In applied sciences, e.g. geology, medicine and material sciences, computer devices like digital cameras, digital microscopes and diverse kinds of tomographic scanners produce a large amount of digital data of some geometric structures. Many applications do not require an actual reconstruction of the underlying geometric structure but only the estimation of basic characteristics, like area, volume, boundary length or particle number, to name just a few. Digital stereology is a mathematical discipline which yields methods for this purpose and describes the quality of such estimation procedures.

It is the purpose of this talk to give a mathematical introduction to this field. We will start with a short history of (continuous) stereology and in particular discuss the Crofton formula, which describes the estimation of geometric characteristics from random planar sections. After clarifying the mathematical notion of digitization, we will discuss how volume and other natural characteristics are commonly estimated from digital images. Many of these estimators are derived by discretization from classic estimators of continuous stereology. The quality of such estimators is therefore not known or only assured under strong and often unrealistic assumptions.

Considering as example the estimation of boundary length in the plane, several new results are presented, which are more suited as they use a geometric approach. They actually allow to approximate the length measure (a local counterpart of the boundary length) of the planar structure. Generalizations, among others to higher dimensions, will conclude the talk.

The lecture will give a short biography of Thiele and a kaleidoscope of some of his many activities and interests, including the design of floor tilings and insurance for unmarried women.

We discuss a stochastic differential equation, as a modelling framework for the turbulent velocity field, that is capable of capturing basic stylized facts of the statistics of velocity increments. In particular, we focus on the evolution of the probability density of velocity increments characterized by a normal inverse Gaussian shape with heavy tails for small scales and aggregational Gaussianity for large scales. In addition, we show that the proposed model is in accordance with Kolmogorov's refined similarity hypotheses.