Title: Homogeneous dynamics and diagonalizable actions
Abstract: Homogeneous dynamics and diagonalizable actions Algebraic actions on quotient spaces G/Gamma give highly interesting dynamical systems that are in many cases intimately linked with number theory. One of the most powerful ways to study these dynamical systems is to study the collection of invariant probability measures.
Such a study forms the core of Ratner's landmark proof of the Raghunathan conjecture regarding orbits of unipotent groups. Already in the 1950's Cassels and Swinnerton Dyner observed (in a dual language) that higher rank diagonal groups have interesting rigidity properties which are still far from being understood. I will survey the progress made in this direction, particularly regarding the measure classification question, as well as the challenges that remain.