FOR 1548: Geometry and Physics of Spatial Random Systems

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Spatially complex structured matter such as foams, gels and porous media is of increasing technological importance due to its material properties. These properties are highly dependent on the shape of the spatial structure. But the shape of disordered matter is a remarkably incoherent concept and the relationship between geometric and physical properties of spatially structured matter is far from being well understood. The interplay between liquid flow through porous rock and the shape of the pores, between growth laws for foams and their cell shape and geometry, and between the mechanical and geometric properties of synthetic or biological materials (e.g. wood) remain active topics of research in the physical sciences. The purpose of the interdisciplinary Research Unit is to develop the stochastic geometry methodology necessary to significantly enhance the current understanding of the relationships between geometric and physical properties of spatial condensed matter. Stochastic geometry provides and analyses mathematical models for random spatial geometric structures and is the only mathematical discipline that can cope with both the geometric and the statistical properties of complex disordered systems. Fundamental mathematical examples are Voronoi tessellations based on random point patterns, random systems of non-overlapping balls or other convex bodies (packings), union sets of randomly scattered (possibly overlapping) particles (Boolean models) and excursion or level sets of (Gaussian) random fields. Stochastic geometry is using and developing a wide range of mathematical techniques, for instance, from probability theory, convex and integral geometry, geometric measure theory and differential and discrete geometry. A successful treatment of the proposed topics requires the synthesis and improvement of existing mathematical tools, the creation of new concepts and techniques at the borderline between geometry and probability theory and a state-of-the-art knowledge in the physics of complex materials. The main topics are tensor valuations, mean value and distributional analysis of tessellations and hard-core models, the exploration of geometric descriptors for Boolean models and geometric properties of random fields and continuum percolation models. One of the six projects is concerned with image analysis and statistical problems raised by the other projects.