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Explore the mathematical foundations of Poisson point processes and their applications to geometric group theory in this 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 such as the hyperbolic plane. Discover how this new random geometric object connects to fundamental questions in mathematics through joint research with Mikolaj Fraczyk that establishes relationships between manifold volume and the number of generators in fundamental groups. Learn how for higher rank semisimple Lie groups, the minimum number of generators in a lattice exhibits sublinear growth relative to covolume, and follow the detailed proof of this significant result. Gain insights into advanced topics in geometric group theory, random geometry, and Lie group theory through clear explanations that assume no prior knowledge of Poisson-Voronoi tessellations, fixed price theory, or higher rank structures, making complex mathematical concepts accessible to researchers and advanced students in mathematics.
