2020-01-09Jaroslav Šupina (Pavol Jozef Šafárik University in Košice, Slovakia)Ideal pseudointersection numbers and topological spaces
Investigations of ideal versions of selection principles pointed our attention to pseudointersection numbers introduced by P. Borodulin–Nadzieja and B. Farkas 2012 and M. Repický 2018. Both modifications of pseudointersection number are parametrized by ideals on natural numbers and related to orderings among such ideals.

We shall focus on these invariants and their connections to selection principles. Their topological characterizations (which resemble Fréchet-Urysohn property) are simple tool to derive inequalities among them. Finally, a result by P. Borodulin–Nadzieja and B. Farkas 2012 shows that ideal version of standard characterization of countable Fréchet-Urysohn property via selection principle by J. Gerlits and Zs. Nagy 1982 is not true.
2020-01-16Corey Bacal Switzer (City University of New York, Graduate Center, USA)Generalized Cardinal Characteristics for Sets of Functions
Cardinal characteristics on the generalized Baire and Cantor spaces \(\kappa^\kappa\) and \(2^\kappa\) have recently generated significant interest. In this talk I will introduce a different generalization of cardinal characteristics, namely to the space of functions \(f:\omega^\omega \to \omega^\omega\). Given an ideal \(\mathcal I\) on Baire space and a relation \(R\) let us define \(f R_{\mathcal I} g\) for \(f\) and \(g\) functions from \(\omega^\omega\) to \(\omega^\omega\) if and only if \(f(x) R g(x)\) for an \(\mathcal I\)-measure one set of \(x \in \omega^\omega\). By letting \(\mathcal I\) vary over the null ideal, the meager ideal and the bounded ideal; and \(R\) vary over the relations \(\leq^*\), \(\neq^*\) and \(\in^*\) we get 18 new cardinal characteristics by considering the bounding and dominating numbers for these relations. These new cardinals form a diagram of provable implications similar to the Cichoń diagram. They also interact in several surprising ways with the cardinal characteristics on \(\omega\). For instance, they can be arbitrarily large in models of CH, yet they can be \(\aleph_1\) in models where the continuum is arbitrarily large. They are bigger in the Sacks model than the Cohen model. I will introduce these cardinals, show some of the provable implications and discuss what is known about consistent inequalities, including new generalizations of well-known forcing notions on the reals to this context. This includes joint work with Jörg Brendle.
2020-01-23Miguel Moreno (Bar-Ilan University, Ramat Gan, Tel Aviv, Israel)Fake Reflection
Motivated from many results in generalized descriptive set theory, Filter Reflection (aka Fake Reflection) is an abstract version of reflection compatible with large cardinals, forcing axioms, but also V=L.

In this talk we will present the motivation and definition of filter reflection, we will explain how to force filter reflection and how to force its failure. We will also show some applications and properties of filter reflection, e.g. the consistency of “\(E_{\omega_1}^\kappa\) filter reflects to a subset of \(E_{\omega}^\kappa\)”.

This is a joint work with Gabriel Fernandes and Assaf Rinot.
2020-01-30Víctor Torres-Pérez (TU Wien)Construction with opposition: Cardinal invariants and games
We consider several game versions of the cardinal invariants \(\mathfrak t\), \(\mathfrak u\) and \(\mathfrak a\). We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \(\mathfrak t<\mathfrak t_{Builder}\) and \(\mathfrak u<\mathfrak u_{Builder}\) are both relatively consistent with ZFC, where \(\mathfrak t_{Builder}\) and \(\mathfrak u_{Builder}\) are the principal game versions of \(\mathfrak t\) and \(\mathfrak u\), respectively. The corresponding question for \(\mathfrak a\) remains open.

This is a joint work with Jörg Brendle and Michael Hrušák.

The [/2020/TorresKGRC2020.pdf slides] for this talk are available.
2020-02-12Vincenzo Dimonte (University of Udine, Italy)Regularity properties in singular generalized descriptive set theory
Generalized descriptive set theory is the study of definable subsets of the space \({}^\kappa 2\) with the bounded topology. Such study has been overwhelmingly focussed on the case with \(\kappa\) regular. Motivated by the theory of rank-into-rank cardinals, we concentrated instead on the case of \(\kappa\) singular of cofinality \(\omega\), painting a picture that is quite similar to the classical descriptive set theory case. This talk is going to center around the generalization of regularity properties (Perfect Set Property and Baire Property) in this context. The PSP is still akin to the classical case, while the BP probably needs more large-cardinal power to be non-trivial.
2020-03-05Yair Hayut (KGRC)$\Pi_1^1$-subcompactness and type omission
Strongly compact cardinals can be characterized in various ways: compactness of \(L_{\kappa,\kappa}\), filter extensions, the existence of fine measures, the strong tree property (+inaccessibility) and many other ways. Localizations of those definitions produce a rich hierarchy. Supercompact cardinals have much fewer parallel characterizations, obtained typically by adding a normality assumption.

In this talk I will present a characterization of supercompact cardinals in terms of compactness of \(L_{\kappa,\kappa}\) with type omission. Using it, I will present a variant of the strong tree property which is (locally) weaker than the ineffable tree property and together with inaccessibility characterize supercompactness. Those characterizations localize to a characterization of \(\Pi^1_1\)-subcomapctness.

This is a joint work with Menachem Magidor.

[/2020/Yair_Hayut_research_seminar_2020_03_05.pdf Slides] for this talk are available.
2020-04-23Noé de Rancourt (KGRC)Weakly Ramsey ultrafilters
Weakly Ramsey ultrafilters are ultrafilters on \(\omega\) satisfying a weak local version of Ramsey's theorem; they naturally generalize Ramsey ultrafilters. It is well known that an ultrafilter on \(\omega\) is Ramsey if and only if it is minimal in the Rudin-Keisler ordering; in joint work with Jonathan Verner, we proved that similarly, weakly Ramsey ultrafilters are low in this ordering: there are no infinite chains below them. This generalizes a result of Laflamme's. In this talk, I will outline a proof of this result, and the construction of a counterexample to the converse of this fact, namely a non-weakly-Ramsey ultrafilter having exactly one Rudin-Keisler predecessor. This construction is partly based on finite combinatorics.

2020-04-30Sandra Müller (KGRC)How to obtain lower bounds in set theory
Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel's analysis of the constructible universe \(L\). Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.

We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals \(\kappa\).

2020-05-14Andrew Brooke-Taylor (University of Leeds, UK)Products of CW Complexes
CW spaces are often presented as the "spaces of choice" in algebraic topology courses, being relatively nice spaces built up by successively gluing on Euclidean balls of increasing dimension. However, the product of CW complexes need not be a CW complex, as shown by Dowker soon after CW complexes were introduced. Work in the 1980s characterised when the product is a CW complex under the assumption of CH, or just \(b = \aleph_1\). In this talk I will give and prove a complete characterisation of when the product of CW complexes is a CW complex, valid under ZFC. The characterisation however involves \(b\); the proof is point-set-topological (I won't assume any knowledge of algebraic topology) and uses Hechler conditions.

The [/2020/ABTProdsCW.pdf slides] for this talk are available.

2020-05-28Diego Mejía (Shizuoka University, Japan)Preserving splitting families
We present a method to force splitting families that can be preserved by a large class of finite support iterations of ccc posets. As an application, we show how to force several cardinal characteristics of the continuum to be pairwise different.

This is a joint work with Martin Goldstern, Jakob Kellner and Saharon Shelah.

The [/2020/20200528_Mejia.pdf slides] for this talk are available.

2020-06-04Stefan Hoffelner (University of Münster, North Rhine-Westphalia, Germany)Forcing the $\Sigma^1_3$-separation property
The separation property, introduced in the 1920s, is a classical notion in descriptive set theory. It is well-known due to Moschovakis, that '''\(\Delta^1_2\)'''-determinacy implies the '''\(\Sigma^1_3\)'''-separation property; yet '''\(\Delta^1_2\)'''-determinacy implies an inner model with a Woodin cardinal. The question whether the '''\(\Sigma^1_3\)'''-separation property is consistent relative to just ZFC remained open however since Mathias' “Surrealist Landscape”-paper. We show that one can force it over L.

There are [/2020/Hoffelner_Talk_Separation_Property.pdf slides] and [/2020/Hoffelner_Notes_Separation_Talk.pdf notes] available for this talk.

2020-06-18Anush Tserunyan (University of Illinois at Urbana-Champaign, USA)Hyperfinite subequivalence relations of treed equivalence relations
A large part of measured group theory studies structural properties of countable groups that hold “on average”. This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on a standard probability space. In this vein, the amenable groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation on a standard probability space, deriving the analogues of structural properties of amenable subgroups (copies of \(\mathbb{Z}\)) of a free group. Most importantly, just like every such subgroup is contained in a unique maximal one, we show that even in the non-pmp setting, every hyperfinite subequivalence relation is contained in a unique maximal one.

The [/2020/hyperfin_in_tree.pdf slides] for this talk are available.

2020-06-25Victoria Gitman (City University of New York (CUNY), New York City, USA)Class forcing in its rightful setting
The use of class forcing in set theoretic constructions goes back to the proof Easton's Theorem that GCH can fail at all regular cardinals. Class forcing extensions are ubiquitous in modern set theory, particularly in the emerging field of set-theoretic geology. Yet, besides the pioneering work by Friedman and Stanley concerning pretame and tame class forcing, the general theory of class forcing has not really been developed until recently. A revival of interest in second-order set theory has set the stage for understanding the properties of class forcing in its natural setting. Class forcing makes a fundamental use of class objects, which in the first-order setting can only be studied in the meta-theory. Not surprisingly it has turned out that properties of class forcing notions are fundamentally determined by which other classes exist around them. In this talk, I will survey recent results (of myself, Antos, Friedman, Hamkins, Holy, Krapf, Schlicht, Williams and others) regarding the general theory of class forcing, the effects of the second-order set theoretic background on the behavior of class forcing notions and the numerous ways in which familiar properties of set forcing can fail for class forcing even in strong second-order set theories.

2020-10-01Jonathan Schilhan (KGRC)Definability of maximal families of reals in forcing extensions
The definability of combinatorial families of reals, such as mad families, has a long history. The constructible universe \(L\) is a good model for definability, for its nice structural properties. On the other hand, as a rule of thumb, the universe can't be too far from \(L\) if it allows for low projective witnesses of such families. Thus it makes sense to look at forcing extensions of \(L\).

We show that after a countable support iteration of Sacks forcing or splitting forcing (or many others) over \(L\), every analytic hypergraph on a Polish space has a \(\mathbf\Delta^1_2\) maximal independent set. This means that in the models obtained by these iterations, most types of interesting “maximal families” have \(\mathbf\Delta^1_2\) witnesses. In particular, this solves an open problem of Brendle, Fischer and Khomskii by providing a model with a \(\Pi^1_1\) mif (maximal independent family) while the independence number \(\mathfrak{i}\) is bigger than \(\aleph_1\).

The [/2020/Jonathan_Schilhan_KGRC_Seminar_Oct_1_2020.pdf slides] for this talk are available.

2020-10-08Colin Jahel (Claude Bernard University Lyon 1, France)Actions of automorphism groups of Fraïssé limits on the space of linear orderings
In 2005, Kechris, Pestov and Todor&ccaron;evi&cacute; exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the actions of some groups. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons.

The [/2020/Colin_Jahel_2020-10-08.pdf slides] for this talk are available.
2020-10-15Ziemowit Kostana (University of Warsaw, Poland)Fraïssé theory, and forcing absoluteness of rigidity for linear orders
During the talk I would like to introduce the theory of Cohen-like first-order structures. These are countable or uncountable structures which are “generic” much in the same sense as the Cohen reals. They can be added to the universe of set theory using finite or, say, countable conditions and exhibit different properties. I will focus on the construction of a rigid linear order, whose rigidity is absolute for ccc extensions.
2020-10-22Philipp Schlicht (KGRC)Tree forcings, sharps and absoluteness
In joint results with Fabiana Castiblanco from 2018, we showed that several classical tree forcings preserve sharps for reals and levels of projective determinacy, and studied their impact on definable equivalence relations (in particular, the question whether they add equivalence classes to thin projective equivalence relations). I will discuss these results and natural open problems on tree forcings and absoluteness that arise from them.

The [/2020/Philipp_Schlicht_Talk_Vienna_2020-10-22.pdf slides] for this talk are available.
2020-10-29Philipp Lücke (University of Barcelona, Spain)Structural reflection and shrewd cardinals
In my talk, I want to present work dealing with the interplay between extensions of the ''Downward Löwenheim–Skolem Theorem'' to strong logics, large cardinal axioms and set-theoretic reflection principles, focussing on the characterization of large cardinal notions through model- and set-theoretic reflection properties. The work of Bagaria and his collaborators shows that various important objects in the middle and upper reaches of the large cardinal hierarchy can be characterized through principles of ''structural reflection''. I will discuss recent results dealing with possible characterizations of notions from the lower part of this hierarchy through the principle \(\mathrm{SR}^-\), introduced by Bagaria and Väänänen. These results show that the principle \(\mathrm{SR}^-\) is closely connected to the notion of ''shrewd cardinals'', introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor's classical characterization of supercompactness.

The [/2020/Philipp_Luecke_2020-10-29.pdf slides] for this talk are available.
2020-11-05Omer Ben-Neria (Hebrew University of Jerusalem, Israel)On Continuous Tree-Like Scales and related properties of Internally Approachable structures
In his PhD thesis, Luis Pereira isolated and developed several principles of singular cardinals that emerge from Shelah's PCF theory; principles which involve properties of scales, such as the inexistence of continuous Tree-Like scales, and properties of internally approachable structures such as the Approachable Free Subset Property.

In the talk, we will discuss these principles and their relations, and present new results from a joint work with Dominik Adolf concerning their consistency and consistency strength.
2020-11-12Hossein Lamei Ramandi (University of Toronto, Ontario, Canada)Can You Take Komjath's Inaccessible Away?
In this talk we aim to compare Kurepa trees and Aronszajn trees. Moreover, we talk about the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem which asserts the same consistency from two inaccessible cardinals. We will briefly sketch the ideas to prove that our large cardinal assumption is optimal. If time permits, we talk about the comparison of Kurepa trees and Aronszajn trees in the presence of no large cardinal.

This is a joint work with Stevo Todorcevic.

The [/2020/Presentation._Aronszajn_in_Kurepa.pdf slides] for this talk are available.
2020-11-19Gabriel Fernandes (Bar-Ilan University, Ramat Gan, Israel)Local club condensation in extender models
Local club condensation is a condensation principle defined by Friedman and Holy. It is a theorem due to Friedman and Holy that local club condensation holds in most of the extender models that are weakly iterable.

We prove that in any weakly iterable extender model with \(\lambda\)-indexing, given a cardinal \(\kappa\), the sequence \(\langle L_\alpha [E] \mid \alpha < \kappa^{++} \rangle\) witnesses local club condensation on the interval \((\kappa^+ , \kappa^{++})\) iff \(\kappa\) is not a subcompact cardinal in \(L[E]\).

We also prove that if \(\kappa\) is subcompact, then there is no sequence \(\langle M_\alpha \mid \alpha < \kappa^{++} \rangle \in L[E]\) with \(M_\kappa = (H_\kappa)^{L[E]}\) and \(M_{\kappa^{++}} = (H_{\kappa^{++}})^{L[E]}\) which witnesses local club condensation in \((\kappa^+ , \kappa^{++})\).

Using the equivalence between subcompact cardinals and \(\neg\square_\kappa\), due to Schimmerling and Zeman, it follows that \(\square_\kappa\) holds iff the sequence \(\langle L_\alpha [E] \mid \alpha < \kappa^{++} \rangle\) witnesses local club condensation on the interval \((\kappa^+ , \kappa^{++})\).
2020-11-26Damian Sobota (KGRC)Convergence of Borel measures and filters on omega
The celebrated Josefson–Nissenzweig theorem asserts, under certain interpretations, that for every infinite compact space K there exists a sequence of normalized signed Borel measures on K which converges to 0 with respect to every continuous real-valued function (i.e. the corresponding integrals converge to 0). We showed that in the case of products of two infinite compact spaces K and L one can construct such a sequence of measures with an additional property that every measure has finite support—let us call such a sequence “an fsJN-sequence” (i.e. a finitely supported Josefson–Nissenzweig sequence). We then studied the case when the spaces K and L are only pseudocompact and we proved in ZFC that if the product of K and L is pseudocompact, then it also admits an fsJN-sequence. On the other hand, we showed that under the Continuum Hypothesis, or Martin's axiom, or even some weaker set-theoretic assumptions concerning weak P-points, there exists a pseudocompact space X such that its square is not pseudocompact and it does not admit any fsJN-sequences. During my talk I will discuss these as well as other results concerning the topic and obtained during a joint work with various combinations of J. Kakol, W. Marciszewski and L. Zdomskyy.

The [/2020/Damian_Sobota_research_seminar_2020-11-26.pdf slides] for this talk are available.

2020-12-03Mirna Džamonja (CNRS & Panthéon Sorbonne, Paris, France and Czech Academy of Sciences, Prague)On logics that make a bridge from the Discrete to the Continuous
The talk starts with a surveys of some recent connections between logic and discrete mathematics. Then we discuss logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence. We compare such logics and discuss some consequences of such comparisons, as well as some hopes for further results in this research project.

The [/2020/Vienna_2020-12-03.pdf slides] for this talk are available.
2020-12-10Michael Hrušák (UNAM, Mexico City, Mexico)Invariant Ideal Axiom
We shall introduce a consistent set-theoretic axiom which has a profound impact on convergence properties in topological groups. As an application we show that consistently (consequence of IIA) every countable sequential group is either metrizable or \(k_\omega\).

The [/2020/hrusak-KGRC.pdf slides] for this talk are available.

2020-12-17Peter Holy (University of Udine, Italy)Ramsey-like Operators
Starting from measurability upwards, larger large cardinals are usually characterized by the existence of certain elementary embeddings of the universe, or dually, the existence of certain ultrafilters. However, below measurability, we have a somewhat similar picture when we consider certain embeddings with set-sized domain, or ultrafilters for small collections of sets. I will present some new results, and also review some older ones, showing that not only large cardinals below measurability, but also several related concepts can be characterized in such a way, and I will also provide a sample application of these characterizations.

The [/2020/Peter_Holy_talk_Vienna.pdf slides] for this talk are available.