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Hamburg Set Theory Workshop 2020

Descriptive Set Theory, Forcing and the Reals. 20 & 21 June 2020Hamburg, Germany Online via Zoom

The Hamburg Set Theory Workshop 2020 is organised by the Arbeitsbereich Mathematische Logik und interdisziplinäre Anwendungen der Logik. It stands in the tradition of the previous workshops HSTW 2014 and HSTW 2016. Everyone is cordially invited to attend.

Registration

Although registration is required for all participants, there is no registration fee for this conference. For registration, write to hstw2020@math.hamburg with your name, affiliation (if applicable). Please do this as soon as possible as places are limited. After registration, you will receive a link to the Zoom webinar and Google doc where we plan for smaller Zoom calls simulating coffee breaks and other features involving social interaction.

Preliminary schedule

(All times given in CEST)
10:00 - 10:25 20th June 2020 Online arrival (and chatting) B. Löwe — Welcome & Opening Yurii Khomskii — Filter-Laver Measurability Wolfgang Wohofsky — The distributivity spectrum Lunch Break Vera Fischer — More $$\mathsf{ZFC}$$ inequalities between cardinal invariants Jindřich Zapletal — Coloring algebraic hypergraphs Informal "bar"-zoom meeting. 21th June 2020 Heike Mildenberger — Miller Forcing with Canjar Ultrafilters Zoltán Vidnyánszky — Countable Borel equivalence relations on co-Ramsey null sets Lunch Break Dorottya Sziráki — The open dihypergraph dichotomy for definable subsets of generalized Baire spaces Dominique Lecomte — Descriptive complexity of Borel relations

Topic

Descriptive set theory discusses the relationship between logical complexity and good behaviour of sets, so-called regularity properties (e.g. Lebesgue measurability, perfect set property, etc.). It is well known that these properties are closely connected to forcing (and in some cases, large cardinals). Modern developments in set theory such as forcing, large cardinals and determinacy give powerful techniques to tackle problems in descriptive set theory. In recent years, set theorists have generalised classical results of descriptive set theory to generalised reals.

Abstracts

Filter-Laver Measurability
Yurii Khomskii

Abstract: We study $$\sigma$$-ideals and regularity properties related to the “filter-Laver” and “dual-filter-Laver” forcing partial orders, with an emphasis on analytic filters which makes sure that the forcings are definable. An important innovation enabling this study is a dichotomy theorem proved recently by Miller.

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The distributivity spectrum
Wolfgang Wohofsky

Abstract: We consider the distributivity spectrum which is the set of possible heights of refining systems of mad families without common refinement; the well known distributivity number $$\mathfrak{h}$$ is the minimal element of this spectrum. If $$\mathfrak{h}$$ is equal to the continuum, the spectrum is trivial, i.e., its only element is $$\mathfrak{h}$$. We want to show that the spectrum can have more than one element. For this, we define a forcing iteration which adds a witness for $$\omega_2$$ being in the distributivity spectrum. To ensure that the distributivity spectrum is not trivial, we have to show that $$\mathfrak{h}$$ is $$\omega_1$$. In fact, we even show that $$\mathfrak{b}$$ is $$\omega_1$$ (and use that $$\mathfrak{h}$$ is below $$\mathfrak{b}$$): the family $$B$$ of ground model reals stays unbounded because our forcing can be represented as a finite support iteration of Mathias forcings with respect to $$B$$-Canjar filters. This is joint work with Vera Fischer and Marlene Koelbing. (Slides)

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More $$\mathsf{ZFC}$$ inequalities between cardinal invariants
Vera Fischer

Abstract: We will discuss some recent $$\mathsf{ZFC}$$ results concerning the generalized Baire spaces, and more specifically the generalized bounding number, relatives of the generalized almost disjointness number, as well as generalized reaping and domination.

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Coloring algebraic hypergraphs
Jindřich Zapletal

Abstract: We discuss independence results in $$\mathsf{ZF}+\mathsf{DC}$$ set theory regarding colorings of algebraic hypergraphs on Euclidean spaces. As a sample result, we show it is consistent with $$\mathsf{ZF}+\mathsf{DC}$$ that there is a partition of the plane into countably many pieces neither of which contains all vertices of a rectangle, and there is no such partition for equilateral triangles.

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Miller Forcing with Canjar Ultrafilters
Heike Mildenberger

Abstract: We show that the parametrised Miller forcings that have been introduced by Guzman and Kalajdzievsky can be used to specifically diagonalise some Canjar ultrafilters and preserves other P-points. We introduce a block structure on the nodes of a tree in such a forcing. This is joint work with Christian Bräuninger. (Slides)

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Countable Borel equivalence relations on co-Ramsey null sets
Zoltán Vidnyánszky

Abstract: The behavior of countable Borel equivalence relations (CBERs) restricted to large subsets of Polish spaces can be strikingly different if one considers various notions of size. For example, every CBER is hyperfinite (that is, can be expressed as the increasing union of Borel equivalence relations with finite classes) restricted to a comeager set, while the analogous result fails badly for restrictions to sets with positive measure. We discuss the behaviour of CBERs on the collection of the infinite subsets of natural numbers, restricted to sets which are co-Ramsey null.

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The open dihypergraph dichotomy for definable subsets of generalized Baire spaces
Dorottya Sziráki

Abstract: The open graph dichotomy for a given set $$X$$ of reals is a generalization of the perfect set property for $$X$$ which can also be viewed as the perfect set version of the Open Coloring Axiom restricted to $$X$$. An $$\omega$$-dimensional generalization of the open graph dichotomy was introduced recently by Raphael Carroy, Benjamin Miller and Daniel Soukup. They obtained several dichotomy theorems regarding the second level of the Borel hierarchy as applications of this $$\omega$$-dimensional open dihypergraph dichotomy.

We extend a theorem of Qi Feng's about the open graph dichotomy for definable subsets of the real line to the $$\omega$$-dimensional open dihypergraph dichotomy restricted to definable dihypergraphs. We also extend this result to the generalized Baire spaces $$\kappa^\kappa$$ for uncountable regular cardinals $$\kappa$$.

More concretely, we show that given an infinite regular cardinal $$\kappa$$, the following statement is consistent relative to the existence of an inaccessible cardinal above $$\kappa$$: the $$\kappa$$-dimensional open dihypergraph dichotomy holds for all subsets $$X$$ of $$\kappa$$ and all $$\kappa$$-dimensional box-open dihypergraphs $$H$$ on $$X$$ such that both $$X$$ and $$H$$ are definable from a $$\kappa$$-sequence of ordinals.

This is joint work with Philipp Schlicht. (Slides)

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Descriptive complexity of Borel relations
Dominique Lecomte

Abstract: We study the descriptive complexity of Borel relations, as relations and not only as sets. So we compare them with the notions of continuous reducibility or injective continuous reducibility, using square maps. In other words, we want to know when a relation is in a given Borel (additive or multiplicative) class. We provide a number of answers with various smallness assumptions, particularly nice for equivalence relations and graphs.

Support

Financially supported by the Verein zur Ausrichtung von Tagungen am Fachbereich Mathematik der Universität Hamburg (VATFBMUHH e.V.).

Organisers

Michel Gaspar, Marcel Krüger, Benedikt Löwe & Deniz Sarikaya. Please do not hesitate to contact us for any further information.

 Impressum 2020-06-28, HSTW2020 OC