Zimmer’s conjecture for some non-split Lie groups.

We will discuss Zimmer’s conjecture on the actions of lattices in higher rank Lie groups on manifolds of small dimension. This conjecture is recently proved by Aaron Brown, David Fisher and Sebastian Hurtado for real split simple Lie groups. We discuss the proof of  this conjecture for simple complex Lie groups. We also give significant improvements for the dimensional bounds for many other non-split Lie groups. This is a joint work with Jinpeng An and Aaron Brown.