Hamburg Workshop on
Set Theory 2015
Generalized Baire Space
20 & 21 September 2015
Hamburg (Germany)
The Hamburg Set Theory Workshop 2015
aims to provide the platform for exchange for the researchers active in the field of the set theory of the generalized Baire space. It is a satellite workshop taking place right before the beginning of the DMV 2015 in Hamburg and it is partially funded by the Deutsche Forschungsgemeinschaft (grant
number LO834/121).
The two days of the workshop will consist of two tutorials, several contributed talks and a concluding discussion session.
Everyone is cordially invited to attend.
Organizers:
Giorgio Laguzzi,
Wolfgang Wohofsky
Location:
20146 Hamburg, Germany. University Main Building in SiemersAllee 1.
How to get there > From Dammtor. (~5 min walk) ;
The seminar talks will take place in the Lectures Hall J, "Magdalene Schoch Hoersaal". (How to reach it.)
Tutorials
Luca Motto Ros (Turin),
Vadim Kulikov(Vienna)
Participants
Andrew BrookeTaylor (Bristol),
David Chodounsky (Prague),
Yurii Khomskii (Hamburg),
Marlene Koelbing (Vienna),
Vadim Kulikov (Vienna),
Giorgio Laguzzi (Freiburg),
Benedikt Löwe (Amsterdam & Hamburg),
Philipp Lücke (Bonn),
Heike Mildenberger (Freiburg),
Diana Carolina Montoya Amaya (Vienna),
Miguel Moreno (Helsinki),
Luca Motto Ros (Freiburg),
Hugo Nobrega (Amsterdam),
Philipp Schlicht (Münster),
Ilya Sharankou (Hamburg),
Dorottya Sziráki (Budapest),
Sandra Uhlenbrock (Münster),
Wolfgang Wohofsky (Hamburg).
Program
20.09.2015, Sunday
 
11:0012:00

Vadim Kulikov  The language M^kappa^+_kappa and Borel sets

12:1514:00
 Lunch Break

14:0014:45
 Dorottya Sziráki  A dichotomy for Sigma^0_2 relations and elementary embeddability at uncountable cardinals.

15:0015:45
 Diana Montoya: On Cichon's diagram for uncountable kappa.

15:4516:15
 Coffee Break

16:1517:00
 Philipp Schlicht  TBA.

17:1518:00
 Philipp Lücke  Lightface Delta^1_1 subsets of omega_1^omega_1.
 

21.09.2015, Monday
 
10:0011:00

Luca Motto Ros  On some classification problems concerning uncountable structures and nonseparable metric spaces.

11:1011:30
 Coffee Break

11:3012:15
 Andrew BrookeTaylor  On large cardinal preservation for generalized Baire space.

12:3014:00
 Lunch Break

14:0014:45
 Giorgio Laguzzi  Amoeba for uncountables.

14:4515:30
 Final discussion session

Abstracts
Andrew BrookeTaylor (Bristol)
On large cardinal preservation for generalized Baire space 
I will discuss some observations regarding the use of large cardinal preservation arguments in the study of generalized Baire space.
Vadim Kulikov (Vienna) The language M^kappa^+_kappa and Borel sets 
The study of generalized Baire spaces is largely motivated by the connections with the theory of uncountable models. In this tutorial we will look at this from the point of view of infinitely deep languages. I will try to focus on the history and motivations behind Borel* sets and the M_{\kappa^+\kappa} language. I will finish by presenting our results with Tapani Hyttinen from 2012 on the relationship between these concepts and putting forward speculations for further research.
Giorgio Laguzzi (Freiburg) Amoeba for uncountables 
Amoeba forcings play a central role in the study of regularity properties and cardinal invariants. In particular they are important for increasing the additivity number of a given ideal. In this talk I will analyze some amoebas for different versions of Sacks forcing in 2^\kappa.
Philipp Luecke (Bonn). Lightface Delta^1_1 subsets of omega_1^omega_1 
We say that a set of functions from omega_1 to omega_1 is Delta^1_1 if it is definable over H(omega_2) by a Delta_1formula with parameters.
It is wellknown that many basic questions about the structural properties of this class are not decided by the axioms of ZFC together with large cardinal axioms.
In my talk, I want to present examples of such questions that are settled by large cardinal axioms when we restrict ourselves to sets of functions defined by Delta_1formulas whose parameters are contained in L(R).
Diana Montoya (Vienna). On Cichon's diagram for uncountable kappa 
Cardinal invariants of the Baire Space omega^omega have been widely studied and understood. In this talk I will
consider higher analogues of these invariants which are naturally associated to the generalized Baire space kappa^kappa,
where kappa is an uncountable cardinal satisfying kappa^<\kappa= kappa. Our research focuses mainly on the analogues to the cardinals in Cichon's Diagram associated with the kappaMeager ideal,
due to the absence of a notion of measure on these spaces. I will present the results that can be easily lifted from the countable
case as well as discuss some differences and open problems that arise
when studying these generalizations.
Luca Motto Ros (Turin). On some classification problems concerning uncountable structures and nonseparable metric 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 talk I will survey some recent results which demonstrate how the socalled generalized descriptive set theory can be used to overcome this limitation and treat analogous classification problems concerning uncountable structures and nonseparable 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.
Dorottya Sziraki (Budapest). A dichotomy for Sigma^0_2 relations and elementary embeddability at uncountable cardinals 
In this talk, we consider the uncountable version of a dichotomy of Wieslaw Kubis': if R is a Sigma^0_2 finitary relation on an analytic subset of the generalized Baire space kappa^kappa, then either all Rindependent sets are of size at most kappa, or there is a kappaperfect Rindependent set. We prove that this statement holds if we assume Diamond_kappa and I^(kappa) (the modification of the assumption I^(kappa) which is suitable for limit cardinals). When kappa is inaccessible or when the relation R is closed, Diamond_kappa is not needed.
We obtain as a corollary the uncountable version of a result of Gabor Sagi's and mine, about the number of kappasized models of a Sigma^1_1(L_{kappa^+,kappa})sentence when considered up to isomorphism (and also up to embeddability, elementary embeddability, and similar notions) by elements of a K_kappa subset of kappa^kappa. This is joint work with Jouko Vaananen.
