FB 6 Mathematik/Informatik

Institut für Mathematik

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WS 2016/2017

25.10.2016 um 16:15 Uhr in 69/125:

Jan-Marten Brunink (Universität Osnabrück)

Volume bounds for lattice polytopes with few interior lattice points

We will discuss the volume of lattice polytopes with few interior lattice points, i.e., polytopes with vertices in Zd and k interior points in Zd for k ≥ 1 sufficiently small. By bounding the barycentric coordinates of the interior points, Hensley showed, that the volume for lattice polytopes with at least one interior lattice point is bounded (while the volume can be arbitrarily large for polytopes without any interior lattice point). Recently, Averkov, Krümpelmann and Nill found sharp bounds for the volume in the case of lattice simplices with exactly one interior lattice point and characterized the volume-maximizing simplices. Generalizing their approach we consider simplices with at least one interior lattice point and improve the best known bound for the volume of such simplices.

01.11.2016 um 16:15 Uhr in 69/125:

Jens Grygeriek (Universität Osnabrück)

Central limit theorem for edge counts in high-dimensional random geometric graphs

We use a Poisson point process as a random set of points in Rd and connect any two different points with an edge whenever their distance is less than or equal to a prescribed distance parameter. This leads us to the well known andom geometric graph and we count the number of edges of this graph that have their midpoint in the d-dimensional unit ball. We derive a central limit theorem for this counting statistic as the space dimension d tends to infitiy.

15.11.2016 um 16:15 Uhr in 69/125:

Patrick Graf (Universität Bayreuth)

Finite quotients of complex tori

A complex torus is the quotient of a finite-dimensional complex vector space by a discrete cocompact subgroup. After reviewing some basic facts about the automorphism group of a complex torus and its finite subgroups, I will discuss how to recognize quotient spaces of complex tori by finite groups in terms of topological data called Chern classes. I will report on work in progress settling the question in complex dimension three. If time permits, I will also discuss the higher-dimensional case. Joint with Tim Kirschner (Essen).

22.11.2016 um 16:15 Uhr in 69/125:

Alexandros Grosdos Koutsoumpelias (Universität Osnabrück)

Gaussian Conditional Independence Ideals and Primary Decompositions

Conditional independence statements express the intuitive idea that knowledge about a random variable does not reveal any information about a second random variable under some assumptions. After translating collections of such statements for normal random vectors into algebraic language we are able to deduce information about them using algebraic tools. Following work by Drton and Xiao, this allows us to identify all complete conditional independence relations on random vectors of small length and test their corresponding varieties for smoothness. We also show that Gaussian conditional independence statements have no finite complete characterisation.

29.11.2016 um 17:15 Uhr in 69/125:

Dr. Alexey Ananyevskiy (St. Petersburg State University)

Algebraic K-theory of some homogeneous varieties

K-theory is a powerful tool that allows to study vector bundles on a variety and provides substantial geometric information. Although algebraic K-theory is universal in a certain precise sense and is very hard to compute in general, it turns out that one can perform some explicit calculations as soon as one constructs enough vector bundles on a variety. In particular, we describe K-theory of twisted forms of smooth affine homogeneous varieties G/H with G and H being split reductive algebraic groups of the same reductive rank. If time permits, I will also discuss some combinatorics of roots, weights and Weyl groups that stays behind the computation.

13.12.2016 um 16:15 Uhr in 69/125:

Dominik Nagel (Universität Osnabrück)

Dual Certificate in Super-Resolution

In this talk we are looking closer to a super-resolution problem which is the reconstruction of an exponential sum on the torus knowing a limited amount of sampled data. After giving an understanding of the problem the solution process of Candès et al. using convex optimization tools is shown briefly. In this context the meaning of a so called Dual Certificate is explained. This mathematical statement guarantees a unique solution to the convex program. A proof is given for our setting.

10.01.2017 um 16:15 Uhr in 69/125:

Carina Betken (Universität Osnabrück)

Stein's Method and the Preferential Attachment Model

The preferential attachment model is a dynamic random graph model, in which new vertices are attached to old vertices with probability proportional to their current degree. We will look at the degree distribution of a randomly chosen vertex in such a graph. The idea to prove convergence to a supposed limit distribution is to useStein's method, which will be explained in the talk.