## Ana Njegomir: A forcing characterization of lambda-ineffable cardinals

Monday, May 29, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Ana Njegomir (Universität Bonn)

Title: A forcing characterization of lambda-ineffable cardinals

## Andrey Morozov: Infinite time Blum-Shub-Smale machines for computability in analysis

Monday, May 15, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Andrey Morozov (Sobolev Institute of mathematics, Novosibirsk)

Title: Infinite time Blum-Shub-Smale machines for computability in analysis

## Chris Lambie-Hanson: Constructions from square and diamond, with an application to super-Souslin trees

Monday, May 8, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Chris Lambie-Hanson (Bar-Ilan)

Title: Constructions from square and diamond, with an application to super-Souslin trees

Abstract. In 1982, Shelah and Stanley proved that, if $\kappa$ is a regular, infinite cardinal, $2^\kappa = \kappa^+$, and there is a $(\kappa^+, 1)$-morass, then there is a $\kappa^{++}$-super-Souslin tree, which is a type of normal $\kappa^{++}$-tree that necessarily has a $\kappa^{++}$-Souslin subtree and continues to do so in any outer model in which $\kappa^{++}$ is preserved and no new subsets of $\kappa$ are present. This result establishes a lower bound of an inaccessible cardinal for the consistency strength of the conjunction of $2^\kappa = \kappa^+$ and Souslin’s Hypothesis at $\kappa^{++}$. In this talk, we will present a method for constructing objects of size $\lambda^+$ from $\square_\lambda + \diamondsuit_\lambda$, where $\lambda$ is a regular, uncountable cardinal. As an application, we will use $\square_{\kappa^+} + \diamondsuit_{\kappa^+}$ to construct a $\kappa^{++}$-super-Souslin tree. For uncountable $\kappa$, this increases Shelah and Stanley’s lower bound from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.

## Ralf Schindler: A Hamel basis for the reals without choice

09 January: 16.30
seminar room 0.008
Mathematisches Institut
Universität Bonn
Endenicher Allee 60

Ralf Schindler (Münster) – A Hamel basis for the reals without choice

The Cohen-Halpern-Levy model N has an infinite set of reals without a countable subset. Answering a question of D. Pincus and K. Prikry from 1975, we show that there is a Hamel basis in N. This is joint work with Liuzhen Wu and Liang Yu, inspired by earlier joint work with Mariam Beriashvili. DC fails in N, and it remains open if in the base theory ZF+DC, the existence of a Hamel basis implies that the reals can be wellordered.

## David Schrittesser: Maximal discrete sets with large continuum

25.01.2016
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: David Schrittesser (University of Copenhagen)

Title: Maximal discrete sets with large continuum

Abstract. In recent joint work with Asger Törnquist, we showed how to construct definable maximal discrete sets in forcing extensions of L, in particular in the Sacks and Miller extension. In particular, the existence of such sets is consistent with V ≠ L. In this talk I shall show the stronger result that the existence of definable discrete sets is consistent with large continuum. In the process, I show an interesting generalization of Galvin’s theorem. In particular, this applies to the example of maximal orthogonal families of measures (mofs). One might hope for a simpler way of constructing a mof in a model with large continuum: to find an indestructible such family in L. While such an approach is possible e.g. for maximal cofinitary groups, this is impossible for mofs.

## Asger Törnquist: Definable maximal orthogonal families and discrete sets in forcing extensions

Thursday, September 10, 2015, 16.30
Seminar room 1.008, Mathematical Institute, University of Bonn

Speaker: Asger Törnquist (Kopenhagen)

Title: Definable maximal orthogonal families and discrete sets in forcing extensions

## Assaf Rinot: Chain conditions of products

Thursday, July 9, 2015, 16.00
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Assaf Rinot (Bar-Ilan University)

Title: Chain conditions of products

Abstract: We shall survey the history of the study of the productivity of the k-cc in partial orders, topological spaces, and Boolean algebras. We shall address a conjecture that tries to characterize such a productivity in Ramsey-type language. For this, a new oscillation function for successor cardinals, and a new characteristic function for walks on ordinals will be proposed and investigated.

## Giorgio Laguzzi: Roslanowski and Spinas dichotomies

Monday, July 14, 2014, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Giorgio Laguzzi (Universität Hamburg)

Title: Roslanowski and Spinas dichotomies

## Otmar Spinas: “Das Problem mit Silver Amoeba”

Monday, June 23, 2014, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Otmar Spinas (Kiel)

Title: “Das Problem mit Silver Amoeba”

Abstract:

Ich werde die offene Frage eroertern, ob ein Amoeba für Silver-Forcing existiert, das keine Cohen reelle Zahlen adjungiert. In der Sprache der kardinalen Invarianten ist dies die Frage, ob konsistenterweise die Ueberdeckungszahl des mageren Ideals kleiner als die Additivitaet des Silver-Ideals ist.

## Mirna Dzamonja: Embeddings of graphs with no large cliques

Monday, June 16, 2014, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Mirna Dzamonja (University of East Anglia, Norwich, UK)

Title: Embeddings of graphs with no large cliques

Abstract:

We shall discuss embeddings between graphs omitting large
cliques and in particular we shall prove that for kappa singular of
cofinality kappa there is no universal graph of size kappa omitting
cliques of size kappa, just in ZFC.