szymon Żeberski.
Seminarium 12.06.2025
Seminarium 26.06.2025
