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| UHH > Fakultäten > MIN-Fakultät > Mathematik > Bereiche > Mathematische Logik |
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TopicDescriptive 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. AbstractsFilter-Laver Measurability 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. The distributivity spectrum 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) More \(\mathsf{ZFC}\) inequalities between cardinal invariants 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. Coloring algebraic hypergraphs 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. Miller Forcing with Canjar Ultrafilters 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) Countable Borel equivalence relations on co-Ramsey null sets 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. The open dihypergraph dichotomy for definable subsets of
generalized Baire spaces 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) Descriptive complexity of Borel relations 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. SupportFinancially supported by the Verein zur Ausrichtung von Tagungen am Fachbereich Mathematik der Universität Hamburg (VATFBMUHH e.V.).OrganisersMichel Gaspar, Marcel Krüger, Benedikt Löwe & Deniz Sarikaya. Please do not hesitate to contact us for any further information. | |||||||||||||||||||||||||||||||
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Impressum | 2020-06-28, HSTW2020 OC |