Die folgenden Vorträge haben im Sommersemester 2013 im Oberseminar reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium reelle Geometrie und Algebra stattgefunden.

 

Montag, 15.04.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Freitag, 19.04.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 22.04.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Freitag, 26.04.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Victor Vinnikov (Ben Gurion University, Israel)

(Gast von Markus Schweighofer)

Determinantal representations of stable and real zero polynomials

Abstract: I will discuss determinantal representations of polynomials that witness stability and symmetry with respect to the polydisc in C^d. I will outline existence proofs for the case d=2, and show how a similar approach leads to a fairly constructive existence proof for positive self-adjoint determinantal representations of real-zero polynomials in two variables.

Montag, 29.04.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Freitag, 03.05.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 06.05.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Artem Chernikov (Hebrew University of Jerusalem)

(Gast von Salma Kuhlmann)

Fields with NTP2

Abstract: NTP2 is a large class of first-order theories introduced by Shelah and generalising both simple and NIP theories. It provides a natural setting for the theory of forking beyond simplicity, and also contains new important algebraic examples: any ultraproduct of p-adics viewed as a valued field is NTP2, as well as certain valued difference fields. But what can be said about groups and fields definable in NTP2 structures? One result is that every field definable in an NTP2 structure has only finitely many Artin-Schreier extensions. In this talk I will give a survey of the area and my recent work with Martin Hils, Itay Kaplan and Pierre Simon.

 

Freitag, 10.05.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

Montag, 13.05.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Krzysztof Krupinski (Uniwersytetu Wrocławskiego, Poland)

(Gast von Salma Kuhlmann)

On superrosy fields

Abstract: There is a long history of results describing the structure of fields under various model-theoretic assumptions. One of the fundamental theorems says that each superstable field is algebraically closed. There are two important conjectures generalizing this theorem: the first one predicts that each supersimple field is pseudo algebraically closed, the second one says that each superrosy field with NIP is either algebraically or real closed. I will discuss certain partial results around the second conjecture. In particular, I will formulate a weaker conjecture that whenever we have a non-trivial valuation on a superrosy field with NIP, then the value group is divisible and the residue field is either algebraically or real closed. I will discuss how to prove this conjecture in the positive characteristic case and what is left for the zero characteristic case.

Freitag, 17.05.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Derya Çıray (Istanbul Bilgi University)

(Gast von Margaret Thomas)

A model-theoretic proof of the Real Nullstellensatz (after Dubois and Riesler)

Abstract: Real Nullstellensatz is an analogue of Hilbert’s Nullstellensatz for real closed fields. They both relate algebraic sets to ideals.
I will present the model theoretic proof of Real Nullstellensatz using model completeness of the Theory of Real Closed Fields (RCF), given by Dubois and Riesler.

Montag, 20.05.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Feiertag, kein Vortrag

Freitag, 24.05.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 27.05.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Freitag, 31.05.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

Montag, 03.06.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Pantelis E. Eleftheriou (University of Waterloo, Canada)

(Gast von Salma Kuhlmann)

Pregeometries and definable groups

Abstract: We describe a recent program for analyzing definable sets and groups in certain model theoretic settings. Those settings include:
(a) o-minimal structures (M, P), where M is an ordered group and P is a real closed field defined on a bounded interval (joint work with Peterzil),
(b) tame expansions (M, P) of a real closed field M by a predicate P, such as expansions with o-minimal open core (work in progress with Gunaydin).
The analysis first goes through a local level, where a pertinent notion of a pregeometry and generic elements is each time introduced.

Freitag, 07.06.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 10.06.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Freitag, 14.06.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

Montag, 17.06.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Will Anscombe (University of Oxford)

(Gast von Salma Kuhlmann)

Definability in fields of power series via an improved Hensel-like Lemma

http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ss-2013-abstracts/#1706Abstract: We detail our Hensel-like Lemma in its original form, its application to the study of definability in power series fields, and then give an indication of present efforts to improve the lemma to reveal more detail about existentially definable sets.

Freitag, 21.06.2013 um 13.30 Uhr (im Raum C427), Oberseminar Reelle Geometrie und Algebra

Tim Netzer (Universität Leipzig)

(Gast von Markus Schweighofer)

How fast do polynomials grow on semialgebraic sets?

Abstract: While the growth of a polynomial function on the whole affine real space is determined by its total degree, the same is obviously not true when restricting to subsets of that space. For example, the function xy^n is of total degree n+1, but grows linearly on a horizontal strip in the plane. On the other hand, modulo bounded polynomials, this phenomenon usually disappears. In the context of moment problems, having a connection between growth and total degree (modulo bounded polynomials) is highly desirable. It has thus been asked whether such connections always exist. To examine these problems, we construct a graded algebra to a given semi-algebraic set, and study all kinds of questions about finite generation. We can then produce examples in dimension two, for which there is no connection between total degree and growth. From the results we also get new examples in dimension three, for which the algebra of bounded polynomials is not finitely generated.

Montag, 24.06.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Nikesh Solanki (Manchester)

(Gast von Arno Fehm)

Uniform Companions For Differential Field Expansions in Characteristic 0

http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ss-2013-abstracts/#2406Abstract: In 2005 M. Tressl presented a theory UC of differential fields in several commuting derivatives of characteristic zero, which serves as a model companion for every theory of differential fields extending a model complete theory in the language of rings whose models are large as pure fields. We call UC a "uniform companion" of theories of large fields in the language rings. In this talk I will show that methods of M. Tressl can be generalised so as to obtain uniform companions for theories in any relational expansion of the language of rings whose models obey a generalised notion of largeness for relational field expansions. This generalised notion of largeness also takes into account the relations satisfied by the rational points, which I shall of course explain explicitly. I will also present some interesting examples where these new uniform companion can be applied to obtain new model companions, in particular, to the case of fields with pseudovaluation rings.

Freitag, 28.06.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Quentin Brouette (Universite de Mons, Belgique)

(Gast von Salma Kuhlmann)

Stellensätze in closed ordered differential fields

Abstract: We will introduce Singer's model completion of ordered differential fields called CODF (for closed ordered differential fields) and give an overview of its most noticeable properties. We will investigate the real radical of differential ideals of the ring of differential polynomials K{X_1,..., X_n} where K is an ordered differential field. Then we will prove a Nullstellensatz for models of CODF.

Dienstag, 02.07.2013 um 13.30 Uhr, Oberseminar Modelltheorie

Franziska Jahnke (Universität Münster)

(Gast von Arno Fehm)

Definable henselian valuations

Abstract: A non-trivial henselian valuation on a field K is often so intrinsic to K that it is already definable in the language of rings. However, by the work of Prestel and Ziegler, there are henselian valued fields which are neither separably nor real closed and which do not admit any non-trivial 0-definable valuation. In this talk, we will give a range of criteria which ensure the existence of a 0-definable henselian valuation on a henselian field. We will discuss work towards a complete classification of those fields in the residue characteristic 0 case.

Freitag, 05.07.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Aurélien Greuet (Université de Versailles-Saint-Quentin)

(Gast von Markus Schweighofer)

Global algebraic optimization

Abstract: We consider the global algebraic optimization problem: given polynomials f, f_1,...,f_p with rational coefficients and n variables, the goal is to compute f^*, the infimum of f subject to the constraints f_1=...=f_p=0.

This problem appears naturally in several areas of engineering science, where it is relevant to have efficient and reliable algorithms.

In this talk, we assume that the constraints f_1,...,f_p define a radical ideal and a smooth and equidimensional variety. Then tools coming from real algebraic geometry can be used. In particular, by adapting the notion of polar variety, we are able to reduce the problem to an optimization problem on a variety of small dimension. Thanks to the dimension reduction, asymptotic phenomena can be managed.

We present two applications in optimization. The first one uses results coming from real algebra to prove the existence of sums of squares certificates of positivity. Using semidefinite programming, this leads to the computation of certified lower bounds on f^*. The second one is a symbolic algorithm computing f^* and testing whether it is attained. It is singly exponential in the number of variables. The implementation is the first one with this complexity and is efficient in practice: it can solve instances that are out of reach of previously known algorithms.

These results are joint work with Mohab Safey El Din and with Feng Guo, Mohab Safey El Din and Lihong Zhi.

Montag, 08.07.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Katharina Dupont (Konstanz)

Applying Keisler Measures on the Axioms of Valuation Topologies

Abstract: For my PhD project I am working on the question "Under which conditions does a dependent field admit a non-trivial definable valuation?".

In his preprint "Definable Valuations" Koenigsmann defines for a subgroup G of a field K the valuation ring O_G.

Assume q ≠ char(K) prime such that G:=(K\{0})^q ≠ K\{0}. Then O_G is (topologically equivalent to) a non-trivial definable valuation ring if and only if the set B_G:={\bigcap_{i=1}^n a_i(G+1) | n ∈ \N, a_i ∈ K\{0}} fulfills the six axioms (V 1)-(V 6) of V-topologies, i.e. B_G is a basis of zero neighbourhoods of a topology induced by an absolute value or a valuation.

Recently we have been able to approach these axioms (for a dependent field K satisfying [K\{0}:(K\{0})^q]

Donnerstag, 11.07.2013 um 17.00 Uhr, Oberseminar Reelle Geometrie und Algebra

José Rodriguez (University of California, Berkeley)

(Gast von Aaron Kunert)

Numerical Algebraic Geometry in Algebraic Statistics

Abstract: Maximum likelihood estimation is a fundamental computational task in statistics and involves beautiful geometry. We discuss this task for determinantal varieties (matrices with rank constraints) and show how numerical algebraic geometry can be used to maximize the likelihood function. Our computational results with the software Bertini led to surprising conjectures and duality theorems. This is joint work with Jan Draisma, Jon Hauenstein, and Bernd Sturmfels.

 

Freitag, 12.07.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

María López Quijorna (Konstanz)

The truncated GNS construction

Abstract: The moment problem asks for the characterization of the linear forms L ∈ \R[x]^*_2t, having a representing measure. In this presentation we start with a given linear form, and with the new point of view of the truncated GNS operators, we will try to decide if such a linear form comes from a measure, and in this case we will try to find the support of such a measure. We will see a few examples where the linear form is not necessarily flat and we can recover the support of such a measure restricted to some space. We will also see examples where we can use the GNS construction to deal with polynomial optimization problems.

 

Montag, 15.07.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Quentin Brouette (Universite de Mons, Belgique)

(Gast von Salma Kuhlmann)

Abstract: We will consider formally real differential fields extensions and their differential Galois groups (that is the group of automorphisms of L in the language {+,-,.,^{-1},',0,1} and fixing K).

Freitag, 19.07.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag