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Explore the mathematical foundations of Poisson point processes and their applications to geometric group theory in this advanced mathematics lecture. 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 geometric group theory through joint research with Mikolaj Fraczyk, specifically examining the relationship between manifold volume and the number of generators in fundamental groups. Learn about the groundbreaking result that for higher rank semisimple Lie groups, the minimum number of generators in a lattice grows sublinearly with respect to covolume. Follow the detailed unpacking of this proof methodology, with no prior knowledge of Poisson-Voronoi tessellations, fixed price theory, or higher rank structures required, making advanced geometric and algebraic concepts accessible to a broader mathematical audience.
