# Schedule of the Workshop: Generalized Baire spaces

## Wednesday, 21. September 2016

 9:00 - 9:30 Registration 9:30 - 10:30 Galeotti, Khomskii & Loewe: „The generalised real numbers (Part I)“ 10:30 - 11:00 Coffee break 11:00 - 12:00 Galeotti, Khomskii & Loewe: „The generalised real numbers (Part II)“ 12:00- 14:00 Lunch break 14:00 - 15:00 Motto Ros: „Generalized descriptive set theory and the classification of uncountable structures and non-separable spaces (Part I)“ 15:00 - 15:30 Moreno: „The Equivalence Modulo Non-stationary Ideals and Shelah’s Main Gap Theorem“ 15:30 - 16:00 Coffee break 16:00 - 16:30 Koelbing: „Gaps in the generalized Baire space (Part I)“ 16:30 - 17:30 Friedman: „Gaps in the generalized Baire space (Part II)“ 17:30 - 18:00 Montoya: „The ultrafilter number for uncountable κ“ 18:30 - 20:00 Informal conference dinner

## Thursday, 22. September 2016

 9:30 - 10:30 Motto Ros: „Generalized descriptive set theory and the classification of uncountable structures and non-separable spaces (Part II)“ 10:30 - 11:00 Group photo and Coffee break 11:00 - 12:00 Aspero: „The Baumgartner topology on the clubs of ω1“ 12:00 - 14:00 Lunch Break 14:00 - 15:00 Laguzzi: „Uncountable trees and Cohen sequences“ 15:00 - 15:30 Fischer: „The spectrum of κ-maximal cofinitary groups“ 15:30 - 16:00 Coffee break 16:00 - 16:30 Sziraki: „A dichotomy for infinitely many Σ02(κ) relations on the κ-Baire space“ 16:30 - 17:00 Wohofsky: „Generalizing Galvin-Mycielski-Solovay's theorem to 2κ for uncountable κ“ 17:00 - 18:00 Discussion session

# Abstracts

## David Asperó: The Baumgartner Topology on the clubs of $\omega_1$

I will take a look at the topology on the set of clubs of $\omega_1$ whose basis is given naturally by the conditions in Baumgartners forcing for adding a club with finite conditions. This topology arises as a natural object to manipulate using forcing with side conditions. I will show connections with cardinal characteristics for the $\omega_1$reals. Finally, I will discuss directions in which to extend the relevant forcing techniques and will ask several questions.

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## Vera Fischer: The Spectrum of $\kappa$-Maximal Cofinity Groups

For $\kappa$ an arbitrary regular uncountable cardinal we define the spectrum of $\kappa$-maximal cofinitary groups (abbreviated k-mcg) as the set of all possible cardinalities of $\kappa$-mcg's. Generalizing Blass's result on the spectrum of mad families on $\omega$, we provide sufficient conditions for a closed set of cardinals to be generically realized as the spectrum of $\kappa$-mcgs. We will conclude with some current directions of research and open questions.

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## Marlene Koelbing: Gaps in the generalized Baire space

In the classical setting, gaps on $\omega$ are considered. There are no $(\omega,\omega)$-gaps, but $(\omega_1,\omega_1)$-gaps exist in ZFC (well-known construction by Hausdorff). Rothberger showed, that $\mathfrak{b}$ is the least cardinal $\lambda$ such that an $(\omega,\lambda)$-gap exists. An $(\omega_1,\omega_1)$-gap is called destructible, if there is an extension of the universe with the same $\omega_1$ in which the gap is not a gap anymore. There is a way to assign to every gap a forcing with the property that it is $ccc$ if the gap is destructible (in this case the forcing witnesses the destructibility).

We generalize the concept of gaps by considering the analogous objects on regular cardinals $\kappa>\omega$ and discuss, to what extent the above results hold for gaps on $\kappa$.

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## Sy-David Friedman: Gaps in the Generalised Baire Space, Part 2

Marlene Koelbing and I have been investigating the destructibility of gaps in the Generalised Baire Space $(\omega_1,\omega_1)$. The natural conjecture is that it is possible to force the nonexistence of destructible gaps, as is the case for the standard Baire Space. This problem ties in with difficult issues concerning the preservation of the $(\omega_2-cc)$ in countable support iterations of $\omega$-closed forcings and in my talk I'll discuss these issues and the possibilities for dealing with them in the case of gaps.

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## Lorenzo Galeotti, Yurii Khomskii and Benedikt Loewe: The generalised real numbers

We discuss the theory of cardinal characteristics for generalised Baire space, give an overview of what is known and list some open problems from the paper "Questions on generalised Baire space" (Khomskii, Laguzzi, Löwe, Sharankou, MLQ 62:4-5, 2016). Combinatorial cardinal characteristics can be directly defined for generalised Baire space, but some of the cardinal characteristics involved in Cicho\'{n}'s diagram involve the topological space $\mathbb{R}$ of the real numbers. This raises the question of what the appropriate analogue for the real numbers is in the generalised setting. We give an overview of Conway's surreal numbers and present a candidate for the generalised real numbers. We close with some results connecting the strength of analogues of classical theorems of analysis in the generalised reals to large cardinals.

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## Giorgio Laguzzi: Uncountable trees and Cohen sequences

We investigate some versions of amoeba for tree forcings in the generalized Cantor and Baire spaces. This generalizes a line of research centered around the study of forcings adding generic trees. Moreover, we also answer some questions posed by Friedman, Khomskii and Kulikov, about the relationships between regularity properties associated with tree-forcings. We show $\mathbf{\Sigma}^1_1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire property which does not occur in the standard case.

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## Diana Carolina Montoya: The ultrafilter number for uncountable $\kappa$

This is a joint work with Andrew-Brooke Taylor, Vera Fischer and Sy-David Friedman in which we provide a model where $\mathfrak{u}(\kappa) = \kappa^+ < 2^\kappa$ holds for a supercompact cardinal $\kappa$. I will present the construction of such a model in the countable case and explain why, the straightforward generalization to an uncountable cardinal does not work, as well as the construction of our model and its main properties. Moreover we study the values of other natural generalizations of classical cardinal characteristics in this model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.

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## Miguel Moreno (joint work with T. Hyttinen and V. Kulikov): The Equivalence Modulo Non-stationary Ideals and Shelah's Main Gap Theorem

One motivation to study the generalized Baire spaces is the connection with model theory. Precisely the connection between the Borel reducibility hierarchy and the classiffcation of theories in Shelah's stability theory. The complexity of a theory is measured by determining the place of the isomorphism relation of models of size $k$ in the Borel-reducibility hierarchy. The different kind of theories (classiffable, unstable, strictly stable, etc) have a total or partial characterization in the Borel-reducibility hierarchy so far (see the work of Friedman, Hyttinen, Kulikov, etc.). The equivalence modulo non- stationary ideal has played an important roll in many of these characterizations. This shows us a connection between the equivalence modulo non-stationary ideal and the classiffcation of theories in Shelah's stability theory. The connection we study is related to the gap between classiffable and non- classiffable theories; stability theory tells us that classiffable theories are less complex than non-classiffable theories and also that their complexity are far apart (Shelah's Main Gap Theorem). In this talk we will see results connecting the equivalence modulo non- stationary ideal and the gap between classiffable and non-classiffable theories. One of those is a Borel-reducibility counterpart of Shelah's main gap theorem: it is consistent that for every classiffable and every non-classiffable theory we can embed the partial order $(P(\kappa), \subset)$ to the Borel-reducibility partial order strictly between the isomorphism relations of these theories.

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## Luca Motto Ros: Generalized descriptive set theory and the classification of uncountable structures and non-separable spaces

In mathematics one frequently deals with the problem of classifying various objects up to some natural notion of equivalence by means of (complete) invariants. In the last three decades, the notion of Borel reducibility in classical descriptive set theory has proven to be an invaluable tool in studying the complexity of many such problems. However, an intrinsic limitation to this method is that it allows us to deal just with classification problems concerning countable (algebraic) structures or separable spaces, such as separable complete metric spaces, separable Banach spaces, and so on.

In this tutorial I will survey some results (by various authors) which demonstrate how the so-called generalized descriptive set theory can be used to overcome this limitation and treat analogous classification problems concerning uncountable structures and non-separable spaces, often leading to interesting results that significantly differ from their countable/separable counterparts. Indeed these applications of generalized descriptive set theory may be collectively viewed as a strong motivation for pursuing the study of this subject and of its surprising connections with combinatorial set theory and model theory.

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## Dorottya Sziraki: A dichotomy for inffnitely many $\mathbf{\Sigma}^2_0 (\kappa)$ relationson the $\kappa$-Baire space

As an initial step in studying dichotomies about independent sets with respect to certain (sets of) definable relations on the $\kappa$-Baire space, I examine the case of a set $\mathcal{R}$ of $\kappa$ many $\mathbf{\Sigma}^2_0 (\kappa)$ relations (of arbitrary finite arity). By considering games introduced by Jouko Väänänen (1991) which allow trees to play, for the $\kappa$-Baire space, a role analogous to that of Cantor-Bendixson ranks in the classical case and by restricting the allowed strategies for these games, I show that the $\kappa$-version of a recent result of Martin Dolezal and Wies law Kubis holds under $\diamond_\kappa$. Namely, for a set $\mathcal{R}$ of relations as above, if $^\kappa\kappa$ has $\mathcal{R}$-independent subsets of "arbitrary Cantor-Bendixson rank", then there exists a $\kappa$-perfect-$\mathcal{R}$-independent subset. For $\kappa$ inaccessible, this is true already in ZFC. As a corollary, we obtain (the slightly more general version of) a recent theorem of Jouko Väänänen and myself. Further related questions are under investigation.

## Wolfgang Wohofsky: Generalizing Galvin-Mycielski-Solovay's theorem to $2^\kappa$ for uncountable $\kappa$
The Galvin-Mycielski-Solovay theorem says that a set on $2^\omega$ is strong measure zero if and only if it can be translated away from each meager set ("meager-shiftable"). Since both the notion of strong measure zero and the notion of meager-shiftable can be generalized to $2^\kappa$, it is natural to ask for which uncountable kappa the Galvin-Mycielski-Solovay theorem holds. I will show that this is indeed the case for weakly compact kappa, but that (it is consistent that) there are counter-examples for successor cardinals. If there is enough time I will also discuss the status of the remaining case, i.e., for kappa being inaccessible but not weakly compact.