Integral geometry in spaces of constant curvature and applications to
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.