Aleksander Cieślak: Cohen-stable families of subsets of integers

Tuesday, June 13, 2017 17:15

Room: D1-215

Speaker:
Aleksander Cieślak

Title: Cohen-stable families of subsets of integers

Abstract. A mad family is Cohen-stable if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-stable. We will find condition necessary and sufficient for mad family to be Cohen-unstabe and investigate when such family exist.

Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017 17:15

Room: D1-215

Speaker:
Jarosław Swaczyna

Title: Haar-small sets

Abstract. In locally compact Polish groups there is a very natural \(\sigma\)-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.

Joanna Jureczko: Some remarks on Kuratowski partitions, new results

Tuesday, May 9, 2017 17:15

Room: D1-215

Speaker:
Joanna Jureczko

Title: Some remarks on Kuratowski partitions, new results

Abstract.K. Kuratowski in 1935 posed the problem whether a function \(f \colon X \to Y\) from a completely metrizable space \(X\) to a metrizable space \(Y\) is continuous apart from a meager set. This question is equivalent to the question about the existence of so called a Kuratowski partition, i. e. a partition \(\mathcal{F}\) of a space \(X\) into meager sets such that \(\bigcup \mathcal{F}'\) for any \(\mathcal{F}' \subset \mathcal{F}\). With any Kuratowski partition we may associate a \(K\)-ideal, i.e. an ideal of the form

\(I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A} F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.\)

It would seem that the information about \(I_{\mathcal{F}}\) would give us full information about the ideal and the world in which it lives.
My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier. During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz - Kunen Theorem (1987) on the existence of measurable cardinals.

Marcin Michalski: Luzin's theorem

Tuesday, April 25, 2017 17:15

Room: D1-215

Speaker:
Marcin Michalski

Title: Luzin's theorem

Abstract. In 1934 Nicolai Luzin proved that each subset of the real line can be decomposed into two full subsets with respect to ideal of measure or category. We shall present the proof of this result partially decoding his work and we will also briefly discuss possible generalizations.

Aleksander Cieślak: Indescructible tower

Tuesday, April 11, 2017 17:15

Room: D1-215

Speaker:
Aleksander Cieślak

Title: Indescructible tower

Abstract. Following the Kunen's construction of m.a.d. family which is indestructible over adding \(\omega_2\) Cohen reals we provide analogous construction for indestructibe tower.