Georgia Institute of Technology College of Sciences

We define and motivate the Poisson point process, which is, informally, a “maximally random” scattering of points in some locally compact, second countable space. We introduce the ideal Poisson--Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove the minimal number of generators of a torsion-free lattice in a higher rank, semisimple Lie group is sublinear in the co-volume of the lattice. We give some intuition for the proof. No prior knowledge on Poisson point processes or symmetric spaces will be assumed.