Vorträge inklusive Abstracts im Wintersemester 2016/2017
Die folgenden Vorträge haben im Wintersemester 2016/2017 im Oberseminar Reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium Reelle Geometrie und Algebra stattgefunden.
Mittwoch 05.10.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Tobias Kaiser (Universität Passau)
(Gast von Patrick Speissegger)
Piecewise Weierstrass preparation and division for o-minimal holomorphic functions
Abstract: The classical Weierstrass preparation theorem and division theorem are a key tool in analytic geometry. Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for the ring of definable holomorphic functions. In the semialgebraic setting and for the structure of globally subanalytic sets and functions we obtain the corresponding results for real analytic functions. As an application we show a definable global Nullstellensatz for principal ideals.
Freitag 28.10.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Daniel Plaumann (Technische Universität Dortmund)
(Gast von Claus Scheiderer)
Gram Spectrahedra
Abstract: The sum-of-squares representations of a given real polynomial in several variables are parametrized by a convex body, its Gram spectrahedron. We discuss results on sums of squares that fit naturally into this context, present some new results on minimal ranks, and highlight related open questions. (Joint work with Lynn Chua, Rainer Sinn, and Cynthia Vinzant)
Montag 07.11.2016 um 15:15 Uhr, Oberseminar Modelltheorie
David Bradley-Williams (Heinrich-Heine-Universität Düsseldorf)
(Gast von Salma Kuhlmann)
Applications of the theory of Jordan groups in describing reducts (of trees)
Abstract: When a structure, M, is presented to us (in a language L), it is a natural problem to try to describe the "reducts" of M, the (relational) structures on the same domain as M which are definable in L, and to do this up to first-order interdefinability. When M is omega-categorical (and countable) this is equivalent to describing the permutation groups H such that Aut(M) < H < Sym(M) that are closed in the natural topology on Sym(M). This characterisation allows us to use the theory of infinite permutation groups to study reducts (and vice-versa). There is a particularly interesting class of permutation groups called Jordan groups which has both a rich theory and the property that if G is a Jordan group then any closed permutation group containing G is also a Jordan group. This fact has already been used to study reducts of combinatorial trees (by Bodirsky and Macpherson) and combinatorial geometries (by Kaplan and Simon). We apply results from the theory of Jordan groups and semilinear orderings to describe the reducts of ordered trees that are sufficiently homogeneous and identify an infinite class of maximally closed subgroups of Sym(omega). We speculate that these methods might one day shed light on an elusive conjecture of Simon Thomas.
(During this talk I will mention joint work with Manuel Bodirsky, Michael Pinsker and András Pongrácz.)
Donnerstag 10.11.2016 um 17:00 Uhr, Allgemeines Mathematisches Kolloquium
Jan Draisma (Universität Bern)
Orthogonal tensor decomposition from an algebraic perspective
Abstract: Every real or complex matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense. Higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. Complementing this existing literature, I will present an algebro-geometric analysis of the set of orthogonally decomposable tensors. This analysis features a surprising connection between orthogonally decomposable tensors and semisimple algebras associative in the case of ordinary or symmetric tensors, and of compact Lie type in the case of alternating tensors.
(Joint work with Ada Boralevi, Emil Horobet, and Elina Robeva.)
Freitag 11.11.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Simon Müller (Universität Konstanz)
Quasi-ordered fields: a uniform approach to orderings and valuations
Abstract: Simply said, a quasi-order on a field K is an ordering, except that the antisymmetry is omitted and the compatibility with plus is weakened. In the note "Quasi-Ordered Fields" [JPPA 45 (1987) 207-210], the author S. M. Fakhruddin shows that there are precisely two kinds of quasi-ordered fields, namely ordered fields and Krull valued fields. Thus, the notion of quasi-orders allows us to treat ordered and valued fields simultaneously
I will introduce the notion of quasi-ordered fields and present the above result by Fakhruddin in detail. This talk will be followed up by a talk on quasi-ordered groups by Gabriel Lehéricy on November 25.
Montag 14.11.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Patrick Speissegger (McMaster University, Hamilton/Zukunftskolleg, Universität Konstanz)
Quasianalytic Ilyashenko algebras
Abstract: In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion.
(Joint work with Tobias Kaiser)
Freitag 18.11.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Cordian Riener (Universität Konstanz)
Quadrature rules of even degree and generalizations in the plane
Abstract: Let μ be a positive Borel measure on Rn. A quadrature rule for μ of strength d∈N is a finite set of points {p1,..., pk}∈Rn together with associated non-negative weights λ1,...,λn∈R such that the integral of any polynomial function f∈R[X1,...,Xn] of degree d can be evaluated as ∫ fdμ = ∑ λi f(pi).
In this talk we address the question, of the maximal number of nodes that are needed to form a quadrature rule and extend previous results from the case of odd d to even d. In particular, we focus of integration in the plane (i.e. n=2) where we show a generalisation of even degree Szegö quadrature to compact curves and we give a simplified argument for a Theorem by Curto and Yoo which asserts that in the plan 6 nodes are sufficient for strength 4 quadrature rules. (Joint work with Markus Schweighofer)
Freitag 25.11.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Gabriel Lehéricy (Universität Konstanz)
A Baer-Krull theorem for quasi-ordered groups
Abstract: The notion of compatibility between orders and valuations is an important topic in real algebra. One can give several characterizations of compatibility via conditions on the residue field, the group of units or the valuation ring. An important result is the Baer-Krull theorem, which describes valuation-compatible orders modulo orders on the residue field. In [1] the authors gave a theorem characterizing v-compatible quasi-orders. During her talk in this seminar, Salma Kuhlmann formulated a Baer-Krull theorem for quasi-ordered fields. The use of quasi-orders enables us to make statements and proofs applicable to both ordered and valued fields.
The goal of this talk is to present analogous results for abelian groups. In particular, we want to consider quasi-orders generalizing both ordered and valued groups, find a characterization of compatibility between a valuation and a quasi-order, and finally state an analog of the Baer-Krull theorem for quasi-ordered abelian groups.
[1] Salma Kuhlmann, Mickael Matusinski, Françoise Point, The valuation difference rank of a quasi-ordered difference field, due to appear in the volume "New Pathways between Group Theory and Model Theory", Proceedings Memorial Conference Rüdiger Göbel (2016), Springer
Freitag 02.12.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Fabian Reimers (Technische Universität München)
(Gast von Cordian Riener)
Separating invariants of finite groups
Abstract: Let X be an affine variety with an action of an algebraic group G (over an algebraically closed field K). A subset (e.g. a subalgebra) of the invariant ring K[X]^G is called separating if it has the same capability of separating the orbits as the whole invariant ring.
In this talk we focus on finite groups and show how the existence of a separating set of small size, or a separating algebra which is a complete intersection, is related to the property of G being a reflection (or bireflection) group. Theorems of Serre, Dufresne, Kac-Watanabe and Gordeev about linear representations are extended to this setting of G-varieties.
Freitag 09.12.2016 um13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Victor Vinnikov (Ben-Gurion University of the Negev, Beer-Sheva)
(Gast von Markus Schweighofer)
Block-diagonalization of matrices over local and graded rings
Abstract: Let R be a local ring over a field. Consider rectangular matrices with entries in R, up to left-right equivalence A -> UAV ,where U and V are invertible matrices over R. When is such a matrix equivalent to a block-diagonal matrix? The cases of dimension zero (matrices over a field) and one (matrices over a discrete valuation ring) can be easily handled explicitly using the classical canonical forms, but the general case is considerably more complicated. An obvious necessary condition is that the ideal of maximal minors of the matrix A factors. This condition is very far from being sufficient, but when the factors are relatively prime ideals we prove a very simple necessary and sufficient condition for block-diagonalization in terms of the Fitting ideals of A. It turns out also that the global question for matrices over a graded ring can be reduced to the local question. As an application, I will discuss reducibility of determinantal representations of reducible hypersurfaces (including positive determinantal representations appearing in the generalized Lax conjecture), and the relation to matrix factorizations. This talk is based on joint work with D. Kerner.
Montag 12.12.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Derya Çiray (Universität Konstanz)
Mild parametrization in o-minimal structures
Abstract: The application of mild parametrization, which is a parametrization with some control on the derivatives, was first introduced in the realm of diophantine geometry by Pila, to obtain results about the density of rational points on the graphs of non-algebraic pfaffian functions. Furthermore he has shown that mild parametrization with sufficient uniformity in parameters would be sufficient to establish Wilkie`s conjecture.
In this talk I will talk about interactions between o-minimal structures and mild parametrization and discuss whether certain o-minimal expansions of the reals admit mild parametrization or not.
Donnerstag 15.12.2016 um 13:30 Uhr, Oberseminar Modelltheorie
Eliana Barriga (Universidad de los Andes, Bogotá/University of Haifa)
(Gast von Pantelis Eleftheriou)
One-dimensional semialgebraic groups over real closed fields
Abstract: Semialgebraic groups over real closed fields have been intensively studied in the last three decades, and it is a field of current research. These groups can be seen as a generalization of the semialgebraic groups over the real field, studied by Madden and Stanton (1992) for the one-dimensional case, and also as a particular case of the groups definable over an o-minimal structure (see Razenj (1991) and Strzebonski (1993) for the one-dimensional case).
In this talk I will present the results we have obtained in my doctoral thesis project on the classification of one-dimensional semialgebraically connected semialgebraic groups over a real closed field. For this, I will show some results on the existence of definable local homomorphisms between definably compact semialgebraic groups and algebraic groups with generic neighborhoods, as well as some results on covering maps in the category of locally definable groups in o-minimal expansions of real closed fields.
This PhD project is supervised by Alf Onshuus and Kobi Peterzil.
Freitag 16.12.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Pawel Gladki (Uniwersytet Śląski, Kattowitz/Akademia Górniczo-Hutnicza, Krakau)
(Gast von Salma Kuhlmann)
Witt rings of accessible posets
Abstract: In this talk we will shall Witt rings on very general and highly abstract structures that we call accessible posets. Examples of such structures include groups of square classes of a field, groups of classes of sums of squares of a formally real field, or, more generally, special groups, reduced special groups, quaternionic schemes, spaces of orderings and the likes. Unlike classical examples, this construction allows conceivable generalizations to exact categories with dualities. This is a joint project with Krzysztof Worytkiewicz.
Freitag 13.01.2017 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Thorsten Theobald (Goethe-Universität Frankfurt)
(Gast von Cordian Riener)
Imaginary projections of polynomials
Abstract: We introduce the imaginary projection I(f) of a multivariate polynomial f ∈ C[z] as the projection of the variety of f onto its imaginary part. Since a polynomial f is stable if and only if I(f) does not intersect the open positive orthant, the notion offers a novel geometric view underlying stability questions of polynomials.
We show that the connected components of the complement of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, we establish structural properties of the components of the complement, such as lower bounds on their maximal number, prove a complete classification of the imaginary projections of quadratic polynomials and characterize the limit directions for polynomials of arbitrary degree.
Based on joint work with Thorsten Jörgens and Timo de Wolff.
Montag 16.01.2017 um 15:15 Uhr, Oberseminar Modelltheorie
Linnéa Sophie Gütlein (Universität Konstanz)
Real exponentiation, Schanuel's conjecture and the generalised Lindemann - Weierstrass theorem
Abstract: In 1996, Macintyre and Wilkie achieved several results about the theory of the ordered field of real numbers with exponentiation. Wilkie managed to prove its model completeness in [6]. Based on this result, in [3] Macintyre and Wilkie developed a recursive subtheory of the theory of real exponentiation and showed that this subtheory axiomatizes the theory of real exponentiation under the assumption of a famous conjecture of transcendental number theory. It is known as Schanuel's conjecture and was first mentioned in the literature by Schanuel's doctoral supervisor Lang in [1]. The conjecture states that for any over Q linearly independent numbers a1,..., am the transcendence degree of a1,..., am, ea1,..., eam over Q is at least m. Its significance lies not only in the fact that it would prove the decidability of the theory of real exponentiation as described above, but also in its ability to deduce other unknown algebraical properties such as the algebraic independence of e and π. The algebraic nature of these numbers had been studied long before the appearance of Schanuel's conjecture. Already in 1882, Lindemann proved that ea is transcendental for every non-zero algebraic number a, from which he deduced the transcendence of π. He published his results in [2], where he also mentioned a more general statement without proof, namely that for arbitrary distinct algebraic numbers a1,..., am, the numbers ea1,..., eam are linearly independent over the algebraic numbers. Some years later, Weierstrass gave a detailed proof of this theorem in [5]. It is therefore known as the Lindemann-Weierstrass theorem. In my talk I will present a proof of the Lindemann-Weierstrass theorem from 1956, given by Niven in [4]. In the end, I am going to point out some yet unproven assumptions of transcendental number theory that could be proved using Schanuel's conjecture, but do not follow from the generalized Lindemann-Weierstrass theorem.
References
[1] S. Lang. Introduction to transcendental numbers. Addison-Wesley series in mathematics. Addison-Wesley Pub. Co., 1966.
[2] F. Lindemann. Über die Zahl π. Math. Ann., 20(2):213-225, 1882.
[3] Angus Macintyre and A. J. Wilkie. On the decidability of the real exponential field. In Kreiseliana, pages 441-467. A K Peters, Wellesley, MA, 1996.
[4] Ivan Niven. Irrational numbers. The Carus Mathematical Monographs, No. 11. The Mathematical Association of America. Distributed by John Wiley and Sons, Inc., New York, N.Y., 1956.
[5] K. Weierstrass. Zu Lindemann's Abhandlung: "Über die Ludolph'sche Zahl". page 1067-1086, 1885.
[6] A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc., 9(4):1051-1094, 1996.
Montag 23.01.2017 um 15:15 Uhr, Oberseminar Modelltheorie
Omar Leon Sanchez (University of Manchester)
(Gast von Pantelis Eleftheriou)
The differential Dixmier-Moeglin equivalence
Abstract: In parallel to the classical Dixmier-Moeglin equivalence (from noncommutative algebra), it is an interesting problem to understand when certain classes of prime ideals in affine complex algebras equipped with a derivation coincide. More precisely, we consider the question: under which circumstances, in such complex algebras, do the differential primitive, differential locally closed, and differential rational ideals coincide? In joint work with Bell, Launois, and Moosa we presented an example where this equivalence does not hold. Recently, with Bell and Moosa, we proved that it does hold when there is a differential-Hopf algebra structure. We used the model theory of differential fields to do this.
Freitag 27.01.2017 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Johannes Rau (Eberhard Karls Universität Tübingen)
(Gast von Cordian Riener)
Real Hurwitz numbers - a tropical approach
Abstract: The study of Hurwitz numbers, despite its long history, has been completely remodeled in the last twenty years with the discovery of deep connections for example to Gromov-Witten theory and matrix integrals, originating in string theory. While classical Hurwitz numbers count certain holomorphic maps, sometimes it is natural to look at the "real" version of the problem (counting holomorphic maps compatible with a given real structure). I will present a tropical/combinatorial approach (i.e. based on pair-of-pants decomposition) to calculate such real Hurwitz numbers. If time permits, we will also discuss an approach, based on work of Itenberg and Zvonkine, to turn these numbers into invariants with respect to the position of the branch points (in the spirit of Welschinger invariants).
Freitag 03.02.2017 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tom Kriel (Universität Konstanz)
Introduction to free spectrahedra
Abstract: Free spectrahedra are a refinement of the concept of spectrahedra. The latter are sets of the form {x ∈ Rn | L(x) ≥ 0 } where L ∈ SR[ X]rxr is a matrix with polynomials of degree at most 1 as entries.
We will explain some basic techniques used in the study of free spectrahedra. After we will give a new proof of a theorem by Helton & McCullough which describes free spectrahedra as the sets which are matrix convex and free basic semialgebraic closed.
Montag 06.02.2017 um 15:15 Uhr, Oberseminar Modelltheorie
Alessandro Berarducci (Università di Pisa)
(Gast von Pantelis Eleftheriou und Salma Kuhlmann)
Arithmetical interpretation of the theory of the real exponential field
Abstract: We show that the ring of integers interprets the prime model of the complete theory T(exp) of the real exponential field. Thus, in particular, true arithmetic interprets T(exp). We also show that every element of the prime model is a recursive real number. More generally, the set of all recursive real numbers is a model of T(exp). (Work in progress with A. Fornasiero and V. Mantova.)
Freitag 17.02.2017 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Patrick Speissegger (McMaster University, Hamilton/ Zukunftskolleg, Universität Konstanz)
Holomorphic extensions of functions definable in R_an,exp
Abstract: I describe a general extension theorem for the germs of functions definable in the o-minimal expansion of the field of real by all restricted analytic functions and the exponential function. The extension theorem can be used to bound the complexity of, for instance, the compositional inverse of a definable germ in terms of the complexity of that germ.
Montag 20.02.2017 um 15:15 Uhr, Oberseminar Modelltheorie
Magdalena Forstner (Universität Konstanz)
Quantifier Elimination Tests
Abstract: A theory T admits quantifier elimination if every formula is equivalent in T to a quantifier-free formula. Quantifier elimination is a very powerful property, as it helps in the question of decidability as well as in the study of definable sets.
There are various tests that one can use to prove quantifier elimination. In this talk I will particularly present one specific quantifier elimination test which Lou van den Dries used in order to show that the theory of (IR 2^Z) in a suitable language eliminates quantifiers. Afterwards I will give a brief overview of the application of this test to prove that this theory, indeed, admits quantifier elimination.
Montag 27.02.2017 um 15:15 Uhr, Oberseminar Modelltheorie
Vincent Grandjean (Universidade Federal do Ceará)
(Gast von Patrick Speissegger)
Mostowski's Proof of the non oscillation conjecture in dimension 3
Abstract: Let F be the germ of a real analytic function at the origin O of Euclidean 3-space. Assume the origin O is a critical point of F. If C is any gradient trajectory of F accumulating at O, it was shown by
Kurdyka, Mostowski and Parusinski that the limit of secants at O along C exists (this claim was known as Thom's Gradient Conjecture). A related conjecture, stated independently by Moussu and Kurdyka around 1995, is that such a gradient trajectory would not oscillate at O. Non-oscillation at 0 means the following: given the germ at O of any semi-analytic set S, the trajectory C must either never intersect S close to O, or being contained in S once close enough to O. Mostowski has a proof of the non-oscillation conjecture in dimension three. In the talk I will present some ingredients and steps of Mostowski's proof.
