Oberseminar mathematische Logik
Advanced talks on mathematical logic by guests and members of the logic group
Time and location
Mondays 16:30-18:00 at room 1.007, Endenicher Allee 60.
The participants of the seminar are welcome for coffee and tea at the Hausdorff-Raum 1.012 from 16:00-16:30 before the talks.
- 19 April Sean Cox (Universität Münster) The Chang Ideal
- Chang's Conjecture is a natural strengthening of the downward Löwenheim-Skolem Theorem. In modern terms, Chang's Conjecture is equivalent to asserting that a certain canonical set S is stationary. If S is stationary, then the Chang Ideal is defined as the nonstationary ideal restricted to S. Generic ultrapowers of the universe by Chang ideals have nice properties.
I will discuss some background, some modest results of mine, and also some interesting results of Foreman (Foreman's results show the power of combining Chang's Conjecture with condensation-like properties).
- 26 April Christoph Weiß (Universität München) Combinatorial characterizations of supercompactness and their applications to forcing axioms
- 03 May Stefan Geschke (Universität Bonn) Basis Theorems for uncountably homogeneous, continuous colorings
- We investigate the class of all continuous colorings of the n-element subsets of a Polish space. Such a coloring c is uncountably homogenous if the underlying space is not the union of countably many c-homogeneous sets. Answering a question of Ben Miller we show that there is a finite family F of uncountably homogeneous, continuous colorings of the n-element subsets of the Cantor space such that every uncountably homogenous, continuous coloring of the n-element subsets of any Polish space X contains an isomorphic copy of one of the colorings from F.
- 10 May no seminar
- 17 May Vassilis Gregoriades (TU Darmstadt) Effective Descriptive Set Theory and Applications in Analysis
- Descriptive Set theory is the area of mathematics which studies the structure of definable sets in separable and complete metric spaces. A central part of this area is Effective Descriptive Set Theory (for shortness Effective Theory) which uses ideas from Logic and in particular from Recursion Theory. This area provides a deeper understanding of the subject and has applications in Analysis. We begin this lecture with a brief summary of the basic concepts and we give a new approach to Effective Theory which yields some surprising results. Then we present some connections with General Topology; all theorems we talk about involve notions from both areas. Finally we give some results in Banach space Theory. Here all theorems involve only classic (i.e., non-effective) notions but still Effective Theory is essential for their proof.
- 24 May no seminar
- 31 May no seminar
- 07 June Marcin Sabok (Polish Academy of Sciences, Warsaw) Idealized forcing and infinite dimensional perfect set theorems
- We will prove that if a Pi^1_1 on Sigma^1_1 definable family of analytic sets is a sigma-ideal generated by closed sets, then it is still a sigma-ideal generated by closed sets in any forcing extension. As a corollary, we get that the countable support iteration of idealized forcings with Pi^1_1 on Sigma^1_1 sigma-ideals generated by closed sets is proper. As an application, we will show some infinite dimensional "perfect set" theorems.
- 14 June Lorenz Halbeisen (Universität Zürich) Ramsey ultrafilters in terms of partitions
- There are different ways to define Ramsey ultrafilters over the natural numbers. For example one can define Ramsey ultrafilters by Ramsey's Theorem, in terms of games, or with Mathias' notion of happy families. Based on these definitions, Ramsey ultrafilters in terms of partitions will be introduced and their existence will be discussed.
- 21 June Lutz Strüngmann (Universität Bonn) Abelian groups that (do not) have an automatic presentation
- 28 June Barbara Csima (University of Waterloo) Turing Degree Spectra of Countable Structures
- Fix a finite or computably infinite language. The Turing degree of a structure over this language, with universe the natural numbers, is the Turing degree of the atomic diagram of the structure. Isomorphic structures may have different Turing degrees. The degree spectrum of a countable structure is the collection of Turing degrees of all isomorphic copies of the structure with universe the natural numbers. Turing degree spectra of structures in general and restricted to particular classes of structures, such as linear orderings, are a well studied topic in Computable Structure Theory. We give a review of basic results about degree spectra of structures, and indicate areas of current research.
- 05 July Toshimichi Usuba (Universität Bonn) A new combinatorial principle on singular cardinals
- 12 July Ioanna Dimitriou (Bonn) Countable sequences of successive singular cardinals
- We will see how to use symmetric forcing to construct models of ZF plus the negation of the axiom of choice, in which there exist countable sequences of successive singular cardinals of cofinality omega. For this we will start from a model of ZFC that contains a countable sequence of strongly compact cardinals. The strongly compacts will be singularised and every cardinal between them will be collapsed. We will prove that in the resulting symmetric model the former strongly compact cardinals still have several combinatorial properties. In the end we will consider some modifications of this construction and see what we can prove in those other situations. This talk is inspired by Moti Gitik's 1980 paper "All uncountable cardinals can be singular". It is a modification of Gitik's construction, in order to prove that certain cardinals indeed collapse, and it is simplified since it does not deal with limits of the sequence of strongly compact cardinals that have uncountable cofinality.
- 19 July no seminar