June 2022
Eliza Jabłońska: From the Steinhaus property to the Laczkovich one
10/06/22 08:16
Tuesday, June 14, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Eliza Jabłońska (AGH University of Science and Technology)
Title: From the Steinhaus property to the Laczkovich one
Abstract: Let \(X\) be a locally compact Abelian Polish group, \(\mathcal{B}(X)\) be the family of all Borel subsets of \(X\) and \(\mathcal{F}\subset 2^{X}\). We consider the following Steinhaus' type properties:
It is known that the family \(\mathcal{M}\) of all meager sets as well as the family \(\mathcal{N}\) of all sets of Haar measure zero satisfy each of these conditions. We prove that the family \(\mathcal{M}\cap\mathcal{N}\) satisfies \((S^-)\), \((D^+)\), \((D^-)\) although it does not satisfy \((S^+)\). We also show that the \(\sigma\)-ideal \(\sigma\overline{\mathcal{N}}\subset \mathcal{M}\cap\mathcal{N}\) generated by closed sets of Haar measure zero satisfies only \((D^+)\) and \((D^-)\) which leads us to the Laczkovich property. This is joint work with T. Banakh, I. Banakh, Sz. Głąb and J. Swaczyna.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Eliza Jabłońska (AGH University of Science and Technology)
Title: From the Steinhaus property to the Laczkovich one
Abstract: Let \(X\) be a locally compact Abelian Polish group, \(\mathcal{B}(X)\) be the family of all Borel subsets of \(X\) and \(\mathcal{F}\subset 2^{X}\). We consider the following Steinhaus' type properties:
- \((S^+)\): \((A+B)\neq\emptyset\) for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((S^-)\): \(0\in (A-A)\) for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((D^+)\): \(A+B\) is non-meager for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((D^-)\): \(A-A\) is non-meager for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\).
It is known that the family \(\mathcal{M}\) of all meager sets as well as the family \(\mathcal{N}\) of all sets of Haar measure zero satisfy each of these conditions. We prove that the family \(\mathcal{M}\cap\mathcal{N}\) satisfies \((S^-)\), \((D^+)\), \((D^-)\) although it does not satisfy \((S^+)\). We also show that the \(\sigma\)-ideal \(\sigma\overline{\mathcal{N}}\subset \mathcal{M}\cap\mathcal{N}\) generated by closed sets of Haar measure zero satisfies only \((D^+)\) and \((D^-)\) which leads us to the Laczkovich property. This is joint work with T. Banakh, I. Banakh, Sz. Głąb and J. Swaczyna.
Jonathan Cancino: Ideal independent families and ultrafilters
02/06/22 09:43
Tuesday, June 7, 2022 17:15
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Jonathan Cancino (Czech Academy of Sciences)
Title: Ideal independent families and ultrafilters
Abstract: A family \(\mathscr{I}\subseteq[\omega]^\omega\) is called ideal independent if no element \(A\in\mathscr{I}\) is almost contained in the union of finitely many other elements in \(\mathscr{I}\). The ideal independence number, denoted by \(\mathfrak{s}{mm}\), is defined as the minimal cardinality of a maximal ideal independent family. We will review some results about ideal independent families and the cardinal invariant \(\mathfrak{s}{mm}\). In particular we will prove that the ultrafilter number is a lower bound for \(\mathfrak{s}{mm}\). Also, we will see that the spectrum of ideal independent families, defined as the family of all cardinalities of maximal ideal independent families, can be quite rich. If time allows, we will sketch a proof that consistently \(\mathfrak{s}{mm}<\mathfrak{a}_T\), where \(\mathfrak{a}_T\) is the minimal cardinality of a family of disjoint compact sets covering the Baire space. This is joint work with V. Fischer and C. B. Switzer.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Jonathan Cancino (Czech Academy of Sciences)
Title: Ideal independent families and ultrafilters
Abstract: A family \(\mathscr{I}\subseteq[\omega]^\omega\) is called ideal independent if no element \(A\in\mathscr{I}\) is almost contained in the union of finitely many other elements in \(\mathscr{I}\). The ideal independence number, denoted by \(\mathfrak{s}{mm}\), is defined as the minimal cardinality of a maximal ideal independent family. We will review some results about ideal independent families and the cardinal invariant \(\mathfrak{s}{mm}\). In particular we will prove that the ultrafilter number is a lower bound for \(\mathfrak{s}{mm}\). Also, we will see that the spectrum of ideal independent families, defined as the family of all cardinalities of maximal ideal independent families, can be quite rich. If time allows, we will sketch a proof that consistently \(\mathfrak{s}{mm}<\mathfrak{a}_T\), where \(\mathfrak{a}_T\) is the minimal cardinality of a family of disjoint compact sets covering the Baire space. This is joint work with V. Fischer and C. B. Switzer.