Die folgenden Vorträge haben im Sommersemester 2014 im Oberseminar Modelltheorie stattgefunden.

Montag, 28.04.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Sudesh Kaur Khanduja (Indian Institute of Science Education and Research, Mohali, India)

(Gast von Salma Kuhlmann)

Irreducible Polynomials

Abstract: The irreducible  polynomials have  a long history. In 1797, Gauss proved that the only irreducible polynomials with real coefficients are linear or quadratic polynomials. However, in view of Eisenstien Irreducibility criterion proved in 1850, for each number n\geq 1, there are infinitely many irreducible polynomials of degree n over rationals. We discuss some generalizations of this criterion discovered in recent years using the theory of valuations, which yield irreducibility criteria by Akira and Tverberg.

Montag, 05.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Arno Fehm (Konstanz)

Fields with almost small absolute Galois group

Abstract: Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly^2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. The main arguments are group theoretic, and most of the talk is accessible without knowledge of model theory. Joint work with Franziska Jahnke.

Montag, 12.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Montag, 19.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Montag, 26.05.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

Montag, 02.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Arthur Paul Pedersen (Center for Adaptive Rationality, Max Planck Institute for Human Development, Berlin)

(Gast von Salma Kuhlmann)

Numerical Representations of Ordered Vector Spaces, with Applications to the Foundations of Probability

Abstract.To motivate my interest in ordered (and more generally valued) vector spaces, I shall first discuss problems in the foundations of probability and expected utility which have prompted a number of authors to consider accounts of probability and expected utility admitting non-Archimedean representations. I shall thereupon introduce a normative theory of subjective probability and expected utility that rests upon qualitative criteria regulating preference judgments (or comparative judgments of expectation). The theory abandons the commitment to the technically convenient but rationally non-obligatory dogma of real-valued representability---the presumption that each agent in any given context is committed to a system of preference judgments (or judgments of probability and expectation) representable by a real-valued indicator. Like other normative theories (e.g., Savage’s theory of personal probability), the theory I advance presumes that an agent is committed to a system of *qualitative, comparative* preference judgments (or judgments of probability and expectation).

I shall then explain a key lemma supporting a full numerical representation of preference (or comparative expectations) in terms of subjective expected utilities formed from possibly non-Archimedean probabilities and utilities. Using Hahn's Embedding Theorem, I shall describe how a simple construction transforming (embedding) a Hahn lexical vector space into a "small" Hahn lexical field can be used to show that expected utilities take a very simple numerical form in terms of power series in a single infinitesimal with addition and multiplication naturally defined by means of the familiar operations of addition and multiplication of power series and with a natural lexicographic ordering. Other accounts of non-Archimedean probability or expectation are insufficiently general and philosophically inadequate. Finally, I shall discuss outstanding mathematical questions to be addressed during my visit.

If time permits, I shall explain a qualitative criterion of coherence reminiscent of de Finetti’s numerical criterion of coherence for his theory of probability (or prevision). The qualitative criterion of coherence is formulated for a truly arbitrary collection of gambles (random quantities), free of structural constraints. Furthermore, the criterion does not require an agent’s comparative judgments to be reflexive, complete, or even transitive. Despite these weak structural constraints, the qualitative criterion of coherence satisfies an analogue of de Finetti’s Fundamental Theorem of Prevision, ensuring that each coherent system of comparative judgments can be extended to a coherent weakly ordered system of comparative judgments over any space of gambles (random quantities). This result furnishes a basis for an account admitting a representation of uncertainty in terms of numerically indeterminate, and possibly non-Archimedean, probabilities and expected utilities in the style of, for example, Levi, Smith, Walley, and Williams.

Montag, 09.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Feiertag, kein Vortrag

Montag, 16.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Derya Çıray (Konstanz)

Counting rational points using mild parametrization

Abstract:This talk is about a method used to estimate the density of rational points (of bounded height) on subsets of the reals.

The Theorem of  Pila and Wilkie (2006) states that the transcendental part of a set X definable in an o minimal expansion of  the real ordered field, has a bound
$C(X,\epsilon)(H)^{\epsilon}$
on the number of its rational points with height smaller or equal to H. It is known that one can not get a better bound for all o-minimal expansions of reals, but Wilkie (2006) has conjectured that this bound can be improved to
$  C_1(X)(logH)^{C_{2}(X)} $
for sets definable in real exponential field.

A set has a mild parametrization, if it can be covered by a finite number of smooth functions with certain bounds on their derivatives (mild functions). I will  explain how mild parametrization is used to achieve  good bounds (of the kind that Wilkie conjectured) on the density of rational points for certain sets, in particular of Pfaffian curves. Wilkie's conjecture and how mild parametrization could be used to prove it, will also be discussed.

Montag 23.06.2014 um 15:15 Uhr, Oberseminar Modelltheorie

(Gast von Alexander Prestel)

Title: Externally definable sets in real closed fields.

Abstract: For a real closed field R, a subset of R^n is called externally definable if it is the intersection of R^n with a semi-algebraic subset of some S^n, where S is a real closed field containing R. For example, convex valuation rings of R are externally definable.
Over the real field, every externally definable set is semi-algebraic by a Theorem of Bröcker.
I will explain how this statement can be phrased for non-archimedean R, what it means geometrically and prove it for fields that are close to being maximally valued. As a by-product we get an algebraic characterization of definable types in real closed valued fields. This is joint work with Francoise Delon.

Dienstag, 24.06.2014 um 13.30 Uhr, Oberseminar Modelltheorie (Room: D406)

 Vagios Vlachos (University of Athens)

 (Gast von Pantelis E.Eleftheriou)

 Title:Prime Models of o-Minimal Theories

Abstract: Consider a first-order structure S=(M,< ,...) such that < is interpreted by a dense linear order on M. S is called o-minimal if every first-order definable subset of M is a finite union of singletons and intervals. A first-order theory T will be called o-minimal if every model of T is an o-minimal structure. The notion of o-minimality was first presented in the ‘80s (van den Dries, Pillay and Steinhorn) and since then has been studied thoroughly by many authors.

In this talk we will show the existence and uniqueness of prime models of o-minimal theories [PS86]. The recipe of the proof is the same with Shelah's analogous result for ω-stable theories which uses Ressayre's result for constructible models [Mar02].

References
[Mar02] David Marker. Model theory: an introduction. Grad. Texts in Math. 217. Springer, 2002.
[PS86] Anand Pillay and Charles Steinhorn. “Definable sets in ordered structures. I”. In: Trans. Amer. Math. Soc. 295.2 (1986), pp. 565–592.

Montag, 30.06.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Nadja Hempel (Université Lyon 1, France)

(Gast von Cédric Milliet)

"Artin-Schreier extensions in generalizations of NIP theories"

Abstract:

The study of algebraic extensions of fields defined in a first order
theory has been of special interest. It is known that NIP fields of
positive characteristic are Artin-Schreier closed. Chernikov, Kaplan
and Simon showed that any field of positive characteristic defined in
an NTP_2 theory admits only finitely many Artin-Schreier extensions.
I will introduce the notion of NIP_n theories due to Shelah and extend
the result of NIP fields to this wider class. Using this, I will also
present some applications to valued fields defined in this setting and
show that non separable closed pseudo-algebraically closed (PAC)
fields lie outside of the hierachie of NIP_n theories, a result due to
Duvet in the NIP context.

Montag, 07.07.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

Montag, 14.07.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

Montag, 21.07.2014 um 15.15 Uhr, Oberseminar Modelltheorie

Katharina Dupont (Konstanz)