Lattices in higher ran k simple Lie groups\, like SL(n\,R) for n>2\, are known to be extremely rigid. Examples of this are Margulis'\; superrigidity theorem\, which show s they have very few linear represenations\, and M argulis'\; arithmeticity theorem\, which shows they are all constructed via number theory. Motiv ated by these and other results\, in 1983 Zimmer m ade a number of conjectures about actions of these groups on compact manifolds. After providing som e history and motivation\, I will discuss a very r ecent result\, proving many cases of the main conj ecture. The proof has many surprising features\, i ncluding that it uses hyperbolic dynamics to prove an essentially elliptic result\, that it uses res ults on homogeneous dynamics\, including Ratner 9\;s measure classification theorem\, to prove res ults about inhomogeneous system and that it uses a nalytic notions originally defined for the purpose s of studying the K theory of C^* algebras. \; LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR