Online activities 22 – 28 May

The announcements are updated continuously. For a list of talks in the coming weeks, please see here.

Kobe Set Theory Seminar
 Monday, 22 May, 16:30 local time (09:30 CEST)
Speaker: Takehiko Gappo, TU Wien
Title: Chang models over derived models with supercompact measures (2/2)
Abstract: The relationship between the Axiom of Determinacy and supercompactness of ω1 has been studied by many people. In 1990’s, Woodin showed that assuming the existence of a proper class of Woodin limits of Woodin cardinals, a generalized Chang model satisfies “ADℝ + ω1 is supercompact.” Recently he also showed that the regularity of Θ in the model follows from determinacy of a long game of length ω1, which is, however, still unknown to be consistent. Based on these results, we conjecture that the following two theories are equiconsistent:
(1) ZFC + there is a Woodin limit of Woodin cardinals.
(2) ZF + ADℝ + Θ is regular + ω1 is supercompact.
Toward this conjecture, we construct a new model of the Axiom of Determinacy, called the Chang model over the derived model with supercompact measures.We then prove that it is consistent relative to a Woodin limit of Woodin cardinals that our model satisfies “ADℝ + Θ is regular + ω1 is < δ-supercompact for some regular cardinal δ > Θ.” This is joint work with Sandra Müller and Grigor Sargsyan.  
Information: Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai for the zoom information in advance.

Bristol Logic and Set Theory Seminar
Time: Tuesday, 23 May, 1.30-2.30pm UK time (14:30-15:30 CET)
Speaker: Beatrice Pitton, University of Lausanne
Title: Definable subsets of the generalized Cantor and Baire spaces
Abstract: Generalized descriptive set theory (GDST) aims at developing a higher analogue of classical descriptive set theory in which ω is replaced with an uncountable cardinal κ in all definitions and relevant notions. In the literature on GDST it is often required that κ<κ = κ, a condition equivalent to κ regular and 2<κ = κ. In contrast, in this paper we use a more general approach and develop in a uniform way the basics of GDST for cardinals κ still satisfying 2<κ = κ but independently of whether they are regular or singular. This allows us to retrieve as a special case the known results for regular κ, but it also uncovers their analogues when κ is singular. We also discuss some new phenomena specifically arising in the singular context (such as the existence of two distinct yet related Borel hierarchies), and obtain some results which are new also in the setup of regular cardinals, such as the existence of unfair Borel∗ codes for all Borel∗ sets. This is joint work with Luca Motto Ros.
Information: The zoom link for this talk is

Baltic Set Theory Seminar
Time: Tuesday, 23 May, 15:00-16:30 CEST
Speaker: Several
Title: Baltic Set Theory Seminar
Abstract: This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:
1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.
2. Paul Larson, A course on AD^+
Information: Please see the seminar webpage.

Bristol Logic and Set Theory Seminar
Time: Wednesday, 23 May, 3.00-4.00pm UK time (16:00-17:00 CET)
Speaker: Bokai Yao, University of Notre Dame
Title: Forcing with Urelements
Abstract: I will begin by isolating a hierarchy of axioms based on ZFCU_R, which is ZFC set theory (with Replacement) modified to allow a class of urelements. For example, the Collection Principle is equivalent to the Reflection Principle over ZFCU_R, while it is folklore that neither of them is provable in ZFCU_R.
I then turn to forcing over countable transitive models of ZFU_R. A forcing relation is full just in case whenever a forcing condition p forces an existential statement, p also forces some instance of that statement. According to the existing approach, forcing relations are almost never full when there are urelements. I introduce a new forcing machinery to address this problem. I show that over ZFCU_R, the principle that every new forcing relation is full is equivalent to the Collection Principle. Furthermore, I show how forcing is able to preserve, destroy and resurrect the axioms in the hierarchy I introduced. In particular, the Reflection Principle is “necessarily forceble” in certain models of ZFCU_R. In the end, I will consider how the ground model definability can fail when the ground model contains a proper class of urelements.
Information: The zoom link for this talk is

Leeds Models and Sets Seminar
Time: Wednesday, 24 May, 13:45-15:00 local time (14:45-16:00 CEST)
Speaker: Adele Padgett, McMaster University
Title: Regular solutions of systems of transexponential-polynomial equations
Abstract: It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function E which satisfies E(x+1) = exp E(x). Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by E could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.
Information: Please see the seminar webpage.

Renyi Institute Set Theory Seminar
 Thursday, 25 May, 10:30 – 12:00 CEST
Speaker: Dorottya Sziraki
Title: Dichotomies for open dihypergraphs on definable subsets of generalized Baire spaces
Abstract: The open graph dichotomy for a subset $X$ of the Baire space $\omega^\omega$ states that any open graph on $X$ either contains a large complete subgraph or admits a countable coloring. It is a definable version of the open coloring axiom for $X$ and it generalizes the perfect set property. The focus of this talk is a recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup. It is motivated by applications to definable sets of reals, in particular to the second level of the Borel hierarchy.
We show that this infinite dimensional dichotomy holds for all subsets of the Baire space in Solovay’s model. Our main results are versions of this theorem for generalized Baire spaces $\kappa^\kappa$ for uncountable regular cardinals $\kappa$. If time permits, we will also look at conditions under which this dichotomy can be strengthened and mention several applications in the setting of generalized Baire spaces.
This is joint work with Philipp Schlicht.
Information: Please see the seminar webpage. This talk will be given in hybrid format.

Vienna Research Seminar in Set Theory
Time: Thursday, 25 May, 11:30-13:00 CEST
Speaker: F. Kaak, Universität Kiel
Title: Set theory of a Suslin line
Abstract: A Suslin line is a linear ordering, which is in some way quite similar to the real line. We will discuss in what ways the set theory of the real line can be adapted to a Suslin line. We give a characterization of Borel sets of the Suslin line, look at a few cardinal characteristics and play games on a Suslin tree.
Information: This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

Vienna Logic Colloquium
 Thursday, 25 May, 15:00 – 15:45 CEST
Speaker: S. Starchenko, University of Notre Dame
Title: On Hausdorff limits of images of o-minimal families in real tori
Abstract: Let {Xs:x∈S} be a family of subsets of Rn definable in some o-minimal expansion of the real field. Let Γ⊆Rn be a lattice and π:Rn/Γ→T be the quotient map. In a series of papers (published and unpublished) together with Y. Peterzil we considered Hausdorff limits of the family {π(Xs):s∈S} and provided their description. In this talk I describe model theoretic tools used in the description.
Information: This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.


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