FOR 1548: Geometry and Physics of Spatial Random Systems

泡沫、凝胶和多孔介质等空间结构复杂的物质由于其材料特性而具有越来越重要的技术意义。这些特性高度依赖于空间结构的形状。但无序物质的形状是一个非常不连贯的概念，空间结构物质的几何和物理性质之间的关系还远未被很好地理解。流体在多孔岩石中的流动与孔隙形状之间的相互作用，泡沫的生长规律与其细胞形状和几何形状之间的相互作用，以及合成或生物材料(例如木材)的力学和几何性质之间的相互作用，仍然是物理科学中活跃的研究课题。跨学科研究股的目的是发展必要的随机几何方法，以大大加强目前对空间凝聚态物质几何性质和物理性质之间关系的了解。随机几何为随机空间几何结构提供和分析数学模型，是唯一能够同时处理复杂无序系统的几何和统计特性的数学学科。基本的数学示例是基于随机点图案的Voronoi细分、不重叠的球或其他凸体(填充)的随机系统、随机分散(可能重叠)粒子的并集(布尔模型)和(高斯)随机场的游程或水平集。随机几何正在使用和发展广泛的数学技术，例如，概率论、凸几何和积分几何、几何测度论和微分和离散几何。要成功处理拟议的专题，需要综合和改进现有的数学工具，在几何学和概率论之间创造新的概念和技术，并掌握复杂材料物理学的最新知识。主要内容包括张量赋值、镶嵌和硬核模型的均值和分布分析、布尔模型的几何描述符的探索以及随机场和连续介质渗流模型的几何性质。六个项目中有一个涉及图像分析和其他项目提出的统计问题。

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.