Institute of Stochastics

Stochastic Geometry II

Time and Place

Lecture

Wednesday, 10:15 - 11:55, E03, Heho 22

Exercise classes

Friday, 12:15 - 13:55, 120, Heho18

 

Type

2 hours lecture, 1 hour exercise classes

4 credit points

Prerequisites

Basic knowledge of probability like taught in "Elementare Wahrscheinlichkeitsrechnung und Statistik". The lecture "Stochastic Geometry" is not necessary.

Content

Stochastic geometry deals with random geometric structures. In this lecture we will treat two selected topics from stochastic geometry:

 Random compact sets

  • The convex hull of random points: We consider the convex hull of finitely many points distributed uniformly within a given compact set (left picture). We will examine stochastic properties (e.g. the expected value) of geometric quantities (e.g. the area) of this convex hull. 
  • The strong law of large numbers: The arithmetic mean convergences to the expected value for random compact sets as well if we define these notions appropriately. 

Random tessellations

  • Definition and elementary properties: Tessellations partition the Euclidean space into so-called cells. 
  • Mean values for geometric quantities of a cell: We will define the notion of the typical distribution of a cell and derive formulas for the expected values of various geometric quantities, e.g. the area or the number of neighboring cells.
  • Models for random tessellations: We will treat the most important models of random tessellations: Voronoi tessellation (middle picture), hyperplane tessellation (right picture) and STIT tessellation.

Exam

There is no prerequisite. Further details will be announced later. 

Exercise sheets

Submitting solutions for the exercise sheets is strongly recommanded.

Literature

  • Schneider, R. and Weil, W.: Stochastic and Integral Geometry. Springer, 2008.
  • Molchanov, I.: Theory of Random Sets. Springer, 2005.
  • (Chiu, S.,) Stoyan, D., Kendall, W. and Mecke, J.: Stochastic Geometry and Its Applications. Wiley, 1995, 2008, 2013.
  • Spodarev, E.: Stochastic Geometry, Spatial Statistics and Random Fields. Springer, 2013.
  • Schmidt, V..: Stochastic Geometry, Spatial Statistics and Random Fields. Springer, 2015.

 

Contact

Lecturer & Teaching assistant

Dr. Jürgen Kampf

Helmholtzstraße 18, Raum 1.46

0731 50 23526

juergen.kampf(at)uni-ulm.de

News

The language of the course has been changed to german.

Only the german version of this page will be updated further.