Die folgenden Vorträge haben im Sommersemester 2012 im Oberseminar Modelltheorie stattgefunden.

 

Montag, 16.04.2012 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 30.04.2012 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Montag, 07.05.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Yimu Yin (Paris 6)

(Gast von Arno Fehm)

Motivic integration in real closed fields

Abstract: I will describe how to construct Hrushovski-Kazhdan style integration in real closed fields. The setting is based on early work by van den Dries-Lewenberg on T-convex theories. If time permits (most likely not), I will also sketch a potential application to real Milnor fibers.

Montag, 14.05.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Lorna Gregory (Konstanz)

Decidability of Theories of Modules over Valuation domains

Abstract: The Ziegler spectrum Zg_R of a ring R is a topological space attached to the module category of R. Its set of points is the set of isomorphism classes of indecomposable pure-injective (aka algebraically compact) modules, pinj_R, and it has a basis of open sets
(φ/ψ) = {N ∈ pinj_R : φ(N) ≠ ψ(N)}
where φ, ψ are pp-1-formulae.

In this talk we will explain how to use Zg_R as a conceptual tool to prove decidability results for the theory of modules. We will apply this method specifically to commutative valuation domains where it yields the following theorem: Let R be an effectively given valuation domain. The following are equivalent:
1. The theory of modules over R is decidable.

2. There is an algorithm which, given a and b in R, answers whether a is in
rad(bR).

This theorem was conjectured by Puninski, Puninskaya and Toffalori.

 

Montag, 21.05.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Philipp Hieronymi (University of Illinois, Urbana-Champaign)

(Gast von Salma Kuhlmann)

Interpreting the projective hierarchy in expansions of the real line

Abstract: We give a criterion when an expansion of the ordered set of real numbers defines the image of the expansion of the real field by the set of natural numbers under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number a, the expansion of the ordered Q(a)-vector space of real numbers by the set of natural numbers defines multiplication on the real numbers. This is joint work with Michael Tychonievich (Ohio State).

Montag, 28.05.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

 

Montag, 04.06.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Karen Lange (Wellesley College)

(Gast von Salma Kuhlmann)

Degrees of orderings on torsion-free abelian groups

Abstract: It is well known that an abelian group admits an ordering if and only if it is torsion-free.  This classical statement is false, however, from a computable perspective.  Downey and Kurtz (1986) showed that there is a computable torsion-free abelian group that admits no computable ordering.  We look at generalizations of this result by examining the collection of orderings X(G) on a given computable torsion-free group G.  Specifically, we are interested in the degree spectrum of X(G), i.e., the   set of  degrees of orderings in X(G). One  way to construct orderings is to use a basis for G, and this relationship between bases and orderings has consequences for the degree spectrum of X(G).  Given these consequences, it is natural to ask whether the  degree spectra of orderings on computable torsion-free  abelian groups are  closed upward.  In joint work with Kach and Solomon, we show the answer is no.

 

Montag, 11.06.2012 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Montag, 18.06.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Danielle  Gondard-Cozette (Institut Mathematique de Jussieu, Universite Paris 6 (Universite Pierre et Marie Curie))

(Gast von Salma Kuhlmann)

On some elementary field theories

Abstract: Orderings have very nice links with valuations and we provide some back- ground on relations between orderings and valuations. But in a formally real field one can also consider objects more general than orderings, which also have a close relationship with valuations: these are called valuation fans. We introduce valuation fans and give some examples.
Then, field theories related to valuation fans are introduced. These are related to valuation fans in the same way that real closed fields are related to orderings. Doing this we get elementary field theories with nice first order axiomatizations and finally study these theories.

Montag, 25.06.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Katharina Dupont (Konstanz)

Definable Valuations and Dependent Fields

Abstract: For my PhD project I am working on definable valuations on dependent fields.
Our aim is to find necessary and sufficient (algebraic or model theoretic) conditions for dependent fields to admit a definable valuation.
In the first part of my talk I will give an introduction to dependent fields. I will give a definition and several theorems which help us to decide whether a given field is dependent. I will talk about the main ideas of how these theorems are used to show that finite fields, algebraically closed fields, real closed fields, Q_p for any prime p and almost real closed fields are dependent fields. I will as well shortly talk about definable valuations on these fields.
In the second part of my talk I will present a promising method which uses definable subgroups to find definable valuations on fields.

 

Montag, 02.07.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Assaf Hasson (Ben-Gurion University of the Negev, Israel)

(Gast von Salma Kuhlmann)

Geometric structures interpretable in o-minimal theories

Abstract: In the 70s of the 20th century Zilber conjectured that if the geometry of a strongly minimal set is not locally modular then the geometric complexity of the structure must arise from an algebraic complexity, namely the existence of an infinite definable field. Though, Zilber's conjecture is far from being true it has been proved - and even generalised - in many special cases of "tame" geometric structures, with far reaching consequences. In 2004 Peterzil asked whether Zilber's conjecture must hold of geometric structures interpretable in o-minimal theories (the question is still open even for the special cases of structures interpretable in compact complex manifolds or even in algebraically closed fields of characteristic 0). This is, probably, the widest general context in which no counter examples are known.
In the talk - beside explaining the different notions needed for the discussion - I will give an overview of the state of the art regarding Peterzil's question, sketching the solution for the unstable case, and discussing the prospects with regard to the stable case.

Montag, 09.07.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Zoe Chatzidakis (CNRS - University Paris 7)

(Gast von Salma Kuhlmann)

Algebraic dynamics, difference fields and model theory

Abstract: (Joint work with E. Hrushovski).
M. Baker proved in 2007 that given a endomorphism f of P^1 of degree >1 and defined over a function field K, either all points of P^1(K) of canonical height 0 are preperiodic, or the endomorphism f descends to the field of constants of K. Szpiro and Tucker asked about generalising the result to higher dimensional varieties. A reformulation by Szpiro of the hypotheses in terms of limited sets allowed model theory to get into the picture, more precisely the model theory of difference fields. We were therefore able to give the correct formulation of the descent result in higher dimension, and prove it.
This is what I will explain in the talk: connections between algebraic dynamics and difference fields; translation of the problem; how model theory of difference fields intervenes.

Montag, 16.07.2012 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag