Poisson-Voronoi Tessellations and Fixed Price in Higher Rank - 1/5

Explore the fundamentals of Poisson point processes and their applications to geometric group theory in this mathematical lecture from Institut des Hautes Etudes Scientifiques. Begin with an introduction to the Poisson point process, described as a "maximally random" distribution of points in space, before delving into the ideal Poisson–Voronoi tessellation (IPVT), a novel random geometric object with fascinating properties when applied to semisimple symmetric spaces such as the hyperbolic plane. Discover how this mathematical framework, developed in collaboration with Mikolaj Fraczyk, provides insights into the relationship between manifold volume and the number of generators in fundamental groups, specifically proving that for higher rank semisimple Lie groups, the minimum number of generators in a lattice grows sublinearly with respect to covolume. Follow along as the proof is systematically unpacked throughout this first installment of a comprehensive minicourse, with no prior knowledge of Poisson–Voronoi tessellations, fixed price theory, or higher rank mathematics required for understanding.