Poisson-Voronoi Tessellations and Fixed Price in Higher Rank - 3/5

Explore the mathematical foundations of Poisson point processes and their applications to geometric group theory in this 58-minute lecture from Institut des Hautes Etudes Scientifiques. Begin with the definition and motivation behind Poisson point processes, understood as maximally random distributions of points in space, before delving into the ideal Poisson–Voronoi tessellation (IPVT) and its fascinating geometric properties on semisimple symmetric spaces like the hyperbolic plane. Discover how this new random geometric object connects to fundamental questions about the relationship between manifold volume and the number of generators in fundamental groups, particularly for higher rank semisimple Lie groups where the minimum number of generators in a lattice exhibits sublinear growth relative to covolume. Follow the detailed unpacking of a proof developed in collaboration with Mikolaj Fraczyk that demonstrates these connections, with no prior knowledge of Poisson–Voronoi tessellations, fixed price theory, or higher rank structures required for understanding.