Die folgenden Vorträge haben im Wintersemester 2014/15 im Oberseminar Modelltheorie stattgefunden.
Montag, 20.10.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Montag, 27.10.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Montag, 03.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Moshe Jarden (Universität Tel Aviv)
(Gast von Arno Fehm)
Starker Approximationssatz für absolut integre Varietäten über galoisschen PSC-Erweiterungen vor globalen Körpern
Abstract: Seien K ein globaler Körper, V eine unendliche eigentliche Untermenge der Menge aller Primdivisoren von K, S eine endliche Untermenge von V und K~ (bzW.K_s) ein fester algebraischer (bzW.separabler) Abschluss von K. Sei Gal(K)=Gal(K_s/K) die absolute galoissche Gruppe von K. Für jedes p in V$ wählen wir einen henselschen Abschluss (bzW, einen reallen oder algebraischen Abschluss) K_p von K bei p, falls p non-archimedisch (bz W archimedisch). Dann ist K_{tot,S}=\inter_{p in S}\inter_{\tau in Gal(K)}K_p^\tau die maximale galoissche Erweiterung von K innerhalb K_s, wo alle p in S total zerfallen. Ist p nonarchimedisch, bezeichnen wir mit O_{K_p,p} den Bewertungsring von K_p und mit O_{K~,p} den ganzen Abschluss von O_{K_p,p} in K_s. Für \sig=(\sig_1,...,\sig_e) in Gal(K)^e sei K_{tot,S} [\sig] die maximale galoissche Erweiterung von K innerhalb K_{tot,S}, der von \sig_1,...,\sig_e festgehalten ist. Dann, für fast alle sig in Gal(K)^e (in bezug auf den haarschen Massese), genügt der Körper M=K_{tot,S}[\sig] den folgenden starken Approximationssatz: Sei V eine affine absolut integre Varietät über K und sei S C T eine endliche Untermenge von V, so dass die Menge V\T nur aus nonarchimedischen Primdivisoren besteht. Für jedes p in S sei \Ome_p eine nicht leere p-offene Untermenge von V_{\simp}(K_p), für jedes p in T\S sei \Ome_p eine nicht leere \Gal(K_p)-invariante p-offene Untermenge von V(K_s). Wir setzen vorauss dass V(O_{K_s,p})<>emptyset für jedes p in V\T ist. Dann, existiert z in V(M), so dass z^{\tau} in\Omega_p für alle p in T und \tau in\Gal(K) und |z^\tau|_{p}
Montag, 10.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Gabriel Lehericy (Universität Konstanz)
Model-theoretic approach to a Galois theory for fields endowed with an operator (Teil I)
Abstract: The purpose of this talk is to introduce the notion of internality between definable sets inside arbitrary structures. These general results will then be applied to the particular cases of differential and difference fields in the context of Galois theory.
Let $\langue$ be a first-order language, $T$ an $\langue$-theory with elimination of imaginaries,
$\model$ a model of $T$ and $Q,C$ two definable subsets of some power of $M$.
If there exists a definable set $X$ and a definable map $f$ from $Q\times X$ to some power of $C$, we say that $Q$ is $(X,f)$-internal to $C$. If $Q$ is $(X,f)$-internal to $C$, and if $\Delta$ is an arbitrary set of formulas in the sorts $(Q,C,X)$, we are particularly interested in the set $Aut_{\Delta}(Q,X,C/C)$ of bijections of the sorts $Q,X,C$ fixing $C$ pointwise and preserving all formulas of $\Delta$.
We shall see that every map of $Aut_{\{f\}}(Q,X,C/C)$ can be obtained by composing maps of the form $f(.,x)$ for some $x\in X$; this will allow us to show that if $f\in\Delta$ then $Aut_{\Delta}(Q,X,C/C)$ is a type-definable group contained in some power of $M$ and we will give explicit formulas to define this group.
These abstract results can be applied to the Galois theory of difference fields. Consider a difference field $(k,\sigma)$ and and equation $AY=Y$ where $A\in GL_n(k)$. The set of solutions of this equation generates an extension $(K,\Sigma)$ of $(k,\sigma)$ and we would like to study the algerbaic structure of the Galois group of this extension.
Set $Q:=\{\text{solutions of the equation}\}$, $C:=\{x\in M\mid \Sigma(x)=x\}$; it is easy to find $(X,f)$ such that $Q$ is $(X,f)$-internal to $C$.
The Galois group associated to the equation can be identified with the set $Aut_{\Delta}(Q,X,C/C)$, where $\Delta:=\{\text{quantifier-free formulas in the sorts $Q,C,X$ }\}$. Since the theory $ACFA$ of algebraically closed fields with automorphisms eliminates imaginaries, we
can apply our previous results by embedding $(K,\Sigma)$ into a model of $ACFA$, which will enable us to give a precise description of the Galois group.
Montag, 17.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Gabriel Lehericy (Universität Konstanz)
Model-theoretic approach to a Galois theory for fields endowed with an operator (Teil II)
Abstract: (siehe oben)
Montag, 24.11.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Salma Kuhlmann (Universität Konstanz)
Real Closed Fields and Models of Peano Arithmetic
Abstract: We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.
References:
[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed elds , to appear in the Journal of Symbolic Logic, arXiv: 1212.6842
[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed elds and
fragments of Peano Arithmetic, arXiv: 1205.2254 (2012)
[3] Conversano, A. - D'Aquino, P. - Kuhlmann, S : -Saturated o-minimal expansions of real closed elds, arXiv: 1112.4078 (2012)
[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
Montag, 01.12.2014 um 13.30 Uhr, Oberseminar Modelltheorie
Pantelis E. Eleftheriou (Universität Konstanz)
Around definable compactness in weakly o-minimal structures
Abstract: We present results stemming from Gil Keren's master thesis on definable compactness for weakly o-minimal structures. We argue that the usual notion of definable compactness from o-minimal structures is not meaningful in a weakly o-minimal structure M, unless M is o-minimal. A byproduct of the argument is the construction of new weakly o-minimal structures that do not have definable choice, extending results from Chris Shaw's PhD thesis "Weakly o-minimal structrues and Skolem functions, University of Maryland, 2008".
Montag, 08.12.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Montag, 15.12.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Daniel Palacin (Universität Münster)
(Gast von Pantelis Eleftheriou)
A Fitting theorem for groups in simple theories
Abstract: A certain amount of model-theoretic ideas for groups in the stable context can be adapted to the more general framework of simple theories. For instance, groups defined in this context satisfy a chain condition on centralizers (up to finite index). In this talk we present some of the main tools and notions of groups in simple theories. Our goal is to show that in a group type-definable in a simple theory, the Fitting subgroup, i.e. the group generated by all normal nilpotent subgroups, is itself nilpotent.
Montag, 22.12.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Im Zeitraum vom 23.12.2014 bis zum 08.01.2015 fanden auf Grund der allgemeinen Betriebsschließung der Universität Konstanz keine Vorträge in den Oberseminaren statt.
Montag, 12.01.2015 um 15.15 Uhr, Oberseminar Modelltheorie
David Masser (Universität Basel)
(Gast von Margaret Thomas)
Zero estimates with moving targets
Abstract: Zero estimates have a long classical history in diophantine approximation and transcendence, as well as more recent applications to counting rational points on analytic varieties. We give some examples showing that the sharpest conceivable results can be false, and in some cases the natural guess has even to be doubled. This is joint work with Dale Brownawell. A by-product is a special case of the (unproved) "Zilber Nullstellensatz".
Montag, 19.01.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Merlin Carl (Universität Konstanz)
Structures Associated with Real Closed Fields and the Axiom of Choice
Abstract: An integer part I of a real closed field K is a discretely ordered subring of K such that every element of K lies between two consecutive elements of I. Mourgues and Ressayre showed that every real closed field has an integer part. Their construction implicitly uses the axiom of choice.
We show that AC is actually necessary to obtain the result by constructing a transitive model of ZF which contains a real closed field without an integer part.
On the way, we demonstrate that a class of questions containing the question whether the axiom of choice is necessary for the proof of a certain ZFC-theorem is algorithmically undecidable. We further apply the methods to show that it is independent of ZF whether every real closed field has a value group section and a residue field section.
This also sheds some light on the possibility to effectivize constructions of integer parts and value group sections which was considered by D'Aquino, Kuhlmann, Knight and Lange.
Montag, 26.01.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Paola D'Aquino (Seconda Università degli Studi di Napoli)
(Gast von Salma Kuhlmann)
Exponential polynomials
Abstract: I will present some recent work on the solution sets of
certain exponential polynomials.
Montag, 02.02.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Mario J. Edmundo (Universidade Aberta, Lissabon)
(Gast von Pantelis Eleftheriou)
On the o-minimal Hilbert's fifth problem
Abstract: The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimal sheaves, which might be useful on it own.
(joint with: M. Mamino, L. Prelli, J. Ramakrishnan and G. Terzo)
Montag, 09.02.2015 um 15.15 Uhr, Oberseminar Modelltheorie
Serge Randriambololona (Galatasaray Üniversitesi, Istanbul)
(Gast von Salma Kuhlmann)
Trichotomy in strongly minimal additive reducts of ACVF_0
Abstract: Zilber's conjecture of trichotomy asserts that the geometry (in the sense of the algebraic closure operator) of a strongly minimal structure 1) is either trivial or constrains the structure to essentially be either 2) a vector space or 3) an algebraic closed field. Though the conjecture was proven false in its general form, the trichotomy principle holds true in many particular instances. After having stated the conjecture in detail (and introduced the vocabulary required for its understanding), I will give an overview of the history of the results related to it and discuss the particular case of additive reducts of ACVF_0, as studied in a joint work with P. Kowalski (Univ. Wrocław).
