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Lattices in higher rank simple Lie groups, like SL(n,R) for n>2, are knownÌýto be extremely rigid.Ìý Examples of this are Margulis' superrigidity theorem, whichÌýshows they have very few linear represenations, and Margulis' arithmeticity theorem,Ìýwhich shows they are all constructed via number theory.Ìý Motivated by these and otherÌýresults, in 1983 Zimmer made a number of conjectures about actions of these groupsÌýon compact manifolds.Ìý After providing some history and motivation, I will discussÌýa very recent result, proving many cases of the main conjecture. While avoidingÌýtechnical matters, I will try to describe some of the novel flavor of the proof. The proofÌýhas many surprising features, including that it uses hyperbolic dynamics to prove anÌýessentially elliptic result, that it uses results on homogeneous dynamics,Ìýincluding Ratner's measure classification theorem, to prove results about inhomogeneousÌýsystem and that it uses analytic notions originally defined for the purposes of studying the K theory ofÌýC^* algebras.Ìý This is joint work with Aaron Brown and Sebastian Hurtado.