Vorträge in der Woche 16.11.2015 bis 22.11.2015

Montag, 16.11.2015: Geometry of unweighted k-nearest neighbor graphs

Prof. Dr. Ulrike von Luxburg (Uni Tübingen, Fachbereich Informatik)

Consider the following game: It is you versus me. You pick a secret probability density function on R^d and sample n points from it. Then you build the k-nearest neighbor graph based on this sample: vertices are the sample points, and you connect two points by an unweighted edge if they are among the k nearest neighbors of each other. Now you give the adjacency matrix of the graph to me, but nothing else (neither the point coordinates, nor any information about the distances between the points). This means that all I get to know about your points is who is among the k nearest neighbors of whom. My task is to find out as much as possible about the secret density function that you had chosen in the beginning. What do you think: Can I reconstruct (some aspects of) the underlying density function, or even your original point configuration? Just think about it for a couple of minutes :-) Answers will be provided in the talk. On a higher level, I will relate my findings to the field of ordinal data analysis - with many open questions somewhere between geometry, topology and statistics.

Dienstag, 17.11.2015: Algebraic Groups and Measure Theory

Katrin Schlegel

Donnerstag, 19.11.2015: Best approximation property of the finite element solutions for parabolic problems

Dimitriy Leykekhman (University of Connecticut)

Finite element error analysis of state constrained optimal control problem or optimal control problems with pointwise controls often require pointwise error estimates in form of the best approximations due to low regularity of the optimal state or adjoint variables. Such best approximation error estimates are well known for elliptic problems, but for the parabolic problems such results are only available for the semidiscrete and not for the fully discrete approximations. In my talk, after reviewing pointwise best approximation properties for elliptic problems and I show how such global and local best approximation results follow from our recently established results on discrete maximal parabolic regularity for a family of discontinuous Galerkin time discretization methods and the stability of the Ritz projection in the maximum norm.

Donnerstag, 19.11.2015: Derivation of the Ginzburg-Landau Theory of Superconductivity via Bifurcation Theory

Dr. Tim Tzaneteas (Tuebingen)