2019-01-10Natasha Dobrinen (University of Denver, Colorado, USA)Ramsey Theory of the Henson graphs
A central question in the theory of ultrahomogeneous relational structures asks, How close of an analogue to the Infinite Ramsey Theorem does it carry? An infinite structure \(\mathbf{S}\) is ultrahomogeneous if any isomorphism between two finitely generated substructures of \(\mathbf{S}\) can be extended to an automorphism of \(\mathbf{S}\). We say that \(\mathbf{S}\) has finite big Ramsey degrees if for each finite substructure \(A\) of \(\mathbf{S}\), there is a number \(n(A)\) such that any coloring of the copies of \(A\) in \(\mathbf{S}\) can be reduced to no more than \(n(A)\) colors on some substructure \(\mathbf{S}'\) of \(\mathbf{S}\), which is isomorphic to the original \(\mathbf{S}\).

The two main obstacles to a fuller development of this area have been lack of representations and general Milliken-style theorems. We will present new work proving that the Henson graphs, the \(k\)-clique free analogues of the Rado graph for \(k\ge 3\), have finite big Ramsey degrees. We devise representations of Henson graphs via strong coding trees and prove Millike-style theorems for these trees. Central to the proof is the method of forcing, building on Harrington's proof of the Halpern-Läuchli Theorem.

There is a [ video] recording of this talk on YouTube.

[ Slides] are available, too.
2019-01-17Moritz Müller (Universitat Politècnica de Catalunya, Barcelona, Spain)Forcing against bounded arithmetic
We study the following problem. Given a nonstandard model of arithmetic we want to expand it by a binary relation that does something prohibitive, e.g. violates the pigeonhole principle in the sense that it is the graph of a bijection from \(n+1\) onto \(n\) for some (nonstandard) \(n\) in the model. The goal is to do so while preserving as much as possible of true arithmetic. More precisely, we want the expansion to model the least number principle for a class of formulas as large as possible. The problem is of central importance in bounded arithmetic and propositional proof complexity. It is not well understood. The talk describes a general method of forcing to produce such expansions.

Slides for this talk are [ available].
2019-01-24Benjamin Vejnar (Charles University, Prague, Czech Republic)Complexity of the homeomorphism relation between compact spaces
We study the complexity of the homeomorphism relation of compact metrizable spaces when restricted to some subclasses such as continua, regular continua or regular compacta. The complexity of an equivalence relation on a Polish space is compared with some others (e.g. with the universal orbit equivalence relation) using the notion of Borel reducibility.

A [ video] recording of this talk is available on YouTube.
2019-03-07Serhii Bardyla (KGRC)Complete topological semigroups
The first part of the talk will be devoted to the investigation of completeness in the class of topological semigroups.

Then we shall discuss a topologization of semigroups of partial isomorphisms between principal ideals of a tree.

There is a [ video] recording of this talk on YouTube.

[ Slides] are available, too.
2019-03-14Christopher Lambie-Hanson (Virginia Commonwealth University, Richmond, USA)Chromatic numbers of finite subgraphs
By the De Bruijn-Erdős Compactness Theorem, if a graph \(G\) has infinite chromatic number, then it has finite subgraphs of arbitrarily large finite chromatic number. We can therefore define an increasing function \(f_G:\omega \rightarrow \omega\) by letting \(f_G(n)\) be the least number of vertices in a subgraph of \(G\) with chromatic number \(n\). We will show in ZFC that, for every function \(f:\omega \rightarrow \omega\), there is a graph \(G\) with chromatic number \(\aleph_1\) such that \(f_G\) grows faster than \(f\). This answers a question of Erdős, Hajnal, and Szemeredi. Time permitting, we will discuss connections between our proof and various diamond and club-guessing principles.

A [ video] recording of this talk is available on YouTube.
2019-03-21Yair Hayut (KGRC)Strong compactness and the filter extension property
The notion of strongly compact cardinal is one of the earliest large cardinal axioms, yet it is still poorly understood.

I will review some classical and semi-classical connections between partial strong compactness, the strong tree property and the filter extension property, getting a level-by-level equivalence and an elementary embedding characterization.

This analysis is especially interesting for the property "every \(\kappa\)-complete filter on \(\kappa\) can be extended to a \(\kappa\)-complete ultrafilter" (where \(\kappa\) is uncountable). This property was isolated by Mitchell and was named "\(\kappa\)-compactness" by Gitik. In his recent paper, Gitik showed that some definable versions of it have a relatively low consistency strength, yet others provide an inner models with a Woodin cardinal. Applying the equivalence above to this case, I will improve the previously known lower bound for \(\kappa\)-compactness.

Then, I'll move to a more speculative area, and conjecture that \(\kappa\)-compactness is equiconsistent with a certain large cardinal axiom in the realm of subcompact cardinals. I will give a few arguments in favour of this conjecture.

A [ video] recording of this talk is available on YouTube.

Slides for this talk are available [ here].
2019-03-28Miha Habič (Czech Technical University in Prague, Czech Republic and Charles University, Prague, Czech Republic)Capturing by normal ultrapowers
Title: If \(\kappa\) is measurable and GCH holds, then any ultrapower by a normal measure on \(\kappa\) will be missing some subset of \(\kappa^+\). On the other hand, Cummings showed that, starting from a \((\kappa+2)\)-strong \(\kappa\), one can force to a model (without collapsing cardinals) where \(\kappa\) carries a normal measure whose ultrapower captures the entire powerset of \(\kappa^+\). Moreover, the large cardinal hypothesis is optimal. I will present an improvement of Cummings' result and show that this capturing property can consistently hold at the least measurable cardinal.

This is joint work with Radek Honzík.

There is a [ video] recording of this talk available on YouTube.
2019-04-04Diana Carolina Montoya (KGRC)The equality $\mathfrak{p}=\mathfrak{t}$ and the generalized characteristics
Marriallis and Shelah solved in the positive the longstanding problem of whether the two cardinal invariants \(\mathfrak{p}\) (the pseudointersection number) and \(\mathfrak{t}\) (the tower number) are equal. In this talk, I will review some essential points in their proof in order to motivate the study of the analogous question for the generalized characteristics \(\mathfrak{p}(\kappa)\) and \(\mathfrak{t}(\kappa)\). I will present some results of Garti regarding this generalization and finally some recent progress (joint work with Fischer, Schilhan, and Soukup) towards answering this question.

2019-04-11Ilijas Farah (York University, Toronto, Canada)See this announcement for the talk at the Sky Lounge

Professor Farah's talk at the KGRC had to be canceled. Please note, however, that his [ other talk] will take place as planned.

2019-05-02Philipp Lücke (Universität Bonn, Germany)Simple definitions of complicated sets
For many types of pathological sets of real numbers (i.e. sets of reals constructed with the help of the Axiom of Choice), it is possible to use results from descriptive set theory to show that these sets cannot be defined by simple formulas in second-order arithmetic. In this talk, I want to present results dealing with the set theoretic definability of pathological objects, i.e. with the question whether objects usually obtained from the Axiom of Choice can be defined in the structure \(\langle\mathrm{V},\in\rangle\) using simple formulas. I will focus on the definability of well-orderings of the reals and bistationary subsets of uncountable regular cardinals.

There is a [ video] recording of this talk available on YouTube.

The slides are also [ available].

2019-05-09Wolfgang Wohofsky (KGRC)A Sacks amoeba forcing preserving distributivity of $P(\omega)/fin$
In my talk, I would like to present joint work with Otmar Spinas, in which we show that it is consistent that \(\mathfrak{h}\) (the distributivity number of \(P(\omega)/fin\), in other words, the least number of maximal almost disjoint families without a common refinement) is strictly smaller than \(\mathrm{add}(s_0)\) (i.e., the least number of Marczewski null sets whose union is not Marczewski null, where a set is Marczewski null if each perfect set has a perfect subset disjoint from it). More explicitly, we show that this relation between \(\mathfrak{h}\) and \(\mathrm{add}(s_0)\) holds in the model obtained by a countable support iteration of length \(\omega_2\) of a specific kind of Sacks amoeba forcing which happens to have the pure decision and the Laver property, and therefore does not add Cohen reals. The model actually satisfies \(\mathfrak{h} = \mathrm{cov}(\mathcal{M}) < \mathfrak{b} = \mathfrak{s} = \mathrm{add}(s_0)\). (If time permits, I would also like to discuss why it seems easy to slightly modify the construction in such a way that the resulting model additionally satisfies the Borel Conjecture, but unclear how to modify it to make the Borel Conjecture fail.)

A [ video] recording of this talk is available on Youtube.
2019-05-23Asaf Karagila (University of East Anglia, Norwich, UK)Preservation theorems for symmetric extensions and Krivine-style results
Jean-Louis Krivine has used methods of realizability to prove several new independence results in ZF+DC. We show how to obtain some of these results using classical methods.

For the proof we also need theorems which lets us preserve some bits of choice in symmetric extensions. One of these theorems is an old folklore result, and the other is a new theorem.
2019-06-06Damian Sobota (KGRC)On convergent sequences of normalised measures on compact spaces
The celebrated Josefson–Nissenzweig theorem—in a special case of a Banach space \(C(K)\) of continuous real-valued functions on an infinite compact Hausdorff space \(K\)—asserts that there exists a sequence of Radon measures \((\mu_n)\) on \(K\) such that the total-variation of each \(\mu_n\) is \(1\) and for every continuous function \(f\in C(K)\) the sequence of the integrals \(\int_Kfd\mu_n\) converges to \(0\). All the recent natural proofs of the theorem start more or less as follows: "Assume there is not such a sequence \((\mu_n)\) but with an additional property that each \(\mu_n\) is a finite linear combination of one-point measures (Dirac's deltas). Then, ..." Although the proofs are correct, it appears that it is not clear at all when this assumption is satisfied. During my talk I will show when (and when not) it is the case that a compact space \(K\) admits a such a sequence of measures. As examples Efimov spaces, products of compact spaces, Stone spaces of some funny Boolean algebras will appear.

This is a joint work with Lyubomyr Zdomskyy.

A [ video] recording of this talk is available on YouTube.
2019-06-13Sy-David Friedman (KGRC)Mighty Mouse
Mighty Mouse is the least iterable structure with "many" strong cardinals of any finite order. Although Mighty Mouse doesn't look very big (she is much smaller than mice with even one Woodin cardinal) she is indeed Mighty: Iterated ultrapowers of Mighty Mouse result in inner models over which the entire universe of sets is generic. The reason for this is that the Stable Core can be captured by such an iteration.

A [ video] recording of this talk is available on YouTube.
2019-10-03Clifton Ealy (Western Illinois University, Macomb, USA)Residue field domination in real closed valued fields
I will talk about some recent results in the model theory of valued fields. No knowledge of valued fields or model theory will be assumed.

Haskell, Hrushovski, and Macpherson show that in an algebraically closed valued field, the residue field and the value group control the rest of the structure: \(tp(L/Ck(L)\Gamma(L)\) will have a unique extension to \(M\supseteq C\), as long as the residue field, \(k(L)\), and value group, \(\Gamma(L)\), of \(L\) are independent from those of \(M\) (and as long as \(C\) is maximal).

This behavior is striking, because it is what typically occurs in a stable structure (where types over algebraically closed sets have unique extensions to independent sets) but valued fields are far from stable, due to the order on the value group.

Real closed valued fields are even further from stable since the main sort is ordered. One might expect the analogous theorem about real closed valued fields to be that \(tp(L/M)\) is implied by \(tp(L/Ck(L)\Gamma(L))\) together with the order type of \(L\) over \(M\). In fact we show that the order type is unnecessary, that just as in the algebraically closed case, one has that \(tp(L/M)\) is implied by \(tp(L/k(L)\Gamma(L))\).

This is joint work with Haskell and Marikova.

A [ video] recording of this talk is available on YouTube.

The [ slides] are also available.
2019-10-10Christina Brech (Universidade de São Paulo, Brazil)Isometries of combinatorial Banach spaces
The Schreier family \(\mathcal{S} = \{F \in [\omega]^{<\omega}: |F| \leq \min+1\}\) induces a structure with no infinite sets of indiscernibles and this can be generalized to the context of Banach spaces. The fact that the canonical basis of the Banach space induced by the Schreier family doesn't have infinite indiscernibles gives a hint on the rigidity of this object, that is, on the fact that isometries of the space fix the basis, up to signs.

In this talk, we will give the background for the previous paragraph and will present a generalized version of it, obtained in a joint work with V. Ferenczi and A. Tcaciuc: given a regular family \(\mathcal{F}\), it is possible to define its corresponding combinatorial space \(X_\mathcal{F}\), which is a Banach space whose norm is defined in terms of the family \(\mathcal{F}\). We prove that every isometry of a combinatorial Banach space \(X_\mathcal{F}\) is induced by a signed permutation of its canonical basis.
2019-10-17Michał Tomasz Godziszewski (Munich Center for Mathematical Philosophy, Universität München, Germany)The Multiverse, Recursive Saturation and Well-Foundedness Mirage
Recursive saturation, introduced by J. Barwise and J. Schlipf is a robust notion which has proved to be important for the study of nonstandard models (in particular, it is ubiquitous in the model theory of axiomatic theories of truth, e.g. in the topic of satisfaction classes, where one can show that if \(M \models ZFC\) is a countable \(\omega\)-nonstandard model, then \(M\) admits a satisfaction class iff \(M\) is recursively saturated). V. Gitman and J. Hamkins showed in A Natural Model of the Multiverse Axioms that the collection of countable, recursively saturated models of set theory satisfy the so-called Hamkins's Multiverse Axioms. The property that forces all the models in the Multiverse to be recursively saturated is the so-called Well-Foundedness Mirage axiom which asserts that every universe is \(\omega\)-nonstandard from the perspective of some larger universe, or to be more precise, that: if a model \(M\) is in the multiverse then there is a model \(N\) in the multiverse such that \(M\) is a set in \(N\) and \(N \models 'M\) is \(\omega\)-nonstandard\(.'\). Inspection of the proof led to a question if the recursive saturation could be avoided in the Multiverse by weakening the Well-Foundedness Mirage axiom. Our main results answer this in the positive. We give two different versions of the Well-Foundedness Mirage axiom -- what we call Weak Well-Foundedness Mirage (saying that if \(M\) is a model in the Multiverse then there is a model \(N\) in the Multiverse such that \(M \in N\) and \(N \models 'M\) is nonstandard\(.'\).) and Covering Well-Foundedness Mirage (saying that if \(M\) is a model in the Multiverse then there is a model \(N\) in the Multiverse with \(K \in N\) such that \(K\) is an end-extension of \(M\) and \(N \models 'K\) is \(\omega\)-nonstandard\(.'\)). I will present constructions of two different Multiverses satisfying these two weakened axioms.

This is joint work with V. Gitman. T. Meadows and K. Williams.

Here are the [/2019/Slides_10-17_a.pdf slides] for this talk.
2019-10-24Yurii Khomskii (Amsterdam University College, Netherlands and Hamburg University, Germany)Symbiosis and Upwards Reflection
In [1], Bagaria and Väänänen developed a framework for studying the large cardinal strength of Löwenheim-Skolem theorems of strong logics using the notion of Symbiosis (originally introduced by Väänänen in his PhD Thesis). Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. We continue the systematic investigation of Symbiosis and apply it to upwards Löwenheim-Skolem theorems (and partially to compactness properties). As an application, we provide some upper and lower bounds of the large cardinal strength of upwards Löwenheim-Skolem-type theorems of second order logic.

This is joint work with Lorenzo Galeotti and Jouko Väänänen.

Here are the [/2019/Khomskii_Seminar_Vienna_2019-10-24.pdf slides] for this talk.

==== Reference ====

[1] Joan Bagaria and Jouko Väänänen, “On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals”, Journal of Symbolic Logic 81 (2) P. 584-604
2019-10-31Vera Fischer (KGRC)Spectrum of Independence
We will consider some recent results concerning the spectrum of independence, i.e. the set of cardinalities of maximal independent families. In particular we will discuss the existence of models in which the spectrum is finite, models in which the spectrum is countably infinite and models in which the spectrum is arbitrarily large. If time permits, we will outline a proof of the consistency of \(\mathfrak{i}<\mathfrak{a}_T\), which brings some light on the long-standing open question of the consistency of \(\mathfrak{i}<\mathfrak{a}\).

Slides for this talk will appear here later.
2019-11-07Jana Ma&rcaron;íková (Western Illinois University, Macomb, USA)Measures and o-minimal structures
O-minimal structures are structures with a dense linear order such that as few subsets of the line are definable as possible. This condition forces definable sets in all dimensions to behave in a topologically tame fashion. But while many properties from semialgebraic geometry carry over to the o-minimal setting, the question whether one has a theory of integration in any o-minimal structure is still open. We shall discuss some partial answers and suggest a possible future direction.
2019-11-14Sy-David Friedman (KGRC)L[Reg]
Gitik and I showed that L[Reg], like L[Card] and V, is a forcing extension of an iterate of a mouse. For L[Card] the mouse is the least one with a measurable limit of measurables (result of Philip Welch) and for V it's Mighty Mouse. For L[Reg] it's the least mouse with a measure concentrating on measurables, but the iterate may have to be truncated (for example if there are no regular limit cardinals). We use Magidor's iteration of Prikry forcings. An unexpected twist is that we need to use old-fashioned mice and not fine-structural mice for the proof.

[/2019/Sy-David_Friedman_2019-11-14_KGRC_research_seminar.pdf Slides] for this talk are available.
2019-11-21Martin Goldstern (TU Wien)Cichoń's Maximum without large cardinals
How many Lebesgue null sets are needed to cover the real line? How many functions (from the natural numbers to the natural numbers) are needed to dominate all functions? The answers to these questions and 10 more related questions are certain uncountable cardinals collected in Cichoń's Diagram, which also shows provable inequalities between these cardinals.

In a recent joint work with Jakob Kellner, Diego Mejía and Saharon Shelah, we constructed a model where the cardinals in Cichoń's Diagram have 10 different values (which is known to be best possible).

The construction first uses a finite support iteration P to get 5 different values for the left side of the diagram, and then finds a subset P' which also separates the values on the right side.
2019-11-28István Juhász (Rényi Mathematical Institute, Budapest, Hungary)Large cardinals in topology
In this talk I intend to deal with a number of problems of general topology that, sometimes rather naturally but sometimes quite surprisingly, lead to or boil down to statements of set theory involving various large cardinals.
2019-12-05Mohammad Golshani (Institute for Research in Fundamental Sciences (IPM), Tehran, Iran)Specializing trees
We review our recent work on specializing trees on higher cardinals. The results are based on several joint works with Aspero, Hayut and Shelah.
2019-12-12Oleksandr Ravsky (National Academy of Sciences of Ukraine)Basic results and open problems on feebly compact paratopological groups
Discussing feebly compact and other compact-like paratopological groups, I am going to present basic results and pose open problems for a forced attack during my stay in Vienna. See for details the talk basic paper at (an updated version should be available in a few days).
2019-12-19Taras Banakh (Ivan Franko National University of Lviv, Ukraine)The Golomb space is topologically rigid
The Golomb space is the space \(\mathcal N\) of natural numbers endowed with the topology generated by the base consisting of arithmetic progressions \(a+b\mathbb N\) with coprime parameters \(a\) and \(b\). The Golomb space is one of the simplest examples of a countable connected Hausdorff space. In this space Topology and Arithmetic are tightly intertwined. We survey topological properties of the Golomb space and prove that this space is topologically rigid, i.e., has trivial homeomorphism group. This resolves a problem posed on Mathoverflow in 2017. Philosophically, the topological rigidity of the Golomb space can be interpreted as the possibility to encode in Topology all Arithmetics (which encodes all Mathematics by the Godel arithmetization).