Stochastic Geometry and Random Graphs

Monday 12:00-14:00 Uhr

Thursday 12:00-14:00 Uhr

This lecture course is devoted to the study of random geometrical objects and structures. Among the most prominent models are random polytopes, random tessellations, particle processes and random graphs. All of these models play an important role in numerous applied problems from natural science and computer science and their study is a modern and actively developing area of probability theory called Sctochastic Geometry and Random Graphs.

In the first part of the lecture course we will get to know some basics concepts of Stochastic and Integral Geometry. In particular we will define such objects as a random closed set, a Poisson point process and a particle process. Later we will consider a few classical models like a random tessellation, which is a random tilling of the space by convex polytopes, and the Boolean model, which is a union of random balls. Some typical questions would be: How does the "typical" polytope of random tessellation look like? Which proportion of space is occupied by the balls of Boolean model in average? In the second part we will get to know a few random graph models, including random geometric graph, Erdös-Renyi graph and preferential attachment model.

The preknowledege in Measure Theory are highly recommended, the preknowledege in Probability Theory are necessary.