Vorträge inklusive Abstracts im Sommersemester 2016
Die folgenden Vorträge haben im Sommersemester 2016 im Oberseminar Reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium Reelle Geometrie und Algebra stattgefunden.
Montag 11.04.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 15.04.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Antonio Lerario (Scuola Internazionale Superiore di Studi Avanzati, Trieste)
(Gast von Markus Schweighofer)
The average topology of an intersection of quadrics, random determinantal representations and Hilbert's Sixteenth problem
Abstract: What is the expected "topology" of the intersection of random quadric hypersurfaces in RP^n? For n=2, the answer follows from a (1993) result of M. Shub and S. Smale: two random conics intersect, on average, in two points (for a natural definition of "random conic").
In the general case n>2, the intersection can have higher order nonzero Betti numbers. In this talk I will combine techniques from Algebraic Topology (spectral sequences) with ideas from Random Matrix Theory to study the asymptotic distribution of these Betti numbers. (The case n=3 is related to a probabilistic approach to Hilbert's Sixteenth problem and admits a dual description using random determinantal representations.)
(This is joint work with E. Lundberg)
Montag 18.04.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 22.04.2015 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Yoshiyuki Sekiguchi (Tokyo University of Marine Science and Technology)
(Gast von Markus Schweighofer)
Perturbation analysis for singular semidefinite programming
Abstract: In this talk, we study behaviors of optimal values and solutions of semidefinite programming under perturbations.
If the feasible set of primal problem has nonempty interior, the optimal values changes continuously when constant terms in constraints are perturbed. However those could change both continuously and discontinuously when we perturb coefficient matrices of variables in constraints. This means that such problem could have some singularity. We will use the Facial Reduction algorithm to find what kind of
perturbations does not raise discontinuity.
(This is a joint work with H. Waki)
Montag 25.04.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 29.04.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Eli Shamovich (Ben-Gurion University of the Negev, Be'er Sheva)
(Gast von Markus Schweighofer)
Determinantal Representations and Ulrich Sheaves
Abstract: In this talk we will describe maximally generated maximal Cohen-Macaulay modules over graded rings, that are also called Ulrich modules. Associated to Ulrich modules there are coherent sheaves on the projective space, called Ulrich sheaves. We will describe the relation between Ulrich sheaves and determinantal representations of subvarieties of \mathbb{P}^d. We will then describe symmetric bilinear forms on Ulrich sheaves and their relation to real symmetric determinantal representations and hyperbolicity.
Montag 02.05.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 06.05.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 09.05.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 13.05.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 16.05.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 20.05.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 23.05.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 27.05.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 30.05.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 03.06.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Mohammad Adm (Universität Konstanz)
Total nonnegativity of some structured matrices and zeros and poles locus of the corresponding polynomials and rational functions
Abstract: In our talk we consider structured matrices which are related to the stability of polynomials and to the localization of the poles and zeros of rational functions. Specifically, in the case of polynomials we focus on the Hurwitz matrix which is used to examine the (Hurwitz) stability of a given polynomial, i.e., to the property that all of its zeros are contained in the open left half of the complex plane. In the case of rational functions we consider matrices of Hurwitz type which are closely related to the Hurwitz matrix and Hankel matrices associated with the Laurent series at infinity of rational functions and focus on R-functions of negative type, i.e., functions which map the open upper half plane of the complex plane to the open lower half plane. Finally, in the polynomial as well as in the rational case we are interested in interval problems which arise when the polynomial coefficients are due to uncertainty caused by, e.g., data uncertainties, but can be bounded in intervals.
(Joint work with Jürgen Garloff)
Montag 06.06.2016 um 15:15 Uhr, Oberseminar Modelltheorie
kein Vortrag
Freitag 10.06.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Rainer Sinn (Georgia Institute of Technology)
(Gast von Markus Schweighofer)
Low-rank sum-of-squares representations on varieties of minimal degree
Abstract: We will give a quantitative proof of the result that every nonnegative quadratic form on a variety X of minimal degree is a sum of dim(X)+1 squares of linear forms in the homogeneous coordinate ring of X. Our proof works the same for all varieties of minimal degree and recovers the bounds that were proved with different methods. The proof also shows that there are only finitely many representations of a general nonnegative quadratic form as a sum of dim(X)+1 squares. We will count the number of representations on all 2-dimensional rational normal scrolls, completing the picture for all surfaces of minimal degree with the results on ternary quartics by Powers, Reznick, Scheiderer, Sottile. Interestingly, smooth quadratic forms on rational normal scrolls might have different numbers of representations: we need stronger assumptions on the generic forms to obtain the general number of representations.
Montag 13.06.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Erik Walsberg (Hebrew University of Jerusalem/ Université Paris VI)
(Gast von Margaret Thomas)
Tame Topology over dp-minimal fields
Abstract: I will discuss the topological properties of definable sets and functions in dp-minimal, non-strongly minimal, expansions of fields.
Freitag 17.06.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tomas Bajbar (Karlsruher Institut für Technologie)
(Gast von Markus Schweighofer)
On Globally Diffeomorphic Polynomial Maps via Newton Polytopes and Circuit Numbers
Abstract: We analyze the global diffeomorphism property of polynomial maps F: R^n -> R^n by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials ||F||_2^2. This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on R^n is equivalent to their Jacobian determinant det JF vanishing nowhere on R^n. In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.
Montag 20.06.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Sebastian Krapp (Universität Konstanz)
On the decidability of the real exponential field
Abstract: The real exponential field $\mathcal{R}_{\exp}=(\mathbb{R}, +, \cdot, 0, 1, <, \exp)$ is the structure of the real field equipped with the unary function of standard exponentiation $\exp(x) = \mathrm{e}^x$. Tarski posed the question whether the theory of $\mathcal{R}_{\exp}$ is decidable, i.e. whether there exists an effective procedure determining whether a given sentence in the language of $\mathcal{R}_{\exp}$ is true in $\mathcal{R}_{\exp}$. Macintyre and Wilkie showed in [1] that under the assumption of Schanuel's Conjecture, a strong conjecture in transcendental number theory, the answer is positive.
In my talk I will firstly present the main steps of Macintyre and Wilkie's proof which are based on the model completeness result of $\mathcal{R}_{\exp}$ in [2] and independent of Schanuel's Conjecture, and secondly explain in more detail how Schanuel's Conjecture eventually resolves the decidability question. No specific knowledge outside basic model theory will be required to follow the talk.
[1] A. Macintyre and A. Wilkie, On the decidability of the real exponential field, in: `Kreiseliana: about and around Georg Kreisel' (Piergiorgio Odifreddi), A. K. Peters, Wellesley, Mass., 1996, pp. 441–467.
[2] A. Wilkie, Model completeness results for expansions of the ordered Field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society 9 (1996), no. 4, 1051–1094.
Freitag 24.06.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Tim Kobert (Universität Konstanz)
Orbitopes of Schur-Horn Type
Abstract: Orbitopes are the convex hull of an orbit of a compact, real algebraic group acting linearly on some real vectorspace. We will first talk about orbitopes in general and then focus on the Symmetric Schur-Horn Orbitopes which are given as the convex hull of an orbit of the special orthogonal group acting by conjugation on the space of symmetric matrices. Then we will present some generalizations to the results about the Symmetric Schur-Horn Orbitopes.
Montag 27.06.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Natalia Walder (Universität Konstanz)
Exponentiation in non-archimedean ordered fields
Abstract: The non-archimedean ordered real closed field of general power series does not admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements. S.Kuhlmann [1] proves a more general result for totally ordered sets which yields that the logarithm on the multiplicative group is never surjective. A sketch of the proof and its consequences will outline its importance on the study of exponentials on the real closed field of power series.
[1] S. Kuhlmann, Ordered Exponential Fields, The Fields Institute Monograph Series, vol. 12 (2000), p. 67- 69
Donnerstag 30.06.2016 um 17:00 Uhr, Allgemeines Mathematisches Kolloquium
Manfred Droste (Universität Leipzig)
Automorphism groups of ordered sets and the Bergman property
Abstract: In this survey, we will present various permutation groups with the Bergman property. Here, a group G is said to have the Bergman property, if for any generating subset E of G, already some bounded power of E ∪ E^-1 ∪ {1} covers G. This property arose in a recent interesting paper of Bergman where it was derived for the infinite symmetric groups. Groups which were, soon after Bergman’s paper, shown to have the Bergman property include automorphism groups of various kinds of homogeneous spaces. Such groups include the homeomorphism groups of the rationals, the irrationals, or Cantor’s set, measure automorphism groups of the reals or of the unit interval, and groups of non-singular or ergodic transformations of the reals. We will concentrate on automorphism groups of ordered sets. The groups of all order automorphisms of the rationals or of the reals have the Bergman property. Also, the order automorphism groups of any weakly 2-transitive countable tree and of the universal homogeneous countable distributive lattice were recently shown to have the Bergman property. However, e.g. groups of bounded order automorphisms of the rationals do not have the Bergman property. The problem arises to find further examples as well as general criteria for classes of groups (or transformation semigroups) acting on structures with the Bergman property. For which of your favorite algebraic structures does the automorphism group (or transformation semigroup) have the Bergman property?
Joint work with R. GÖBEL, C. HOLLAND and G. ULBRICH, resp. with J.
TRUSS.
Freitag 01.07.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
Montag 04.07.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Simon Müller (Universität Konstanz)
The O-minimal Point Counting Theorem
Abstract: The O-minimal Point Counting Theorem is originally due to Pila and Wilkie ([1]) and deals with the distribution of rational points on certain subsets of \mathbb{R}^n. In simple terms, it states that any set S ⊆ \mathbb{R}^n, which has no infinite semialgebraic subset and is definable in some O-minimal expansion of the ordered field of real numbers \overline{\mathbb{R}} := (\mathbb{R},<,+,×,0,1), contains only a few rational points. In 2013, Wilkie sketched a new proof of the same theorem ([2]) that differs from the original one mainly in its number theoretic side. In particular, it relies on the Lemma of Thue-Siegel instead of the Bombieri-Pila determinant method. The subject of this talk is to present the main steps of this new approach by Wilkie.
[1] Pila, J. and Wilkie, A. J., The rational points of a definable set, Duke Mathematical Journal, Vol. 133, No. 3, 2006, 591-616.
[2] Wilkie, A. J., Rational points on definable sets, Lectures by Prof. Alex Wilkie, University of Manchester, 2013.
Donnerstag 07.07.2016 um 17:00 Uhr, Allgemeines Mathematisches Kolloquium
Mikhail Tyaglov (Shanghai Jiao Tong University)
Hurwitz stable polynomials and a dual class of polynomials
Abstract: In the talk, I survey basic properties of Hurwitz stable polynomials and provide their relations to special matrices and orthogonal polynomials. I also introduce the class of the so-called selfinterlacing polynomials dual to the class of Hurwitz stable polynomials and having some similar properties. Self-interlacing polynomials have only real and simple roots, and have certain applications to the theory of orthogonal polynomials. I will also consider the extensions of Hurwitz stable and self-interlacing polynomials to entire functions.
Freitag 08.07.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Charu Goel (Indian Institute of Science Education and Research, Mohali)
(Gast von Maria Infusino)
Test sets for positivity of invariant forms and applications to sum of squares
Abstract: In this talk, we first explain test sets for positivity of symmetric quartics given by Choi-Lam-Reznick in 1980 and review their generalizations to symmetric and even symmetric polynomials given by Timofte in 2003. We then discuss how these test sets play an important role in establishing analogues of Hilbert's characterization from 1888, namely the pairs (n; 2d) for which a positive semidefinite n-ary 2d-ic form can be written as sum of squares of other forms, for symmetric and even symmetric forms respectively. Finally we discuss our idea to a wider genealization to analogues of Hilbert's characterization for forms invariant under the action of other groups, in particular reflection groups and Lie groups.
Montag 11.07.2016 um 15:15 Uhr, Oberseminar Modelltheorie
Nathalie Regnault (Université de Mons, Université libre de Bruxelles)
On Topological Exponential Differential Fields
(Gast von Pantelis Eleftheriou)
Abstract: Question: Does Th(R,<,exp,D), the theory of the real ordered field with exponentiation and derivation, have a model-completion?
To answer it, we work in a more general setting, dealing with topological fields of caracteristic 0 with an exponential subring on which the exponential is continuous. These structures, which we equipp with an exponential derivation D on which there isn't any continuity hypothesis, also encompass the p-adics endowed with a valuation and a partially defined exponential.
Freitag 15.07.2016 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
kein Vortrag
