Titles and Abstract of J. Yukich's course
(click on the lecture titles for the lecture notes)



Title: Probabilistic Analysis of Geometric Structures

Lecture 1: Probabilistic analysis of Euclidean optimization problems

Lecture 2: Central limit theorems for statistics of geometric structures

Lecture 3: Limit theory for statistics of geometric structures via stabilizing score functions

Lecture 4: Statistics of random polytopes

Lecture 5: Rates of multivariate normal approximation for statistics of geometric structures


Abstract:
Many questions arising in stochastic geometry and applied probability, as well as in random graphs, spatial statistics, and statistical physics, may be understood in terms of the behavior of statistics of large random geometric structures. Here the randomness often comes from an underlying point process, i.e., a random collection of points in $\R^d$, and the structure is generated from the underlying point process and heavily depends on the geometry of the points. Examples for such structures are random graphs such as nearest neighbor graphs where the vertices are the points of a point process and edges are drawn according to deterministic rules that take the geometry of the points into account, random polytopes generated as convex hulls of the underlying point process, or random tessellations such as Voronoi tessellations. Models and structures of this type are used in physics, materials science, telecommunications, and statistics of large data sets.

This course will survey methods for establishing the limit theory of statistics of geometric structures. The five lectures will focus on classical sub-additive methods as well as more recent stabilization methods. The latter tool allows one to study statistics which may be expressed as a sum of spatially dependent terms having short range interactions but complicated long range dependence.