Die folgenden Vorträge haben im Wintersemester 2011/12 im Oberseminar Modelltheorie stattgefunden.
Montag, 17.10.2011 um 16.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Montag, 24.10.2011 um 16.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Montag, 31.10.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Immanuel Halupczok (Münster)
(Gast von Arno Fehm)
Whitney stratifications in valued fields
Abstract: In geometry over ℝ or ℂ, Whitney stratifications are a very useful toolto describe singularities of (semi-)algebraic sets. I will present avariant of Whitney stratifications in Henselian valued fields withresidue field of characteristic zero. This is not just a straightforward translation; however, using methods from non-standard analysis,we will see that the existence of our valued-field stratificationsactually implies the existence of the classical Whitneystratifications.
The talk will not require any prior knowledge of Whitney stratificationsor non-standard analysis.
Montag, 07.11.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Lorna Gregory (Konstanz)
Are Ziegler Spectra always sober?
Abstract: The (right) Ziegler spectrum, Zg_R, of a ring R is a topological space
attached to the module category of R. The aim of this talk is to present two equivalent definitions of the Ziegler spectrum of a ring and to motivate the question "Are Ziegler spectra always sober?". The first (and original) definition is model theoretic and the second is in terms of the category of additive functors from the category of finitely presented left modules to abelian groups. We will see that, from the point of view of functor categories, Zg_R is analogous to the Hochster dual of a spectrum of a ring, where the ring is replaced by the category of finitely presented modules.
I will begin this talk with an introduction to the model theory of modules. There will be a follow on seminar where I specialise to the Ziegler spectra of valuation domains and show that in this case Zg_R is indeed sober.
Montag, 14.11.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Lorna Gregory (Konstanz)
Are Ziegler Spectra always sober? II
Abstract: This talk will be a continuation of my talk on 7.11.2011. I will define the left and right Ziegler spectra of a ring and explain why soberness of both the left and right Ziegler spectra would imply that the right and left spaces are in fact homeomorphic (after identifying topologically indistinguishable points).
Montag, 21.11.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Salma Kuhlmann (Konstanz)
Primes and Irreducibles in Truncation Integer Parts
Abstract: An integer part (IP) Z of an ordered field F is a discretely ordered subring, with 1 as least positive element, and such that for every x ∈ F, there is a z ∈ Z such that z ≤ x < z+1. Shepherdson (1964) showed that IP's of real closed fields are exactly the models of a fragment of Peano Arithmetic called Open Induction. This motivated very active research in studying the arithmetic properties of these rings. For a field of generalized power series ℝ((G)) (with exponents in an arbitrary ordered abelian group G ≠ 0), a canonical IP is the ring ℝ((G<0))⊕ℤ. In this talk, I will discuss its primes and irreducibles, and more generally those of the IP's of the special class of truncation closed subfields of ℝ((G)).
Montag, 28.11.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Johanna Harde (Konstanz)
Über die Wertegruppe einer differentiellen Bewertung
Abstract: Zwei Ansätze die Theorie der Hardykörper zu verallgemeinern sind durch die differentiell bewerteten Körper nach M. Rosenlicht und durch die spezielleren H-Körper nach M. Aschenbrenner und L. v. d. Dries gegeben. Zu diesen Körpern sind jeweils sogenannte asymptotische Paare assoziiert. Umgekehrt lassen sich unter bestimmten Bedingungen aus asymptotischen Paaren zugehörige differentiell bewertete Körper beziehungsweise H-Körper gewinnen. Bei dieser Konstruktion gehen Rosenlicht und Aschenbrenner/ v.d. Dries auf unterschiedliche Weise vor. Ich möchte die beiden verschiedenen Konstruktionsweisen vergleichen und Gemeinsamkeiten aufdecken.
Montag, 05.12.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Merlin Carl (Konstanz)
A new fine structure for simplifying applications of constructibility
Abstract: ZFC, Zermelo-Fraenkel set theory with choice, is now widely accepted as “the” formal framework for common mathematics. Independence from ZFC hence means undecidability on the basis of the current view of mathematics. One way to obtain such results is the use of inner models, definable sub-classes of the set-theoretical universe V. The subset-smallest such model is Gödel’s constructible universe L. due to its very concrete nature, it can be analyzed very precisely, leading to strong combinatorial principles in L that are hence known to be consistent with ZFC. However, the classical apparatus for such results, Jensen’s fine structure theory, is technically rather complex already in the context of L, and even more so when one considers generalizations, in particular core models. Therefore, there have been several attempts do develop simplified fine structures for L and its relatives. I will give an introduction into constructibility theory in general and then proceed to present one of these approaches to a simplified fine structure, the F-hierarchy, which was invented and exhibited by van Eijmeren, Koepke and myself.
Montag, 12.12.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Merlin Carl (Konstanz)
A new fine structure for simplifying applications of constructibility II
Abstract: The constructible universe L is arguably the most concrete model of set theory. Its homogenous structure allows proofs of many conjecture undecidable on the basis of ZFC. The definition of the model, along with some of its most crucial properties was given in the first part of my talk. In the second part, we will study the question of how similar L is to the complete set-theoretical universe V. It turns out that there is a strong dichotomy, in that L is either very similar or very different from V, which depends on the existence of an elementary embedding from L to itself. I will introduce the necessary concepts to make the intuition of a similarity between V and L precise, describe the tools for simplifying the analysis of such questions I studied in my thesis and sketch the proof of a variant of a theorem of Magidor: If there is no elementary embedding from L to itself, then every set of ordinals closed under the basic constructible operations is a union of at most countably many sets in L.
Montag, 19.12.2011 um 16.15 Uhr, Oberseminar Modelltheorie
Marcus Tressl (Manchester)
(Gast von Salma Kuhlmann)
Real and p-adically closed rings
Abstract: I will talk about generalisations of the notions of real closed fields and p-adically closed fields to commutative rings. Real closed rings were introduced by Niels Schwartz in the early 1980's in the context of so-called abstract semi-algebraic functions, as rings of global sections of a certain sheaf on the real spectrum of a ring (this will all be explained in the talk). Examples include: real closed fields, convex valuation rings of real closed fields and rings of real valued continuous functions on a topological space. In the 1990's there were attempts to transfer the real theory to the p-adics, but a direct approach to define abstract p-adic spaces has failed so far. In the talk I will introduce (from scratch) real closed rings in a model theoretic framework that can be literally copied to the p-adic case. In particular, p-adically closed fields and rings of p-adic valued continuous functions on a topological space are p-adically closed rings. I will discuss examples and properties of these rings, most importantly - as in the case of (ordered/p-valued) fields - every ring has a real closure and a p-adic closure, uniquely determined up to isomorphism.
Montag, 09.01.2012 um 16.15 Uhr, Oberseminar Modelltheorie
kein Vortrag, sonst am Freitag, 13.01.2012 um 14.15 Uhr.
Arno Fehm (Konstanz)
Decidability of fields of algebraic numbers
Abstract: I will talk about decidability results for theories of large algebraic fields, for example certain fields of totally real algebraic numbers. The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out connections with the work of Ax on the theory of finite fields.
Montag, 16.01.2012 um 16.15 Uhr, Oberseminar Modelltheorie
Margaret Thomas (Konstanz)
The density of rational points on certain Pfaffian sets
Abstract: This talk will be an overview of a number of results and methods used in the research program of bounding the density of rational and algebraic points lying on sets definable in o-minimal expansions of the real field. We will review the historical background to this question from transcendental number theory, including Pila and Wilkie's influential work in this area. Following on from this, Wilkie has conjectured an improvement to their main result for sets definable in the real exponential field. We shall outline some results in this direction, including the proven one-dimensional case of the conjecture and some partial results for certain surfaces.
These results are joint work with Gareth O. Jones.
Montag, 23.01.2012 um 16.15 Uhr, Oberseminar Modelltheorie
Prof. Dr. Paola D'Aquino (Seconda Universitá di Napoli)
(Gast von Salma Kuhlmann)
Real closed fields and models of Peano Arithmetic
Abstract: Shepherdson proved that a discretely ordered ring is a model of IOpen iff it is an integer part of a real closed field. I will present a joint result with J. Knight and S. Starchenko where we characterize the real closed fields which have an integer part which is a model of full Peano Arithmetic.
Montag, 30.01.2012 um 16.15 Uhr, Oberseminar Modelltheorie
Prof. Dr. Paola D'Aquino (Seconda Universitá di Napoli)
(Gast von Salma Kuhlmann)
Real closed fields and models of Peano Arithmetic, part II
Abstract: I will continue to talk on the result jointly obtained with J. Knight and S. Starchenko.
Montag, 06.02.2012 um 16.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
