Die folgenden Vorträge haben im Wintersemester 2011/12 im Oberseminar reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium reelle Geometrie und Algebra stattgefunden.
Montag, 17.10.2011 um 16.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 21.10.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Katrin Tent (Münster)
(Gast von Salma Kuhlmann)
Über die Automorphismengruppe des Urysohn-Raumes
Abstract: Sehr homogene Strukturen wie zB. symmetrische Räume oder der universelle Graph haben häufig einfach Automorphismengruppen. Ich werde ein allgemeines Kriterium erklären, dass es einem erlaubt die Einfachheit oder relative Einfachheit einer Automorphismengruppe zu beweisen. Dieses lässt sich auf den Urysohn-Raum, aber auch viele andere homogene Strukturen anwenden.
Montag, 24.10.2011 um 16.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 28.10.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
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.
Donnerstag, 03.11.2011 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Marie Francoise Roy (Université de Rennes 1)
(Gast von Markus Schweighofer)
Effective 17'th Hilbert's problem
Abstract: Artin's beautiful solution to Hilbert 17th problem proves that a positive polynomial is always a sum of squares of rational functions, but it does not give an explicit method for producing the sum of squares.
This is a very puzzling situation and a very challenging question.
I shall explain the method of my joint work (in progress) with Henri Lombardi and Daniel Perrucci on the topic. Our aim is to give an explicit construction of the sums of squares with elementary recursive degree bounds.
Freitag, 04.11.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Louis Theran (FU Berlin)
(Gast von Cordian Riener)
Generic planar rigidity with forced symmetry
Abstract: A periodic framework is a planar structure, periodic with respect to a lattice, made of fixed length bars connected by joints with full rotational freedom. The allowed motions preserve periodicity and the lengths and connectivity of the bars. When all the allowed motions areisometries, a framework is rigid.
I'll talk about an eficiently-checkable combinatorial characterization of which generic frameworks are rigid and some generalizations to other kinds of symmetry.
This is joint work with Justin Malestein.
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.
Freitag, 11.11.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Valerio Capraro (Université de Neuchâtel)
(Gast von Markus Schweighofer)
Introduction to Connes' embedding conjecture and its applications
Abstract:Connes' embedding conjecture was stated by Alain Connes in 1976. Since then, many mathematicians have been trying to approach it from different point of view. The result is that now there is a number of equivalent problems both in Operator Algebras and other branches of mathematics, that make the problem one of the most interesting open problems in Operator Algebras and neighborhoods.
The first part of the talk is devoted to the formulation of the conjecture and some of its equivalent conjectures. In the second part I wish to present a new application: the construction of a geometric invariant associated to those factors that verify the conjecture. This construction was begun by Nate Brown and completed in two joint works, one with Tobias Fritz and one with Nate Brown himself.
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).
Freitag, 18.11.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Sebastian Gruler (Konstanz)
Summen 2m-ter Potenzen von reellen Polynomen
Abstract: In diesem Vortrag stelle ich meine Diplomarbeit mit dem Titel ”Summen 2m-ter Potenzen von reellen Polynomen“ vor. Im ersten Teil geht es dabei um die Frage unter welchen Bedingungen ein f ∈ ℝ[X1, . . . , Xn], das auf der basisch-abgeschlossenen Menge W = W(h1,...,hs) strikt positiv ist, in der von h1,...,hs erzeugten Prärdnung der Stufe 2m liegt. Es stellt sich heraus, dass für ungerades m die Kompaktheit der Menge W genügt. Im zweiten Teil geht es um die Frage unter welchen zusätzlichen Bedingungen solch ein f im univariaten Fall auch für gerades m eine solche Darstellung besitzt und unter welchen Bedingungen f sogar in dem von h1,...,hs erzeugten Modul der Stufe 2m liegt.
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)).
Freitag, 25.11.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
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.
Donnerstag, 01.12.2011 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Thomas Kahle (EPDI)
(Gast von Daniel Plaumann)
Binomial ideals and reality
Abstract: In algebraic statistics many problems feature binomial equations like xπ=y2. If the exponents of the binomials are rational and one is working over an algebraically closed field, then methods of commutative algebra and classical algebraic geometry can be applied to attack these problems. Probabilities, however, are non-negative real numbers. We will show how to use combinatorics and linear algebra to overcome these restrictions. For instance, a variant of Hilbert's basis theorem holds: The solutions of infinitely many binomial equations are cut out by finitely many equations derived from an underlying oriented matroid. If the original equations happen to have integer exponents, then one can apply Hilbert's original theorem and compare the two outcomes using binomial primary decomposition.
Freitag, 02.12.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Alexander Merkurjev (University of California, Los Angeles)
(Gast von Karim Johannes Becher)
Essential dimension in algebra
Abstract: Essential dimension of an algebraic object is the smallest number of algebraically independent parameters required to define the object. This notion was introduced by Buhler, Reichstein and Serre. Relations to algebraic geometry and representation theory of algebraic groups will be discussed.
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.
Freitag, 09.12.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
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.
Freitag, 16.12.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
Christoph Hanselka (Konstanz)
Real Zero Polynomials, Complete Monotony and the BMV Conjecture
Abstract: A polynomial is called 'real zero' if it has only real zeros on every real line through the origin. One of the basic facts that I will talk about is that the connected component of the origin in the set of nonzeros of such a polynomial is convex. (Those sets are called rigidly convex). The methods used to show this will give a connection to the BMV conjecture (due to Bessis, Moussa, and Villani) saying that for psd matrices A and B and a natural number m the polynomial tr((A+tB)m) has only nonnegative coefficients.
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.
Freitag, 23.12.2011 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 09.01.2012 um 16.15 Uhr, Oberseminar Modelltheorie
kein Vortrag, sonst am Freitag, 13.01.2012 um 14.15 Uhr.
Freitag, 13.01.2012 um 14.15 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag. Bitte beachten Sie, dass es gibt einen Vortrag an diesem Freitag im Oberseminar Modelltheorie.
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.
Donnerstag, 19.01.2012 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Merlin Carl (Konstanz)
Unendliche Spiele - eine kurze Geschichte des Determiniertheitsaxioms
Abstract: Determiniertheitsaxiome wie AD und seine Varianten verallgemeinern auf natürliche Weise grundlegende Resultate über die Existenz von Gewinnstrategien für endliche Spiele mit vollständiger Information. Als intuitiv plausible Alternative zum Auswahlaxiom AC mit zahlreichen attraktiven Konsequenzen etwa für Maßtheorie und Analysis ist die Untersuchung von AD inzwischen ein zentrales Thema der Mengenlehre. Die Analyse seiner Konsistenzstärke verbindet auf elegante Weise einige der stärksten großen Kardinalzahlhypothesen mit konkreten Gegenständen der mathematischen Praxis.
Im Vortrag werde ich AD mit einigen seiner wichtigsten Konsequenzen vorstellen und exemplarisch demonstrieren, wie es zur Behandlung von Fragen der Analysis verwendet werden kann.
Freitag, 20.01.2012 um 14.15 Uhr, Oberseminar reelle Geometrie und Algebra
Prof. Christian Haase (Frankfurt)
(Gast von Cordian Riener)
Permutation Polytopes
Abstract: A permutation polytope is the convex hull of a subgroup of the permutation matrices. The superstar among the permutation polytopes is the Birkhoff polytope of doubly stochastic matrices – the convex hull of the full symmetric group. In this talk I will report on results about combinatorial properties of permutation polytopes, starting from examples which clarify the connections between the group and the polytope.
In order to obtain further results, it seems necessary to get a hand on the inequality description of these polytopes. But this description turns out surprisingly complex already for innocent cyclic groups. This class of polytopes contains the family of marginal polytopes studied in optimization and statistics.
This is joint work with Barbara Baumeister, Benjamin Nill and Andreas Paffenholz.
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.
Freitag, 27.01.2012 um 14.15 Uhr, Oberseminar reelle Geometrie und Algebra
Charu Goel (Konstanz)
Gram matrices of Symmetric forms
Abstract: In 1888 Hilbert proved that in the ring ℝ[X1,X2,...,Xn]: Pn,m = Σn,m iff n = 2, m = 2, or (n, m) = (3, 4), where Pn,m and Σn,m are respectively the cones of positive semidefinite (psd) and sum of squares (sos) forms of degree m. Thus in all other cases Σn,m ⊂ (strictly) Pn,m (I will call them non-Hilbert cases).
I will start by introducing our problem; Finding necessary/sufficient conditions on the coefficients of a symmetric form to be psd or sos, via the entries of a corresponding gram matrix. Then I will survey further results by Choi, Lam, Reznick and Harris for even symmetric forms in some of the non-Hilbert cases. Finally I will present the main questions that I work on and some relevant worked out examples.
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.
Freitag, 03.02.2012 um 14.15 Uhr, Oberseminar reelle Geometrie und Algebra
Samuel Volkweis Leite (Konstanz)
On p-adic Representation theorems
Abstract: In the 1940 Stone give an algebraic characterization of the ring of continuous real valued functions on a compact Hausdorff space. We shall give an algebraic characterization of the ring of p-adic valued functions on a compact Hausdorff space.
This will be a continuation of a preliminary talk given by Prof. Prestel on 01.02.2012 in the Quadratic forms seminar.
Montag, 06.02.2012 um 16.15 Uhr, Oberseminar Modelltheorie
Kein Vortrag
Freitag, 10.02.2012 um 14.15 Uhr, Oberseminar reelle Geometrie und Algebra
Lior Bary-Soroker (Tel Aviv)
(Gast von Arno Fehm)
On composition of polynomials and towers of fields
Abstract: Let f be a univariate polynomial with coefficients in a field K. If f is separable irreducible then it induces a field extension L/K, where L=K(x) and x is a root of f. The primitive element theorem tells us that every finite separable extension L/K is generated in such a way. Now if f,g are two univariate polynomials with coefficients in K s.t. the composition is separable irreducible, then they generate a tower of finite separable extensions E/L/K. Namely E=K(x) and L=K(g(x)), where f(g(x))=0. Does the converse hold?
We do not know the answer in general. As a baby case we study this question and some of its variants over pseudo algebraically closed fields, where we get surprising answers.
