Title: Finitary isomorphisms of Poisson point processes
Abstract: As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary. This is joint work with Terry Soo.
Title: Geometry of integral vectors
Abstract: Given an integral vector, there are several geometric and arithmetic objects one can attach to it. For example, its direction (as a point on the unit sphere), the lattice obtained by projecting the integers to the othonormal hyperplane to the vector, and the vector of residues modulo a prime p to name a few. In this talk I will discuss results pertaining to the statistical properties of these objects as we let the integral vector vary in natural ways.