Integral geometry in spaces of constant curvature and applications to stochastic geometry

常曲率空间中的积分几何及其在随机几何中的应用

• 批准号: 443914364
• 负责人: Professor Dr. Daniel Hug
• 金额: --
• 依托单位: Matematicko-fyzikální fakulta
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/443914364?language=en
• 关键词: Integral; geometry; spaces; constant; curvature

Integral geometry provides indispensable tools for the mathematical analysis of random geometric systems in Euclidean space, and conversely new developments in integral geometry are often triggered by natural tasks in stochastic geometry and statistical physics. Recent advances in translative integral geometry and tensor valuations have already been successfully applied to density functional theory in physics. We plan to explore further the integral geometry of tensor-valued functions and more general homogeneous geometric valuations in Euclidean and spherical spaces. Translative integral geometry naturally involves mixed functionals of finite sequences of convex bodies. It is an important long term goal to understand how these mixed functionals are related to mixed volumes and to express all these functionals in terms of common fundamental measures such as the flag measures. Furthermore, we will study key models of stochastic geometry including random tessellations, Boolean models or random graphs in hyperbolic space. Our previous work on Poisson hyperplanes in hyperbolic space has shown that several unexpected phenomena occur in spaces of constant negative curvature. We will develop the required tools from integral geometry in hyperbolic space needed to analyze random geometric systems in hyperbolic space and deal with specific applications for instance to random tessellations and Boolean models in non-Euclidean spaces.

积分几何为欧几里德空间中的随机几何系统的数学分析提供了不可或缺的工具，相反，积分几何的新发展往往是由随机几何和统计物理中的自然任务触发的。平移积分几何和张量估值的最新进展已成功应用于物理学中的密度泛函理论。我们计划进一步探索张量值函数的积分几何以及欧几里德空间和球面空间中更一般的齐次几何估值。平移积分几何自然涉及凸体有限序列的混合泛函。了解这些混合泛函如何与混合体积相关，并用常见的基本度量（例如标志度量）来表达所有这些泛函，是一个重要的长期目标。此外，我们将研究随机几何的关键模型，包括双曲空间中的随机镶嵌、布尔模型或随机图。我们之前对双曲空间中泊松超平面的研究表明，在恒定负曲率的空间中会出现一些意想不到的现象。我们将从双曲空间中的积分几何开发所需的工具，以分析双曲空间中的随机几何系统并处理特定应用，例如非欧几里得空间中的随机镶嵌和布尔模型。