David Schrittesser: News on mad families

KGRC Research Seminar  – 2017-06-22 at 4pm.

Speaker: David Schrittesser (University of Copenhagen, Denmark)

Abstract: This talk is about two results on mad families (dating from this year): Firstly, in joint work with Asger Törnquist and Karen Bakke Haga, we link madness of certain definable sets to forcing and use this to show that under the Axiom of Projective Determinacy there are no projective mad families. Moreover, the results generalize: we may replace “being almost disjoint” by “being $J$-disjoint”, for certain ideals $J$ on the natural numbers including, e.g., Fin $\times$ Fin. The other result is an improvement of Horowitz and Shelah’s construction of a Borel maximal eventually different family of functions. We obtain a closed such family, and the result even generalizes to certain compact spaces.

 

Diana Carolina Montoya Amaya: Some cardinal invariants of the generalized Baire spaces

The successful PhD defense of Diana Carolina Montoya Amaya took place Wednesday, June 14 at the KGRC. Congratulations!

Abstract: The central topic of this talk is the well-known Cardinal invariants of the continuum and it is divided in two parts: In the first one we focus on the generalization of some of these cardinals to the generalized Baire spaces $\kappa^\kappa$, when $\kappa$ is a regular uncountable cardinal. First, we present a generalization of some of the cardinals in Cichon’s diagram to this context and some of the provable ZFC relationships between them. Further, we study their values in some generic extensions corresponding to $<\!\!\kappa$-support and $\kappa$-support iterations of generalized classical forcing notions. We point out the similarities and differences with the classical case and explain the limitations of the classical methods when aiming for such generalizations. Second, we study a specific model where the ultrafilter number at $\kappa$ is small, $2^\kappa$ is large and in which a larger family of cardinal invariants can be decided and proven to be $<\!2^\kappa$.

The second part deals exclusively with the countable case: We present a generalization of the method of matrix iterations to find models where various constellations in Cichon’s diagram can be obtained and the value of the almost disjointness number can be decided. The method allows us also to find a generic extension where seven cardinals in Cichon’s diagram can be separated.

Board of examiners:

Professor Mirna Džamonja (University of East Anglia)
o.Univ.-Prof. Sy-David Friedman (Universität Wien)
ao.Univ.Prof. Martin Goldstern (TU Wien)

Dima Sinapova: Iterating Prikry Forcing

Monday, June 26th, 2017, 10.30-12.00

Aula Lagrange, Palazzo Campana, Università di Torino

Speaker: Dima Sinapova (University of Illinois at Chicago)

Title: Iterating Prikry Forcing

Abstract:

We will present an abstract approach of iterating Prikry type forcing. Then we will use it to show that it is consistent to have finite simultaneous stationary reflection at $\kappa^+$ with not SCH at $\kappa$. This extends a result of Assaf Sharon. Finally we will discuss how we can bring the construction down to $\aleph_{\omega}$. This is joint work with Assaf Rinot.

Stefan Hoffelner: NS saturated and Delta_1-definable

Monday, June 19, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Stefan Hoffelner (University of Vienna)

Title: NS saturated and Delta_1-definable

Abstract:

Questions which investigate the interplay of the saturation of the nonstationary ideal on $omega_1$, NS, and definability properties of the surrounding universe can yield surprising and deep results. Woodins theorem that in a model with a measurable cardinal where NS is saturated, CH must definably fail is the paradigmatic example. It is another remarkable theorem of H. Woodin that given $omega$-many Woodin cardinals there is a model in which NS is saturated and $omega_1$-dense, which in particular implies that NS is (boldface) $Delta_1$-definable. S.D. Friedman and L. Wu asked whether the large cardinal assumption can be lowered while keeping NS $Delta_1$-definable and saturated. In this talk I will outline a proof that this is indeed the case: given the existence of $M_1^{#}$, there is a model of ZFC in which the nonstationary ideal on $omega_1$ is saturated and $Delta_1$-definable with parameter $K_{omega_2^K}$ (note that $omega_2^K$ is of size $aleph_1$ in that model). In the course of the proof I will present a new coding technique which seems to be quite suitable to obtain definability results in the presence of iterated forcing constructions over inner models for large cardinals.

Ari Brodsky: ​Constructing free Souslin trees from a proxy principle

BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, June 13th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: ​Ari Brodsky (BIU)

Title: ​Constructing free Souslin trees from a proxy principle

Abstract. ​More than 40 years ago, Jensen constructed a free Souslin tree of height $\omega_1$ from $\diamondsuit(\omega_1)$.
We show how to construct a free $\kappa$-Souslin tree, where $\kappa$ is an arbitrary regular uncountable cardinal.
This is joint work with Assaf Rinot.​

Aleksander Cieślak: Cohen-stable families of subsets of integers

Tuesday, June 13, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wroclaw University of Science and Technology)

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.

Yizheng Zhu: Iterates of M_1

Monday, June 12, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Yizheng Zhu (University of Münster)

Title: Iterates of M_1

Abstract:

Assume Delta^1_3-determinacy. Let L_{kappa_3}[T_2] be the admissible closure of the Martin-Solovay tree and let M_{1,infty} be the direct limit of$M_1 via countable trees. We show that L_{kappa_3}[T_2]cap V_{u_{omega}} = M_{1,infty} | u_{omega}.

Thilo Weinert: Avoiding Quadruples Using a Scale

BIU seminar in Set Theory

On 08/06/2017, 10-12, Building 604, Room 103

Speaker: Thilo Weinert (BGU)

Title: Avoiding Quadruples Using a Scale

Abstract. In 1971, Hajnal showed that the continuum hypothesis implies omega_1^2 -|-> (omega_1^2, 3)^2 and in the same year, together with Erdős, that GCH implies that for every infinite cardinal kappa and every alpha < (kappa^+)^2 we have alpha -|-> (kappa^+ * kappa, 3)^2. In the same paper they showed that for infinite cardinals kappa and alpha < (kappa^+)^2 we have (kappa^+)^2 —> (alpha, 3)^2. In 1987, together with Baumgartner, he showed that for regular kappa satisfying 2^kappa = kappa^+ = lambda we have lambda^2 -|-> (lambda * kappa, 4)^2.

In 1998, Jean Larson showed that for regular kappa and lambda = kappa^+ the existence of a scale of length lambda of functions f : kappa — > kappa implies the failure of the aforementioned partition relations shown to
be unprovable from ZFC in the seventies, i.e. lambda * kappa —> (lambda * kappa, 3)^2 and lambda^2 —> (lambda^2, 3)^2. She commented that it would be interesting to know whether this hypothesis also suffices to prove lambda^2 -|-> (lambda * kappa, 4)^2.

It does.

Zoltán Vidnyánszky: Borel chromatic numbers: finite vs infinite

KGRC Research Seminar – 2017-06-08 at 4pm.

Speaker: Zoltán Vidnyánszky (York University, Toronto, Canada)

Abstract: One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that a Borel graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs with infinite Borel chromatic number. Using descriptive set theoretic methods we answer this question and a couple of similar questions negatively, showing that one cannot hope for the existence of a Borel graph whose embeddability would characterize Borel (or even closed) graphs with infinite Borel chromatic number. We will also discuss a positive result and its relation to Hedetniemi’s conjecture.

Assaf Hasson: Strongly dependent henselian fields and ordered abelian groups

​​BGU Seminar in Logic, Set Theory and Topology

Time: Tuesday, June 6th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Assaf Hasson (BGU)

Title: Strongly dependent henselian fields and ordered abelian groups

Abstract:
The strong non-independence property was introduced by Shelah in order to capture, within the class of theories without the independence property (aka dependent theories), an analogue of the class of super-stable theories. Shelah conjectured (roughly) that any infinite field with the strong non-independence property (aka strongly dependent) is either real closed, algebraically closed or supports a definable (henselian) valuation. The conjecture was solved (Johnson) in the very special case of dp-minimal fields, and otherwise remains wide open. In fact, most experts believe the conjecture (replacing “algebraically closed” with “separably closed”) to be true of all fields without the independence property, and the algebraic division line between the two classes of fields remains unclear.

In the talk we will show that strongly dependent ordered abelian groups do have a simple algebraic characterisation, and suggest the interpretability of ordered  abelian groups which are not strongly dependent as a new (not yet fully satisfactory) conjectural division line.

If time allows we will draw from the classification of strongly dependent ordered abelian groups some conclusions concerning strongly dependent henselian fields (e.g., if K is strongly dependent then any henselian valuation v — not necessarily definable — on K has strongly dependent residue field and value group).

The talk will aim to be, more or less, self-contained and little use (if any) will be made of technical model theoretic terms.

Based (mostly) on joint work with Yatir Halevi.