Die folgenden Vorträge haben im Wintersemester 2013/14 im Oberseminar Modelltheorie stattgefunden.

 

Montag, 21.10.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Montag, 28.10.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Cédric Milliet (Konstanz)

Variations on a theme by Aldama

Abstract: Let G be a group, and f(x,y) a formula in the language of groups. We say that f(x,y) has the independence property if for all natural number n, we can find elements a0, a1, …, an and (bI) I ⊆ n+1 in G such that f(ai,bI) holds if and only if i ∈ I. The group G is said to be ‘without the independence property’ if no formula has the independence property in G. Such groups include stable groups. We shall review recent results of Shelah and Aldama concerning infinite abelian and nilpotent subgroups of a group G without the independence property, and address the following open question: is a soluble subgroup S of G contained in a soluble definable subgroup of G?

 

Montag, 04.11.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Alexander Prestel (Konstanz)

Model theoretic criteria for uniform definability (with fixed quantifier complexity) of valuation rings 

Abstract: (see abstract for Oberseminar Reelle Geometrie und Algebra on Nov. 8, i.e.) We shall prove the (pure) existence of formulas defining uniformly the valuation rings O in different classes of henselian fields (K,O). For instance, we prove that there is an EA-formula in the ring language defining O uniformly for all henselian fields (K,O) where the residue class field is finite, pseudo-finite,or hilbertian. (Note that all function fields and number fields are hilbertian)

 

Montag, 11.11.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Pantelis E. Eleftheriou (Konstanz)

Pregeometries and definable groups

Abstract: We describe a recent program for analyzing definable sets and groups in certain model theoretic settings. Those settings include:

(a) o-minimal structures (M, P), where M is an ordered group and P is a real closed field defined on a bounded interval (joint work with Peterzil),

(b) tame expansions (M, P) of a real closed field M by a predicate P, such as expansions with o-minimal open core (work in progress with Gunaydin and Hieronymi).

The analysis of definable groups first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.

This talk is a natural continuation of the one given last semester, but no previous knowledge will be assumed.

 

Montag, 18.11.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 25.11.2013 um 15.15 Uhr, Oberseminar Modelltheori

kein Vortrag

 

Montag, 02.12.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 09.12.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Merlin Carl (Konstanz)

Infinitary computations with random oracles

Abstract: Turing computability provides an intuitive and attractive formal framework for the analysis of the question which mathematical objects and functions can be generated by applying a certain procedure finitely many times. However, in mathematical practice, we often encounter examples of objects generated by infinite processes involving limits, such as adding reals, computing with limits of real sequences, forming direct limits of directed systems, hulls, closures etc. This motivates the study of infinitary computations that can account for the limit phenomenon in such processes and provide a framework for extending results from recursion theory to infinitary mathematical objects.

A fascinating theorem of classical recursion theory due to G. Sacks is that a function f: N -> N is Turing-computable from all oracles in a set X of positive Lebesgue measure iff f is recursive. Intuitively, this means that the use of random generators does not enrich the set of computable functions, not even when computability is weakened to computability with positive probability.

We consider the analogous problem for various infinite time machines: If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? Exploiting the strong connection between generalized computability and models of Kripke-Platek set theory, we show that the answer is positive answer for most other machine types. Moreover, we show that this is independent from ZFC for ordinal Turing machines (OTMs) with and without ordinal parameters. In particular, the parameter-free version is false in Gödel’s constructible universe L. This is joint work with Philipp Schlicht.

 

Montag, 16.12.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 06.01.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Feiertag, kein Vortrag

 

Montag, 13.01.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 20.01.2014 um 13.30 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 27.01.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Will Ascombe (University of Oxford)

(Gast von Arno Fehm)

Asymptotic classes and measurable structures

Abstract: A well-known theorem of Chatzidakis, van den Dries, and Macintyre gives a nice asymptotic description of the cardinalities of sets drawn from uniformly definable families in finite fields. The description extends the Lang-Weil estimates from irreducible varieties to arbitrary definable sets.

Taking this theorem as a definition, Macpherson and Steinhorn developed a notion of a 1-dimensional asymptotic class of finite structures. They gave a variety of examples, both algebraic and combinatorial, and proved that non-principal ultraproducts of these classes are measurable, and in turn supersimple of S₁-rank 1. In subsequent work Elwes generalised this notion to an N-dimensional asymptotic classes. This definition still lives in the supersimple world, but there are interesting new examples.

Much more recently Macpherson and Steinhorn have yet again generalised the definition to `multidimensional asymptotic class' (`mac'). In this talk I will survey this subject, hopefully providing motivation for the definitions, and explain the ongoing work on macs.

 

Montag, 03.02.2014 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 10.02.2014 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag