11.12.2019 um 16:15 Uhr (!) in 69/125
Prof. Dr. Heidemarie Bräsel (Otto-von-Guericke-Universität Magdeburg)
Ethnomathematik – Geometrie auf drei Kontinenten
10.12.2019 um 13:00 Uhr in Raum 69/E15 -Uni-GK-
An exploration of the truncated cylinder model
Consider a Boolean model and a random direction movement scheme. Given a fixed time horizon T>0, we model these movements via cylinders reaching from the original plane of the boolean model at time 0 to an affine plane at time T. We study the asymptotic behavior of functionals of the union set of cylinders Z restricted to a window of observation W, which has its volume tending to infinity. Considered functionals include the 3-dimensional volume of the intersection of Z and W and its Euler characteristic.
17.12.2019 um 12:00 Uhr in Raum 69/E15
Random Geometric Structures
In the first part, we use a stationary Poisson point process to construct two random simplicial complexes, the random Vietoris-Rips complex and the random Cech complex, that are rich enough to realize any compact topological manifold, at least up to homotopy equivalence.
In the second part, we investigate the random Vietoris-Rips complex constructed from a stationary Poisson point process on a d-dimensional compact set using the uniform norm.
Opposed to most of the existing literature in which the focus lies on random geometric graphs in the fixed dimensional case, we will generalize this model by investigating the asymptotic distributional behavior of the f-vector if the intensity as well as the space dimension d tend to infinity simultaneously.
It turns out, that the phase-transition phenomenon occurs also in the components of the f-vector of the random Vietoris-Rips complex in the fixed and high-dimensional cases.