April 2016

Barnabas Farkas: Towers in filters and related problems

Tuesday, May 10, 2016 17:15

Room: D1-215

Speaker:
Barnabas Farkas

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

Monday, April 25, 2016 17:15

Room: 604 IM

Speaker:
Magdalena Nowak

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.