Jonathan Cancino
Jonathan Cancino: On nwd-MAD families
18/04/23 14:28
Tuesday, April 18, 2023 17:00
Location: room A.4.1 C-19
Speaker: Jonathan Cancino (Czech Academy of Sciences)
Title: On nwd-MAD families
Abstract: The cardinal invariant a(nwd) is defined as the minimal cardinality of an uncountable maximal antichain of the power set of the rational modulo the nowhere dense ideal. This cardinal invariant was introduced by J. Steprans, and he proved that in the Laver's model it is \(\omega_1\), and the pseudointersection number p is a lower bound for it. In this talk we will prove some related results, for example, the additivity of the meager ideal is a lower bound for a(nwd), thus improving Steprans theorem, as well as some facts about the structure of nwd-MAD families.
Location: room A.4.1 C-19
Speaker: Jonathan Cancino (Czech Academy of Sciences)
Title: On nwd-MAD families
Abstract: The cardinal invariant a(nwd) is defined as the minimal cardinality of an uncountable maximal antichain of the power set of the rational modulo the nowhere dense ideal. This cardinal invariant was introduced by J. Steprans, and he proved that in the Laver's model it is \(\omega_1\), and the pseudointersection number p is a lower bound for it. In this talk we will prove some related results, for example, the additivity of the meager ideal is a lower bound for a(nwd), thus improving Steprans theorem, as well as some facts about the structure of nwd-MAD families.
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.