**Title**: Distributional Lattices in Symmetric spaces.

**Speaker**: Elliot Paquette (Ohio State University)

**Abstract**: A Riemannian symmetric space X is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. A lattice in such a space can be considered as a discrete subgroup G of isometries so that a Borel fundamental domain of the quotient space G/X has finite Riemannian volume. Lattices mirror the structure of the ambient space in many ways: for example, X is amenable if and only if the the ambient space is amenable. We introduce the notion of a distributional lattice, generalizing the notion of lattice, by considering measures on discrete subsets of X having finite Voronoi cells and certain distributional invariance properties. Non-lattice distributional lattices exist in any Riemannian symmetric space: the Voronoi tessellation of a stationary Poisson point process is an example. With an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. We give some open problems related to these processes and some pretty pictures.