April 2016

# Barnabas Farkas: Towers in filters and related problems

26/04/16 14:02

Tuesday, May 10, 2016 17:15

*Room:*D1-215*Barnabas Farkas*

Speaker:Speaker:

*Title*: Towers in filters and related problems*Abstract*. I am going to present a survey on my recently finished joint work with J. Brendle and J. Verner. In this paper we investigated which filters can contain towers, that is, a \(\subseteq^*\)-decreasing sequence in the filter without any pseudointersection (in \([\omega]^\omega\)). I will present Borel examples which contain no towers in \(\mathrm{ZFC}\), and also examples for which it is independent of \(\mathrm{ZFC}\). I will prove that consistently every tower generates a non-meager filter, in particular (consistently) Borel filters cannot contain towers. And finally, I will present the "map'' of logical implications and non-implications between (a) the existence of a tower in a filter \(\mathcal{F}\), (b) inequalities between cardinal invariants of \(\mathcal{F}\), and (c) the existence of a peculiar object, an \(\mathcal{F}\)-Luzin set of size \(\geq\omega_2\).# Magdalena Nowak: Counterexamples for IFS-attractors

20/04/16 15:22

**Monday**, April 25, 2016 17:15

*Room:*

**604 IM**

*Magdalena Nowak*

Speaker:

Speaker:

*Title*: Counterexamples for IFS-attractors

*Abstract*. I deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS). An attractor is a compact set which remains invariant for such a family. Thus, I consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which I will refer to as Banach and topological fractals, respectively. I present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.