Mathematical Conversations

The classical Birkhoff individual ergodic theorem states that in the presence of an ergodic invariant measure, almost every orbit is uniformly distributed with respect to the measure. For many applications (in particular to number theory), it is necessary to understand the topological and statistical properties of every (and not only almost every) orbit. I will briefly survey the history of this subject, starting with the foundational work of Ratner and Margulis, and hopefully get to some recent developments which changed my perspective of the area.