School of Mathematics/Computer Science

Institute for Mathematics

SS 2018

11.04.2018 um 17:15 Uhr in Raum 69/125

Dr. Lutz Kämmerer (Universität Osnabrück)

Spatial Discretizations for Fast Fourier Transforms of Sparse Multivariate Trigonometric Polynomials

Based on considerations on three different concepts - random sampling, sparse grids, rank-1 lattices - for constructing spatial discretizations for arbitrary sparse multivariate trigonometric polynomials, we develop a new approach called multiple rank-1 lattices that combines the advantages of the aforementioned

concepts with respect to
- efficient algorithms for evaluation and reconstruction
- oversampling
- approximation properties

Subsequently, we present the dimensional-incremental sparse fast Fourier transform as one application, where the recently developed discretization concept leads to a significant reduction of the computational complexity and reliable results.

18.04.2018 um 17:15 Uhr in Raum 69/125

Prof. Dr. Oliver Röndigs (Universität Osnabrück)

24

During my sabbatical at the Institut Mittag-Leffler and the Hausdorff-Institut, I obtained
a result in which the number 24 appears. This colloquium talk will mention several properties
of this interesting number, in the hope of being entertaining for at least 24 minutes.

26.04.2018 um 17:15 Uhr in Raum 69/125

Prof. Dr. Bernd Sturmfels (MPI Leipzig)

Learning Algebraic Varieties from Samples

This lecture discusses the role of algebraic geometry in data science.
We report on recent work with Paul Breiding, Sara Kalisnik and Madeline Weinstein.
We seek to determine a real algebraic variety from a fixed finite subset of points.
Existing methods are studied and new methods are developed. Our focus lies on
topological and algebraic features, such as dimension and defining polynomials.
All algorithms are tested on a range of datasets and made available in a Julia package.

02.05.2018 um 17:15 Uhr in Raum 69/125

Dr. Yurii Kolomoitsev (Universität zu Lübeck)

On the Growth of Lebesgue Constants for Convex Polyhedra

The talk is devoted to the Lebesgue constants of polyhedral partial sums of Fourier series. New upper and lower
estimates of the Lebesgue constant in the case of anisotropic dilations of general convex polyhedra will be presented.
These estimates are applied to analyze the absolute condition number of the multivariate polynomial interpolation
on the Lissajous-Chebyshev node points.
This is joint work with P. Dencker, W. Erb (University of Lübeck), and T. Lomako (Institute of Applied Mathematics
and Mechanics of NAS of Ukraine).

23.05.2018 um 17:15 Uhr in Raum 69/125

Prof. Dr. Tài Huy Hà (Tulane University, New Orleans, USA)

Combinatorial structures through algebraic lenses

We shall discuss an algebraic approach to investigate a number of important invariants and structures in graph theory and integer linear programming. Particularly, we shall focus on how to compute the chromatic numbers, how to recognize the existence of odd cycles in graphs, and how to detect integer linear programming systems with packing and max-flow-min-cut properties.

30.05.2018 um 17:15 Uhr in Raum 69/125

Prof. Dr. Ngô Việt Trung (Institute of Mathematics Hanoi, Vietnam)

Depth of powers of homogeneous ideals

Let I be an arbitrary homogeneous ideal in in a polynomial ring R. Brodmann showed that the function depth R/It = dim R - pd R/It (where pd is the projective dimension) is always a constant for t >> 0. In other word, it is a convergent function. Subsequent works have indicated that this function may behave wildly. Herzog and Hibi conjectured that any convergent non-negative numerical function is the depth function of powers of a homogeneous ideal. We will solve this conjecture by showing that any convergent non-negative numerical function is the depth function of powers of a monomial ideal. The lecture is about the basic ideas of the solution, which are rather elementary and should be understandable for everybody.

06.06.2018 um 17:15 Uhr in Raum 69/125

Prof. Dr. Sylvie Roelly (Universität Potsdam)

Random Sphere Packing

We consider a system of hard balls in Euclidean space, undergoing random dynamics and interacting via a mutual attraction force. Such stochastic evolutions converge asymptotically in time towards equilibrium states, which are connected with the famous geometry problem of close-packing of equal spheres.

13.06.2018 um 17:15 Uhr in Raum 69/125

Jun.-Prof. Dr. Daniel Rudolf (Universität Göttingen)

Perturbation Theory for Markov Chains

Perturbation theory for Markov chains addresses the question how small differences in the transition mechanism of Markov chains are reflected in differences between their distributions after a certain number of steps. In the statistical analysis of datasets the approximate simulation/sampling of posterior distributions is of particular interest. For this goal approximations of transition kernels can be used (and considered as perturbations) which might lead to implementable algorithms and a reduction of the computational cost. However, they also might change the invariant distribution, such that posterior and stationary distribution do not coincide anymore. By using the perturbation theory we derive estimates of the bias of such approximations of geometrically ergodic Markov chains. We illustrate the result by considering a Monte Carlo within Metropolis algorithm.

20.06.2018 um 17:15 Uhr in Raum 69/125

Cone Angles - An Interplay of Geometry and Combinatorics

Euler’s Polyhedron Formula and it’s generalization, the Euler-Poincare formula, is a cornerstone of the combinatorial theory of polytopes. It states that the number of faces of various dimensions of a convex polytope satisfy a linear relation and it is the only linear relation (up to scaling). Gram’s relation generalizes the fact that the sum of (interior) angles at the vertices of a convex $n$-gon is $(n-2)\pi$. In dimensions $3$ and up, it is necessary to consider angles at all faces. This gives rise to the interior angle vector of a convex polytope and Gram’s relation is the unique linear relation (up to scaling) among its entries. In this talk, we will consider generalizations of “angles” in the form of cone valuations. It turns out that the associated generalized angle vectors still satisfy Gram’s relation and that it is the only linear relation, independent of the notion of “angle”! To proof such a result, we rely on a very powerful connection to the combinatorics of zonotopes and hyperplane arrangements. If time permits, we will ponder the notion of flag-angle vectors, a semi-discrete counterpart to flag-vectors of polytopes, and the linear relations satisfied by flag-angle vectors. This is joint work with Spencer Backman and Sebastian Manecke.

04.07.2018 um 17:15 Uhr in Raum 69/125

Prof. Dr. Kristina Reiss (Technische Universität München)

ALICE:Bruchrechnen: Arbeiten mit dem Tablet PC im Unterricht der Sekundarstufe

Bruchrechnen ist ein schwieriger Teilbereich der mathematischen Grundbildung, so dass die meisten Schülerinnen und Schüler hier Unterstützung beim Lernen benötigen. Bisherige Forschungsarbeiten gehen davon aus, dass ein handlungsorientiertes Arbeiten und folglich die aktive Veränderung mathematischer Darstellungen durch die Schülerinnen und Schüler den Lernprozess unterstützen kann. In diesem Zusammenhang können mobile elektronische Geräte eine Möglichkeit zur Umsetzung geeigneter Lernumgebungen sein.

Im Forschungsprojekt ALICE:Bruchrechnen wird untersucht, inwieweit der Einsatz von Tablet-PCs einen Einfluss auf den Anfangsunterricht in der Bruchrechnung hat. Dafür wurde ein digitales Lehrbuch zur Verwendung auf iPads entwickelt, das besonderen Wert auf interaktive und adaptive Aufgaben legt, die einen Darstellungswechsel nicht nur erlauben, sondern zumeist auch fordern. Zur Evaluation nahmen Sechstklässlerinnen und Sechstklässler aus unterschiedlichen Schularten in drei Gruppen an einer vierwöchigen Interventionsstudie teil. Gruppe 1 arbeitete mit dem iBook auf iPads, Gruppe 2 mit der gedruckten Version des iBooks in Buchform (Experimentalgruppen). Zur Kontrolle von Effekten durch das Unterrichtsmaterial arbeitete Gruppe 3 mit konventionellen Schulbüchern.

Im Vortrag wird die Entwicklung der Lernumgebung mit Hilfe von iBooks Author in Bezug auf theoretische Überlegungen und die praktische Umsetzung geschildert. Darüber hinaus werden Ergebnisse der empirischen Studie vorgestellt. Insbesondere werden einerseits Produkte des Lernens in Form von Leistungsdaten und andererseits Prozesse des Lernens in Form von Lösungswegen betrachtet.

Das Projekt ALICE:Bruchrechnen wird durch die Heinz-Nixdorf-Stiftung gefördert.

Leitung: Kristina Reiss und Jürgen Richter-Gebert / Mitarbeiter: Stefan Hoch, Frank Reinhold und Bernhard Werner