Recent Progress on Zimmer's Conjecture

Lattices in higher rank simple Lie groups are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds and in a recent breakthrough with Brown and Hurtado we have proven many of them. The talk will begin with a statement of a special case of our results that requires no background. I will then provide history, context and motivation. Towards the end, I will give some indication of the proof with emphasis on surprising connections to other areas of mathematics that arise.