Die folgenden Vorträge haben im Sommersemester 2015 im Oberseminar Modelltheorie stattgefunden.
Montag 13.04.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Montag 20.04.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Samaria Montenegro (Université Paris Diderot - Paris 7)
(Gast von Salma Kuhlmann)
PRC and their stability theoretic properties
Abstract: The notion of PAC fields has been generalized by Basarab and by Prestel to ordered fields. Prestel calls a field M pseudo real closed (PRC) if M is existentially closed (in the language of rings) in every regular extension L to which all orderings of M extend. Equivalently, if every absolutely irreducible variety defined over M that has a rational point in every real closure of M, has an M -rational point.
In this talk we will study the class of bounded PRC fields from a model theoretical point of view. We work with the complete theory of a fixed bounded PRC field M in the language of rings expanded with enough constant symbols. The boundedness condition implies that M has only finitely many orders. Our main theorem is a positive answer to a conjecture of Chernikov, Kaplan and Simon that says: A PRC field is NTP2 if and only if it is bounded. This also allows us to explicitly compute the burden of types, and to describe forking. Some of these results generalize to bounded PpC fields, using the same kind of techniques.
Montag 27.04.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Arno Fehm (Universität Konstanz)
On the existential theory of equicharacteristic henselian valued
fields
Abstract: The first order theory of a henselian valued field of residue characteristic zero is well-understood through the celebrated Ax-Kochen-Ershov principle, which states that it is completely determined by the theory of the residue field and the theory of the value group. For henselian valued fields of positive residue characteristic, no such general principle is known. I will report on joint work with Will Anscombe in which we study (parts of) the theory of equicharacteristic henselian valued fields and prove an Ax-Kochen-Ershov principle for existential (and slightly more general) sentences. I will also discuss applications to the definability of henselian valuation rings and to the existential decidability (Hilbert's tenth problem) of the local field F_q((t)), which was proven by Denef and Schoutens
assuming resolution of singularities.
Montag 04.05.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Françoise Point (FNRS-FRS, Université de Mons)
(Gast von Salma Kuhlmann)
Transfer results in topological differential fields
Abstract: In the late seventies, M. Singer axiomatised the class of closed ordered differential fields. Following on one hand his approach and the one of M. Tressl on large differential fields, with N. Guzy, we gave a general scheme for axiomatising certain classes of topological differential fields. Here we will focus on the question of which properties transfer from the class of their existentially closed reducts (forgetting about the derivation), such as the NIP property, existence of "good" bounds for VC-density of definable sets, existence of a fibered dimension, density of definable types. We will give partial answers in particular in the case of closed ordered differential fields. (Part of this work is joint with Quentin Brouette.)
Montag 11.05.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Piotr Kowalski (Uniwersytet Wrocławski, Breslau)
(Gast von Pantelis Eleftheriou)
Strongly minimal reducts and Zilber's trichotomy
Abstract: This is joint work with Serge Randriambololona.
I will briefly recall Zilber's trichotomy conjecture about strongly minimal structures and its current status. Then I will focus on strongly minimal structures which are reducts of fields (with a possible extra structure). Time permitting, I will discuss the situation for algebraically closed valued fields.
Montag 18.05.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Haydar Göral (Université Claude Bernard Lyon 1)
(Gast von Panthelis Eleftheriou)
Mann Property and Definable Groups
Abstract: In this talk, we study the pair (K,G) where K is an algebraically closed field and G is an infinite multiplicative subgroup of K* with the Mann property. The main examples of this property come from number theory. In 1965, H. Mann showed that the set of complex roots of unity has the Mann Property. Later, it was proven that any multiplicative group of finite rank in any field of characteristic zero has the Mann property. The theory of the pair (K,G) is axiomatized by L. van den Dries and A. Günaydin, and they prove that the pair (K,G) is stable. We first characterize the independence in the pair and this allows us to characterize definable groups in (K,G) by applying the group con figuration theorem, and the tools used by T. Blossier and A. Martin-Pizarro for pairs of algebraically closed fields. It turns out that, up to isogeny, a definable group in (K,G) is an extension of a type-interpretable group in G by an algebraic group defined in K.
Montag 25.05.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Montag 01.06.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Florian Pop (University of Pennsylvania)
(Gast von Arno Fehm)
A minimalistic p-adic Analog of Artin-Schreier
Abstract: The talk is about a minimalistic p-adic analog of the famous Artin-Schreier Theorem on the Galois characterization of the real closed fields. There are already full Galois p-adic analogs of that theorem, but our aim in this talk is to use just a "minimalistic" Galois information, precisely, the Z/p metabelian Galois group of the fields. We notice that in the case of p-adically closed fields, the Z/p metabelian Galois group is finite (and has a well understood structure). I will explain in detail the terms and present some ideas about the (quite technical) proof.
Montag 08.06.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Montag 15.06.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Benjamin Druart (Université Claude Bernard Lyon 1)
(Gast von Gabriel Lehéricy)
Linear groups definable in P-minimal structures
Abstract: P-minimal structures were introduced by Haskell and Macpherson in 1997 following the example of o-minimality. The aim is to describe expansions of $Q_p$ such that definable sets are similar to semialgebraic sets. In this talk, we will show, that, roughly speaking, a linear commutative group definable in P-minimal structure is definably isomorphic to a semialgebraic group. One essential tool is the p-adic exponential function. We will also introduce a notion of p-connexity.
Montag 22.06.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Philipp Hieronymi (University of Illinois at Urbana-Champaign)
(Gast von Pantelis Eleftheriou)
A tame Cantor set
Abstract: Let R denote the real ordered field. Our focus here is on expansions of R by Cantor sets. For our purposes, a Cantor set is a non-empty, compact subset of the real line that has neither interior nor isolated points. We consider the following question due to Friedman, Kurdyka, Miller and Speissegger: is there a Cantor set K and a natural number N such that every set definable in (R,K) is Sigma_N^1? I will answer this question positively. In addition to using techniques from model theory, o-minimality and descriptive set theory and previous work of Friedman et al., the work presented in this talk depends crucially on well known results about the monadic second order theory of one successor due to Buechi, Landweber and McNaughton.
Montag 29.06.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Paola D'Aquino (Seconda Università degli Studi di Napoli)
(Gast von Salma Kuhlmann)
Roots of exponential polynomials
Abstract: I will present classical results on roots of exponential polynomials over the complex field into a more general setting of Zilber exponential fields.
Montag 06.07.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 10.07.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 13.07.2015 um 15:15 Uhr, Oberseminar Modelltheorie
Mickaël Matusinski (Université de Bordeaux)
(Gast von Salma Kuhlmann)
On the algebraicity of Puiseux series
Abstract: Joint work with M. Hickel (U. of Bordeaux).
Our goal is to understand what distinguishes an algebraic Puiseux series (over K(x) the one-variable rational function field) among formal Puiseux series. More precisely, we solve the following problems:
- given a bivariate polynomial equation P(x,y)=0, give a formula for the coefficients of a Puiseux series y(x) solution in terms of the coefficients of the equation ;
- given an algebraic Puiseux series, reconstruct a vanishing polynomial from its coefficients.
There are many contributions to these questions. I will report on them before describing our contributions.
