## Oberseminar Mathematische Logik

Advanced talks on mathematical logic by guests and members of the logic group

### Organized by

- Prof. Dr. Peter Koepke
- Prof. Dr. Stefan Geschke
- Dr. Philipp Schlicht

### Time and location

Mondays 16:30-18:00 at room 107.

The members of the seminar are welcome for coffee and tea at the Hausdorff-Raum 109 (this is the "old" room number above the doorframe; it's next to 111) from 16:00-16:30 before the talks.

### Programme

**27 July**Jana Flašková (University of West Bohemia, Pilsen)*Some ultrafilters on omega*- In my talk I want to prove the following theorem: "There exists (in ZFC) an ultrafilter u on omega such that for every one-to-one function f there exists a set U in u such that f[U] belongs to the summable ideal i.e. sum_{n in f[U]} 1/n < infty." The construction of such an ultrafilter is based on Gryzlov's construction of 0-points and involves construction of some special linked families of subsets of natural numbers. Some connection to other topics such as the more general concept of I-ultrafilters (introduced by Baumgartner) or a topological question concerning covering omega^* by nowhere dense sets (posed by van Douwen) will be mentioned.

**13 July**Toshimichi Usuba (Universität Bonn)*Splitting general stationary sets*- The notion of general stationary set was defined by Woodin in the context of his stationary tower forcing. For the classical stationary set of a regular uncountable cardinal, there is a well-known Solovay's result that every stationary set of kappa can be split into kappa disjoint stationary sets. A similar result for stationary sets of P_kappa lambda is known as well. In this talk, we will consider splitting problems about general stationary sets. We will outline a proof of a splitting theorem which is an analogue of Solovay's result. We will also discuss splitting stationary sets related to Jonsson cardinals.

**29 June**Vera Fischer (Universität Wien)*Bounding, splitting and almost disjoint families.*- In 1984 S. Shelah obtained the consistency of $\mathfrak{b}=\omega_1<\mathfrak{s)=\omega_2$ using a proper forcing notion, which adds a real not split by the ground model reals and satisfies the almost bounding property. In this talk, we will consider some recent results regarding the splitting, bounding and almost disjointness number. We will briefly outline some iteration techniques, which have been used in or developed for their study, and conlcude with open questions and further directions of research.

**25 May**Samuel Coskey (CUNY and MPIM)*Countable Borel equivalence relations*- Borel equivalence relations is an area of descriptive set theory which concerns the complexity of equivalence relations on a standard Borel space (i.e., a Polish space equipped just with its sigma-algebra of Borel sets). There are interesting examples from within logic, such as the Turing equivalence relation on P(omega). Moreover, many classification problems for other areas of mathematics can be regarded as equivalence relations on standard Borel spaces. For instance, the classification problem for torsion-free abelian groups of rank n corresponds to the isomorphism equivalence relation on a suitable subspace of P(Q^n). Both of these examples are instances of *countable Borel* equivalence relations, that is, equivalence relations that are Borel as subsets of the plane and which have the property that every equivalence class is countable. After giving the definitions, I'll discuss what structure theory exists for these objects. I'll pay special attention to the example of torsion-free abelian groups, where there are several key applications.

**18 May**Mirna Džamonja (University of East Anglia)*Trees of singular height and chain models*- We present joint work with Jouko Väänänen in which we show how combinatorics of trees of singular height can be used to understand the model theory of chain models (as studied by Carol Karp). We shall define the appropriate Ehrenfeucht-Fraïssé games and see that we can obtain phenomena very close to those for L_omega,omega, including consequences in the descriptive set theory of 2^kappa for singular kappa.

**11 May**Arthur Apter (CUNY)*Can the least uncountable regular cardinal be the least Jonsson cardinal?*

**04 May**Philipp Doebler (Universität Münster)*The semiproperness of stationary set preserving forcings*- We show that if for all regular theta above omega_1 there is a semiproper partial ordering that adds a generic iteration of length omega_1 with last model H_theta, then all stationary set preserving forcings are semiproper. This is joint work with Ralf Schindler.

**27 April**Philipp Schlicht (Universität Bonn)*Thin equivalence relations in scaled pointclasses*- A thin equivalence relation on the reals has the property that there is no perfect set of pairwise inequivalent reals. I will present an inner-model theoretic proof that thin equivalence relations which are Sigma_1 over an initial segment of L(R) at the beginning of certain gaps are Delta_1 over the initial segment, assuming sufficient determinacy. This is joint work with Ralf Schindler.

**20 April**Jörg Brendle (Kobe University)*Kardinalzahlinvarianten analytischer Quotienten*- Viele der klassischen Kardinalzahlinvarianten des Kontinuums, wie die "tower number", die "splitting number", die "almost disjointness number", etc. beschreiben die kombinatorische Struktur der Booleschen Algebra P(N) / fin. Für ein definierbares Ideal I auf N kann man den Quotienten P(N) / I betrachten und die zu P(N) / fin gehörigen Invarianten auf analoge Weise einführen. Dabei stellt sich ganz natürlich die Frage, in wie weit man für diese neuen Invarianten die Ungleichungen, die für die klassischen Analoga gelten, beweisen kann, bzw. wie die neuen Invarianten mit ihren klassischen Analoga zusammenhängen. Dies ist besonders interessant für klassische Ideale wie das "summable ideal" oder das "density zero ideal", oder -- allgemeiner -- für analytische P-Ideale, die nach Soleckis Charakterisierung eine simple Beschreibung haben. In meinem Vortrag möchte ich einen Abriss bekannter Resultate und offener Probleme in diesem Themenkomplex geben.

Last changed: 24 July, 2009