Vorträge inklusive Abstracts im Sommersemester 2015
Die folgenden Vorträge haben im Sommersemester 2015 im Oberseminar Reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium Reelle Geometrie und Algebra stattgefunden.
Montag 13.04.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 17.04.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Salma Kuhlmann (Universität Konstanz)
Quasi-Orderings: a uniform approach to orderings and valuations
Abstract: In the note "Quasi-Ordered Fields" by S. M. Fakhruddin [JPAA 45 (1987) 207-210] the author introduces the notion of a quasi-ordered field and shows that such a field is either an ordered field or a Krull valued field. In this talk we take this approach further to exhibit a theory of (quasi-order) convex valuations.
Classical results on (order) convex valuations such as: the characterization of order convexity via the residue field, the Theorem of Baer Krull, etc, can be reformulated for quasi order convex valuations in a natural way.
In particular, this provides an elegant and uniform treatment of lifting of orderings, coarsening and composition of valuations.
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.
Freitag 24.04.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Grey Violet (Zukunftskolleg, Universität Konstanz)
Polynomials, control, parametric stability
Abstract: Stability problems for parametric families of polynomials and matrices are among the oldest in control, being examined in lots of different settings. Despite that, mathematical structure of that kind of problems are far from being well understood even in for the simplest important cases such as PI or PID-controllers.
In that talk I am going to start from a classical geometric approach of $D$-decomposition of the parameter space going back to works of Yu.I. Neimark and D.Mitrovic and then developed by B.Polyak, E.Gryazina et al., proceed with an applications of real algebraic geometry to that problem due to author of the talk, and then discuss a work-in-progress devoted to a building an unified algebro-geometric approach to different kinds of stability theories based on study of symmetric powers of semialgebraic sets, their affine sections and stratifications of the space of polynomials.
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.
Freitag 01.05.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
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.)
Freitag 08.05.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Daniel Dadush (Centrum Wiskunde & Informatica, Amsterdam)
(Gast von Markus Schweighofer)
On the Shadow Simplex Method for Curved Polyhedra
Abstract: Linear Programming (LP), which captures continuous optimization problems with linear constraints and objectives, is one of the main modelling languages used within Operations Research. The most popular technique for solving LPs in practice is the simplex method, which prescribes a rule for moving from solution to solution along the boundary of the feasible region until a global optima is reached. Despite its practical success, our theoretical understanding of simplex style algorithms remains relatively limited: the best worst case runtime achieved by a simplex algorithm (random facet rule) is subexponential in the number of variables, and polynomial runtimes are only known for random or restricted families of LPs.
In this work, we study the performance of the simplex method for a class of LPs whose feasible regions are suitably "curved", roughly speaking where the constraints of the feasible region meet at "sharp" angles. This class vastly generalizes that of totally unimodular LPs, which model many important combinatorial problems such as network flow. For our main results, we give a new variant and analysis of the shadow simplex algorithm -- which follows a path over solutions induced by the boundary of a 2 dimensional projection (the so-called "shadow") of the feasible region -- that achieves fast polynomial runtimes for "curved" LPs and substantially improves upon, both in terms of simplicity and efficiency, all previous works for this class.
Joint with Nicolai Hahnle (Bonn University)
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.
Freitag 15.05.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
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.
Freitag 22.05.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Christoph Schulze (Universität Leipzig)
(Gast von Claus Scheiderer)
Homogeneous versions of Schmüdgen's and Putinar's theorems
Abstract: Schmüdgen's and Putinar's theorems are important results of Real Algebraic Geometry. I will present a homogeneous version of Schmüdgen's theorem and a projective version of Putinar's theorem.
First I will show an inductive property which is based on the Stone-Weierstraß theorem. This will lead us, together with Polya's theorem, to Putinar's theorem and a first projective version of Putinar's theorem. A homogeneous version of Stengle's and Krivine's Positivstellensatz of Zeng provides the basis for a homogeneous version of Schmüdgen's theorem. Another application of the inductive property (Stone-Weierstraß theorem) yields to a more general version of the projective version of Putinar's theorem.
Montag 25.05.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Donnerstag 28.05.2015 um 17:00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Jaka Cimpric (Univerza v Ljubljani)
(Gast von Markus Schweighofer)
Archimedian Classes of Matrices over ordered Fields
Freitag 29.05.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Aljaž Zalar (Univerza v Ljubljani)
(Gast von Markus Schweighofer)
Matrix Fejéer-Riesz theorem with gaps
Abstract: Two equivalent versions of the matrix Fejér-Riesz theorem characterize positive semidefinite matrix polynomials on the complex unit circle T and on the real line R. We extend the characterization to arbitrary closed basic semialgebraic sets K\subset T and K\subset\R by the use of matrix preorderings from real algebraic geometry. In the T-case the characterization is the same for all sets K, while in the R-case the characterizations for compact and non-compact sets K are different. Furthermore, we study a complexity of the characterizations in terms of a bound on the degrees of the summands needed. We prove, for which sets K, K the degrees can be bounded by the degree of the given matrix polynomial and provide counterexamples for the sets, where this is not possible. At the end we give an application of results to a matrix moment problem.
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.
Freitag 05.06.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 08.06.2015 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Donnerstag 11.06.2015 um 17:15 Uhr, Allgemeines Mathematisches Kolloquium
Frank Vallentin (Universität zu Köln)
(Gast von Markus Schweighofer)
Packings in two, three and more dimensions
Abstract: How densely can one pack a given body into a given container? This is a fundamental question in discrete geometry with applications in mathematics, information theory, physics and materials science. The bodies to pack are often spheres or polytopes, the geometric container can be a compact (like the sphere), or a noncompact space (like Euclidean space). In this talk I will recall some history of geometric packing problems. Then I will introduce universal methods to solve these problems computationally but rigorously, using tools from convex optimization, real algebraic geometry, and harmonic analysis. I will report on progress on the kissing number problem, the Kepler problem and the packing of general convex bodies.
Freitag 12.06.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Antonio Lerario (Université Claude Bernard Lyon 1)
(Gast von Markus Schweighofer)
Complexity of intersections of real quadrics and topology of discriminant varieties
Abstract: In this talk I will focus on the problem of understanding the topology of an intersection X of real quadrics. I will introduce a new notion of geometric complexity (inspired to fewnomials and related), using the discriminant in the space of quadratic forms. I will discuss a sort of "duality" between X and the set of singular quadrics in the linear system defining it; in the case of intersections of three quadrics this picture offers a "dual" point of view on Hilbert's Sixteenth Problem.
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.
Freitag 19.06.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Sandra Lang (Technische Universität Darmstadt)
(Gast von Salma Kuhlmann, Claus Scheiderer und Markus Schweighofer)
Approximation of Projective Tensor Norms with Convex Algebraic Geometry
Abstract: Entanglement is a key concept in quantum physics. To deal with this phenomenon, many mathematical models to measure entanglement of states had been developped. At this point, projective norms as functions on topological multipartite tensor products play an important role.We want to investigate a method to approximate the projective norm by the concept of so-called theta bodies. These objects were proposed earlier in order to approximate the convex hull of a real algebraic variety by a chain of convex semialgebraic supersets. We show that the theta bodies associated to the product vectors converge to the unit sphere of the projective norm. We arrive at the conclusion that already the first theta body constitutes the unit sphere in case of bipartite tensor products. In all other cases, we use semidefinite programming to perform the approximation.
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.
Freitag 26.06.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Sebastian Gruler (Universität Konstanz)
Lower bounds on the size of positive-semidefinite lifts for some families of polytopes
Abstract: One says a polytope P admits a positive-semidefinite (psd) lift of size k, if P is the image of the intersection of the psd-cone S_+^k with an affine subspace under a linear map. The question of whether low-dimensional lifts exists is very interesting in optimization. Lee, Raghavendra and Steurer proved in a recent and celebrated work the first super-polynomial lower bounds on the size of psd-lifts for explicit families of polynomials.
This work is the main topic in my talk. I will give an idea of the proof and present some used tools and a small improvement of the bounds. But first, I present a theorem of Grigoriev (2001), that is very useful for the proof. I also give a new easy proof of it with the help of a recent result of Blekherman and Sinn.
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.
Freitag 03.07.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Rainer Sinn (Georgia Institute of Technology)
(Gast von Daniel Plaumann)
Generische projizierte Spektraeder
Abstract: Projizierte Spektraeder sind Projektionen von affin-linearen Schnitten des Kegels positiv semidefiniter Matrizen. Wir charakterisieren das Polynom, das auf dem Rand eines projizierten Spektraeders verschwindet, unter der Voraussetzung, das sowohl die Projektion als auch der Schnitt generisch sind.
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.
Donnerstag 16.07.2015 um 17:00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Yosef Yomdin (Weizmann Institute of Science, Rehovot, Israel)
(Gast von Margaret Thomas)
Counting zeroes of analytic functions: Taylor domination via Bautin ideals and recurrence relations
Abstract: The problem of bounding a possible number of zeroes of analytic functions appears in many areas of mathematics. In this talk I plan to discuss three of them: counting closed trajectories of polynomial plane vector-fields (Hilbert’s 16-th problem), Approximation theory (Remez-type inequalities), and counting rational points on analytic varieties. In each of these domains its own approaches have been developed. In particular, one of the strongest results in (largely open) Hilbert’s 16-th problem, asking for upper bounds on the number of isolated closed trajectories of polynomial plane vector-fields, was obtained by N. Bautin in 1935: at most three closed trajectories can bifurcate from an equilibrium point in a family of quadratic vector fields.
This result was obtained via analyzing the ideals, generated by the Taylor coefficients of the Poincare “first return mapping”. (This Taylor coefficients turn out to be polynomials in the parameters of the vector field). Computing Bautin ideals remains a notoriously difficult problem, widely studied in the research around Hilbert’s 16-th problem.
Bautin’s approach relates the algebra of Taylor coefficients a_k(v) in C[v] of an analytic family f_v(z) (v being the parameters) with the ``Taylor domination'', i.e. bounding all the Taylor coefficients a_k(v) through the few initial ones, uniformly in the parameters v. Then Taylor domination easily produces bounds on the number of zeroes of f_v(z). Many recent results demonstrate applicability of Bautin ideals in various problems far beyond the study of closed trajectories of ODE’s. I plan to present some of these results, and to relate them (if time allows) with recurrence relations on a_k(v), and with Remez-type inequalities.
Freitag 17.07.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tobias Kuna (University of Reading)
(Gast von Maria Infusino)
The realizability problem: an infinite dimensional moment problem
Abstract: The realizability problem is an old problem still essentially open in several areas of physics and other sciences. The problem will be introduced and it will be explained how it can be considered as a particular instance of the infinite dimensional moment problem. The aim of the talk is to explain the motivations in diverse applications that led to the realizability problem, which are the desired results and what has been achieved for this particular problem till today. This work is in collaboration with M. Infusino, J. Lebowitz, E. Speer and E. Caglioti.
