Abstract in Wintersemester 2013/14
Die folgenden Vorträge haben im Wintersemester 2013/14 im Oberseminar reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium reelle Geometrie und Algebra stattgefunden.
Montag, 21.10.2013 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Donnerstag, 24.10.2013 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Bernd Sturmfels (University of California, Berkeley und MPI Bonn)
(Gast von Claus Scheiderer und Markus Schweighofer)
The Euclidean Distance Degree
Abstract: The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points of this optimization problem. We focus on varieties seen in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition.
This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.
Freitag, 25.10.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Bernd Sturmfels (University of California, Berkeley and MPI Bonn)
(Gast von Claus Scheiderer und Markus Schweighofer)
Non-Negative Rank of Matrices and Tensors
http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201314-abstracts/#2410Abstract: An m x n matrix has non-negative rank r if it factors as a product of non-negative matrices of formats m x r and r x n. Non-negative factorization of matrices and tensors has numerous applications in engineering and the sciences. This talk concerns the semi-algebraic sets of tensors with a given non-negative rank. We present work with Allman, Rhodes and Zwiernik on tensors of non-negative rank two, and ongoing work with Kubjas and Robeva on matrices of non-negative rank three. We also discuss the geometry of expectation maximization in statistics.
Montag, 28.10.2013 um 15.15 Uhr, Oberseminar Modelltheorie
Cédric Milliet (Konstanz)
Variations on a theme by Aldama
Abstract: Let G be a group, and f(x,y) a formula in the language of groups. We say that f(x,y) has the independence property if for all natural number n, we can find elements a0, a1, …, an and (bI) I ⊆ n+1 in G such that f(ai,bI) holds if and only if i ∈ I. The group G is said to be ‘without the independence property’ if no formula has the independence property in G. Such groups include stable groups. We shall review recent results of Shelah and Aldama concerning infinite abelian and nilpotent subgroups of a group G without the independence property, and address the following open question: is a soluble subgroup S of G contained in a soluble definable subgroup of G?
Donnerstag, 31.10.2013 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Cédric Milliet (Konstanz)
The Compactness Theorem of first order logic
http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201314-abstracts/#2410Abstract: The Compactness Theorem of first order logic states that a set of formulas has a model, provided that every finite subset of it have a model. It is usually attributed to Kurt Gödel who derived it as an immediate consequence of its Completeness Theorem. It has been given many other proofs since then, and has nowadays a central place in a graduate course of Model theory, being probably the tool most frequently used by a Model theorist. It is indeed a powerful instrument to provide the existence of structures having `limit' properties. In the talk, I will try to give a comprehensive introduction needed to understand the statement of the theorem, sketch a proof of how to construct this `limit model', and outline some of its surprising consequences in Algebra, Analysis and Combinatorics.
Freitag, 01.11.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Feiertag, kein Vortrag
Montag, 04.11.2013 um 15.15 Uhr, Oberseminar Modelltheorie
Alexander Prestel (Konstanz)
Model theoretic criteria for uniform definability (with fixed quantifier complexity) of valuation rings
http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201314-abstracts/#2410Abstract: see abstract for Nov. 8
Donnerstag, 07.11.2013 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Murray Marshall (University of Saskatchewan, Saskatoon)
(Gast von Markus Schweighofer)
Application of localization to the multivariate moment problem
Abstract: It is explained how the localization technique introduced by M. in 2003 leads to a useful reformulation of the multivariate moment problem in terms of extension of positive semidefinite linear functionals to positive semidefinite linear functionals on the localization of the real polynomial ring R[x_1, \dots,x_n] at p = (1+x_1^2) \dots (1+x_n^2) or p' = (1+x_1^2) \dots (1+x_{n-1}^2). It is explained how this reformulation can be exploited to prove new results concerning existence and uniqueness of the measure \mu and density of the complex polynomial ring C[x_1, \dots, x_n] in the space of measurable functions L^s(\mu), 1 \le s < \infty, and, at the same time, to give new proofs of old results of Fuglede, Nussbaum, Petersen and Schmuedgen, results which were proved previously using the theory of strongly commuting self-adjoint operators on Hilbert space.
Freitag, 08.11.2013 um 13.30 Uhr (im Raum G309), Oberseminar Reelle Geometrie und Algebra
Alexander Prestel (Konstanz)
Definable valuations
Abstract: We shall prove the (pure) existence of formulas defining uniformly the valuation rings O in different classes of henselian fields (K,O). For instance, we prove that there is an EA-formula in the ring language defining O uniformly for all henselian fields (K,O) where the residue class field is finite, pseudo-finite,or hilbertian. (Note that all function fields and number fields are hilbertian)
Montag, 11.11.2013 um 15.15 Uhr, Oberseminar Modelltheorie
Pantelis E. Eleftheriou (Konstanz)
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 and Hieronymi).
The analysis of definable groups first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.
This talk is a natural continuation of the one given last semester, but no previous knowledge will be assumed.
Freitag, 15.11.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 18.11.2013 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 22.11.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 25.11.2013 um 15.15 Uhr, Oberseminar Modelltheori
kein Vortrag
Freitag, 29.11.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Mario Kummer (Konstanz)
Determinantal Representations and a Matrix Positivstellensatz
Abstract: A major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral, i.e. the solution sets of linear matrix inequalities. We will use sum-of-squares decompositions of certain matrix polynomials to approach this problem. More precisely, we will prove that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation.
Montag, 02.12.2013 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 06.12.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Johannes Rau (Universität des Saarlandes, Saarbrücken)
(Gast von Daniel Plaumann)
Topological classification of real rational curves in the plane
Abstract: The study of the topological shape of real algebraic curves in the plane has a long history. In the past, it was mainly focused on smooth curves. In this talk, however, we consider singular curves, namely rational curves of degree 5 (and 4) with only ordinary double points. We will discuss how to obtain restrictions on the possible topological types from knowledge of the smooth classification and how to construct the remaining types by using patchworking techniques.
(Joint with Ilia Itenberg and Grigory Mikhalkin.)
Montag, 09.12.2013 um 15.15 Uhr, Oberseminar Modelltheorie
Merlin Carl (Konstanz)
Infinitary computations with random oracles
Abstract: Turing computability provides an intuitive and attractive formal framework for the analysis of the question which mathematical objects and functions can be generated by applying a certain procedure finitely many times. However, in mathematical practice, we often encounter examples of objects generated by infinite processes involving limits, such as adding reals, computing with limits of real sequences, forming direct limits of directed systems, hulls, closures etc. This motivates the study of infinitary computations that can account for the limit phenomenon in such processes and provide a framework for extending results from recursion theory to infinitary mathematical objects.
A fascinating theorem of classical recursion theory due to G. Sacks is that a function f: N -> N is Turing-computable from all oracles in a set X of positive Lebesgue measure iff f is recursive. Intuitively, this means that the use of random generators does not enrich the set of computable functions, not even when computability is weakened to computability with positive probability.
We consider the analogous problem for various infinite time machines: If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? Exploiting the strong connection between generalized computability and models of Kripke-Platek set theory, we show that the answer is positive answer for most other machine types. Moreover, we show that this is independent from ZFC for ordinal Turing machines (OTMs) with and without ordinal parameters. In particular, the parameter-free version is false in Gödel’s constructible universe L. This is joint work with Philipp Schlicht.
Freitag, 13.12.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Christoph Hanselka (Konstanz)
Maximal Signature Determinantal Representations of Plane Curves.
Abstract: By the theorem of Helton and Vinnikov, every ternary hyperbolic polynomial admits a positive definite real symmetric determinantal representation. More generally one can tell by the topology of the real locus of any plane curve, what maximal signature a symmetric determinantal representation can attain at a given point. I will sketch this relation and how to obtain these representations and also point out the main technical difference to the definite case.
Montag, 16.12.2013 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 20.12.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 06.01.2014 um 15.15 Uhr, Oberseminar Modelltheorie
Feiertag, kein Vortrag
Freitag, 10.01.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag, 13.01.2013 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 17.01.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Franziska Jahnke (Universität Münster)
(Gast von Arno Fehm)
On almost real closed fields
Abstract: A field is called almost real closed if it admits a henselian valuation with real closed residue field. These fields were studied by Delon and Farre in the nineties. They proved that if a field is almost real closed but not real closed, then it admits a definable henselian valuation (in the language of rings) and showed on the other hand that not every henselian valuation on such a field is definable. We will give an alternative proof of the first result, more specifically, we will use p-henselian valuations to show that there is always a parameter-free definable henselian valuation.
Montag, 20.01.2014 um 13.30 Uhr, Oberseminar Modelltheorie
kein Vortrag
Donnerstag, 23.01.2014 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Karim Johannes Becher (Universität Antwerpen)
(Gast von Arno Fehm)
Algebren mit Involution und ihre Zerlegbarkeit
http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201314-abstracts/#2410Abstract: Ich berichte über eine gemeinsamen Arbeit mit N. Grenier-Boley und J.-P. Tignol, in der wir Algebren mit Involution von kleinem Grad betrachten, genauer solche, die eine maximale symmetrische etale Teilalgebra vom Grad 4 enthalten.
Solchen Algebren mit Involution können wir eine Invariante in Gestalt einer Pfisterform zuordnen, und das unabhängig von der Charakteristik.
Diese Invariante ist trivial genau dann, wenn sich die Algebra mit Involution vollständig zerlegen lässt. Durch diesen neuen Zugang lassen sich bereits bekannte Zerlegbarkeitskriterien für verschieden je nach Charakteristik und Typ der Involution unterschiedliche Kriterien unter einen Hut fassen.
Außerdem erhalten wir einen neuen Beweis für einen Satz von Rowen über die Existen triquadratischer Erweiterungen in zentraleinfachen Algebren vom Grad 8 und Exponenten 2.
Freitag, 24.01.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Eli Shamovich (Ben-Gurion University of the Negev)
(Gast von Salma Kuhlmann)
Determinantal Representations and Hyperbolicity of Space Curves
Abstract: In the talk I will introduce the notion of hyperbolicity and determinantal representations of hypersurfaces and then move on to define an extension of those notions for subvarieties of Pn of higher codimension. Then using the incidence correspondence we will transfer those notions to the Grassmannian. A case of particular interest is the case of curves in Pn, for n > 2, on which I will elaborate further on the new phenomena in this setting and present a construction for determinantal representations of certain form. This talk is based on a joint work with Victor Vinnikov.
Montag, 27.01.2014 um 15.15 Uhr, Oberseminar Mo
delltheorie
Will Ascombe (University of Oxford)
(Gast von Arno Fehm)
Asymptotic classes and measurable structures
http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201314-abstracts/#2410Abstract: A well-known theorem of Chatzidakis, van den Dries, and Macintyre gives a nice asymptotic description of the cardinalities of sets drawn from uniformly definable families in finite fields. The description extends the Lang-Weil estimates from irreducible varieties to arbitrary definable sets.
Taking this theorem as a definition, Macpherson and Steinhorn developed a notion of a 1-dimensional asymptotic class of finite structures. They gave a variety of examples, both algebraic and combinatorial, and proved that non-principal ultraproducts of these classes are measurable, and in turn supersimple of S1-rank 1. In subsequent work Elwes generalised this notion to an N-dimensional asymptotic classes. This definition still lives in the supersimple world, but there are interesting new examples.
Much more recently Macpherson and Steinhorn have yet again generalised the definition to `multidimensional asymptotic class' (`mac'). In this talk I will survey this subject, hopefully providing motivation for the definitions, and explain the ongoing work on macs.
Freitag, 31.01.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Salma Kuhlmann (Konstanz)
An application of T. Jacobi's Positivstellensatz to locally multiplicatively convex topological R-algebras
Abstract: Let A be a commutative (unitary) R-algebra, and τ a topology on A induced by a submultiplicative semi-norm (or more generally by a family of such). Let S ⊆ A be a ∑A^{2d}-module. We compute the τ-closure of S in terms of the Gelfand Spectrum of τ, and apply our result to approximate positive polynomials by sums of squares. The proof is based on an application of the archimedean Positivstellensatz to Banach algebras. Joint work with M. Ghasemi and M. Marshall.
Montag, 03.02.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 07.02.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
Rainer Sinn (Konstanz)
Extreme Rays of the Spectrahedron of Moment Matrices for Ternary Forms
Abstract: The moment spectrahedron is the dual cone to the sums of squares cone. It has extreme rays of rank one, which correspond to point evaluations. For ternary forms, beginning in degree 6, it must have more. We review bounds on the ranks of extremal moment matrices which do not come from point evaluations due to Blekherman and present a constructive approach related to classical algebraic geometry of plane curves revolving around the Theorem of Cayley-Bacharach. We will arrive at a description of the Zariski closure of the extreme rays in terms of Gorenstein ideals and discuss applications to the algebraic boundary of the sums of squares cone.
Montag, 10.02.2014 um 15.15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag, 14.02.2014 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Donnerstag, 13.03.2014 um 17.00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Marie Francoise Roy (University of Rennes 1)
(Gast von Charu Goel)
Computing the topology of a real algebraic curve
Abstract: Computing the topology of a real algebraic curve is a classical problem in algorithmic real algebraic geometry. Most papers are based on some variant of Cylindrical Decomposition : decompose the X-axis into a finite number of open intervals and points above which the curve has a cylindrical structure. We discuss the history of this topic and two recent results.
References:
On the computation of the topology of plane curves Niang Diatta D., Rouillier F., Roy M.-F. [hal-00935728 - version 1]
Improved Complexity Bounds for Computing with Planar Algebraic Curves, Kobel A., Sagraloff M.
