#### The workshop will be held at Ludwig-Windthorst-Haus Lingen in Germany from 22nd to 27th March 2015.

## Invited Speakers

Imre Bárány (Alfréd Rényi Mathematical Institute, Budapest)

**Longest Convex Chains**

Assume *X_n* is a random sample of n uniform, independent points from a triangle T with vertices *A, B, C*. A subset *Y ⊂ X_n* is a convex chain if every point from *Y* is a vertex of the convex hull of *A, C*, and *Y*. A longest convex chain, *Y,* of *X_n* is a convex chain with largest cardinality. The length *L_n=|Y|* of a longest convex chain is a random variable which is a distant relative of the much studied longest increasing subsequence. In this talk we determine the order of magnitude of the expectation of *L_n*. We show further that *L_n* is highly concentrated around its mean, and that the longest convex chains have a limit shape.

Anne Estrade (Université Paris Descartes)

**A Central Limit Theorem for the Euler Characteristic of Gaussian Excursions**

Abstract: We study the Euler characteristic of an excursion set of a real stationary isotropic Gaussian random field defined on **R**^d. Let consider a fixed level *u * and also the excursion set above this level, *A(T,u)=*{t ∈ T: X (t) ≥ u} where *T* is a bounded rectangle in **R**^d. The aim of this paper is to establish a central limit theorem for the Euler characteristic of* A(T,u)* as *T* grows to **R**^d, as conjectured by R. Adler more than ten years ago.

The required assumption on *X* is having trajectories in C³. It is stronger than Geman’s assumption traditionally used in dimension one. Nevertheless, our result extends to higher dimension what is known in dimension one. In that case the Euler characteristic of *A(T,u)* equals the number of up-crossings of *X* at level *u.* This is a joint work with Jose Rafael Leon, Universidad Central de Venezuela.

Markus Kiderlen (Aarhus University)

**Rotational Crofton Formulae**

Abstract: The invariator principle is a Blaschke-Petkantschin-type result. It states, roughly speaking, how a flat in an isotropic subspace must be chosen in order to obtain an invariant at in *n*-dimensional space. A combination of this principle with the (kinematic) Crofton formula leads to rotational Crofton formulae. After giving an overview of classical Crofton formulae for intrinsic volumes and certain other valuations on convex bodies, we formulate known and new rotational Crofton formulae for these functionals. From a special case of these results one can obtain an unbiased estimator of surface area in local stereology. The calculation of this estimator is based on a renement of the classical tangent count method for IUR (isotropic uniform random) planes. As an illustrating example, we estimate the surface area of the nuclei of giant-cell glioblastoma from microscopy images.

This talk is based on joint work with E.B.V. Jensen, A.H. Rafati and O. Thorisdottir.

Wolfgang König (Weierstraß-Institut Berlin)

**Cluster Size Distribution in Classical Many-Body Systems with Lennard-Jones Potential**

Abstract: We study a classical many-body system with pair-interaction given by a stable Lennard-Jones potential. This interaction has a repellent term that prevents the particles from collapsing, and an attractive term, which induces the formation of clusters of the particles. For fixed inverse temperature and fixed particle density, we derive a large-deviation principle for the distribution of the cluster sizes in the thermodynamic limit. Afterwards, we show that the rate function Gamma-converges, in the low-temperature dilute limit towards some explicit rate function. This function has precisely one minimising cluster size configuration, which implies a kind of law of large numbers for the cluster sizes in this decoupled limit. This is joint work with S. Jansen and B. Metzger. We also report on ongoing work with S. Jansen, B. Schmidt and F. Theil on details of the low-temperature behaviour of the system for fixed particle density in one dimension.

Raphaël Lachièze-Rey (Université Paris Descartes)

**Voronoi Approximation of Irregular Sets
**Abstract: Let K be an unknown set lying in, say, [0,1]^d, and suppose that one disposes of the information of which points belong to K among a random sample of points X in [0,1]^d, typically Poisson or binomial. A possible way to approximate K is to take the union of all Voronoi cells based on X with center in K, yielding the Voronoi approximation of K. We present here some results about the asymptotic quality of this approximation. We give in particular Berry-Esseen bounds for the Kolmogorov distance between the volume of K and that of its approximation, as well as an almost sure result of convergence for the renormalised Hausdorff distance.

One of our aim was to determine the minimal regularity assumptions K has to verify to obtain a good approximation. Like in many works about set estimation, our results require that the boundary density of K is bounded from below as one zooms in on the boundary. Apart from that, no convexity or smoothness assumption is needed, and we are able for instance to apply the results to sets with a self-similar boundary, such as the Von Koch flakes and antiflakes in dimension 2.

In collaboration with Giovanni Peccati, Sergio Vega.

Klaus Mecke (Universität Erlangen)

**Geometry and Physics of Random Spatial Structures
** A morphometric analysis of stochastic geometries is introduced by using tensorial valuations, i.e., tensor-valued Minkowski functionals. Tensorial physical properties such as elasticity, permeability and conductance of microstructured heterogeneous materials require quantitative measures for anisotropic characteristics of random spatial structure. Tensor-valued Minkowski functionals, defined in the framework of integral geometry, provide a concise set of descriptors. The talk provides an overview on the application on stochastic geometries used in physics. A robust computation of these measures is presented for microscopy images and polygonal shapes by linear-time algorithms. Their relevance for shape description, their versatility and their robustness is demonstrated by applying them to experimental datasets, specifically microscopy datasets. Applications are shown in two dimensions on Turing patterns and on sections of ice grains from Antarctic cores. In three dimensions Minkowski tensors have been used to quantify the anisotropy of fluids and granular matter, of confocal microscopy images of sheared biopolymers and of triply-periodic minimal surface models for amphiphilic self-assembly.

Christoph Schnörr (Universität Heidelberg)

**Segmentation of Random Structures in Image Data**

Segmentation of image structure requires model-based contextual decisions in order to cope with noise and local ambiguities. This talk surveys current techniques of image segmentation, with a focus on spatial Markov random fields and their shortcomings, as regards the representation of extended shapes and large-scale inference for analyzing 3D image data. As a consequence, variational inference (as opposed to MCMC) in connection with more involved marked point process models, and more generally the interplay of stochastic geometry and image analysis, constitute important directions of research.

Dominic Schuhmacher (Universität Göttingen)

**Old and New Covergence Rates for Thinnings of Point Processes
**Abstract: We consider random thinnings of point processes via retention probabilities supplied by a [0,1]-valued random field. It is well known that, under conditions, an increasingly dense process thinned by an asymptotically vanishing retention field approximates a Cox or even Poisson process distribution. We give an overview of various old and new rates for this convergence and present some new directions.

Wolfgang Weil (Karlruher Institut für Technologie)

**Valuations, Flag Measures and Boolean Models**

Abstract: Valuations (additive functionals) on convex bodies comprise a very active area of current research. Flag measures are local analogs of the intrinsic volumes and generalize curvature and surface area measures. Both notions have interesting applications in Integral and Stochastic Geometry, in particular to Boolean models. In the talk we study the interrelations between these notions, we present some recent results and discuss open problems.