Alle Vorträge im Sommersemester 2017 inklusive Abstracts
Montag 10.04.17 um 15:15 Uhr, Oberseminar Modelltheorie
Gareth Jones (University of Manchester)
(Gast von Patrick Speissegger)
Pfaffian functions and elliptic functions
Abstract: After giving some motivation, I will discuss work in progress with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result.
Freitag 28.04.17 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Didier Henrion (LAAS-CNRS, Toulouse)
(Gast von Maria Infusino und Salma Kuhlmann)
Moment problems for polynomial differential equations
Abstract: The Lasserre hierarchy of semidefinite relaxations allows to solve non-convex semi-algebraic optimization problems at the price of solving a family of convex optimization problems of increasing size. The key idea consists of modeling a linear programming problem in the space of measures supported on the feasibility set and then dealing with the corresponding problems of moments.
In this talk, following a joint work with Lasserre, Prieur and Trélat, we explain how this idea can be extended to optimal control of polynomial ordinary differential equations with semi-algebraic constraints. Trajectories are modeled by occupation measures, standard objects in Markov decision processes and dynamical systems. We argue that it may be worthwhile to pursue an alternative approach consisting of using measures supported on infinite-dimensional Banach spaces of solutions. This would readily allow for extensions to nonlinear stochastic differential equations or partial differential equations.
Freitag 12.05.17 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Patrick Speissegger (McMaster University / 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.
Freitag 19.05.17 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Cordian Riener (Universität Konstanz)
Semi-definite programming and arithmetic progression
Abstract: Let p be a prime number and Fp be the field with p elements. Then a subset P⊆Fp is called a k-term arithmetic progression, if it is of the form P = {u + jv : j = 0,..., k-1} with u∈Fp, v∈Fp*.
In the talk I will present a approach based semi-definite programming and symmetry reduction to obtain bounds on the minimal number of k-term arithmetic progressions every fixed-density subset of Fp contains. (Joint work with Aron Rahman and Frank Vallentin)
Donnerstag 08.06.17 um 17:00 Uhr, Schwerpunktskolloquium Reelle Geometrie und Algebra
Niels Schwartz (Universität Passau)
(Gast von Markus Schweighofer)
Spectral spaces and localic spaces
Abstract: The spectral spaces (with the spectral maps as morphisms) form a subcategory, Spec, of the category Top of topological spaces. Spectral spaces can be defined by a list of topological properties. As Hochster showed, they are the Zariski spectra of rings (commutative with unit), equivalently the prime ideal spectra of bounded distributive lattices (Stone). The lecture focuses on the relationship between Spec and Top. It is a rather simple fact that Spec is a reflective subcategory of Top. The
spectral reflection of a topological space belongs to a special class of spectral spaces, which are called localic spaces. These also arise naturally in the area of pointfree topology. The relationship between a topological space and its spectral reflectionwill be explained.
Freitag 23.06.17 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Mario Kummer (Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig)
(Gast von Claus Scheiderer)
Separating Morphisms from Real Algebraic Curves
Abstract: Given a real algebraic curve we consider the set of all morphisms to the projective line with the property that the preimage of every real point consists entirely of real points. It turns out that this generalises the notion of interlacing polynomials on the real line to projective curves. Using this theory, we will answer a question raised by Shamovich and Vinnikov on hyperbolic curves as well as a question by Fiedler-LeTouzé on totally real pencils on plane curves. (Joint work with Kristin Shaw)
Freitag 07.07.17 um 13:30 Uhr, Oberseminar Reelle geometrie und Algebra
Jan Rolfes (Universität zu Köln)
(Gast von Cordian Riener)
A greedy approach to cover compact metric spaces by balls
Abstract: In this talk I give a general algorithmic framework to construct coverings of compact metric spaces X by balls. This framework is based on of Chvátal's approximation algorithm for the weighted set cover problem. We prove a logarithmic bound in terms of the size of the balls for every compact metric space. When X=Sd we give a constructive covering of Sd with a density comparable to the currently best one by Böröczky and Wintsche.
Freitag 14.07.17 um 13:30 Uhr, Oberseminar Reelle Geometrie und Algebra
Christoph Schulze (Universität Konstanz)
Asymptotic Behaviour of the Dimensions of the Sets of Non-Negative Extremal Forms
Abstract: We study the sets of extremal forms of the cone of nonnegative forms Pn,2d in n+1 (n≥2) homogeneous variables of degree 2d and the projective dimensions of their Zariski-closures inside R[x0,...,xn]2d. We will show lower and upper bounds for these dimensions for d≥d0(n). If n=2 and 2d≥24 we obtain that this dimension lies between 2⁄3 {2d+2 \choose 2}-20⁄3 and 2⁄3 {2d+2 \choose 2}+4⁄3. These estimations are based on the construction of sets of low-dimensional faces which generate Pn,2d. The Alexander-Hirschowitz theorem and implications considering the sets of special points of interpolation problems will play a central role.
