Thomas Ohlson Timoudas: Pointwise Ergodic Theory on Groups and Ratner’s Theorems

Ergodic theory studies the long-time behaviour of dynamical systems. The strongest results are pointwise, they say something about the long-time behaviour of a point under some transformation.

What happens if we restrict our studies to dynamical systems with an algebraic origin? This is the area of Homogeneous dynamics, with applications in number theory. Some very strong results have been obtained for such systems, and I will present perhaps the most striking of them: Ratner's theorem(s).