Our aim in this course is to review some results and techniques in ergodic theory of action of Lie groups and their discrete subgroups and then to sketch the proof of a profound theorem of G. A. Margulis known as "superrigidity".

Margulis' superrigidity theorem says that under some conditions on Lie groups and their discrete subgroups, any isomorphism between discrete subgroups extends to isomorphism of the ambient groups, and roughly speaking these discrete subgroups determines the Lie groups completely.
To this end, we need to talk about some backgrounds from the structure theory of semisimple Lie groups and Algebraic groups together with some tools from Homogeneous dynamics, for instance Moore's theorem on ergodicity of action of certain closed subgroups of Lie groups on their quotients by lattices.
Finally, we shall briefly discuss some applications of superrigidity in Riemannian geometry and also in characterization of lattices in higher rank simple Lie groups.

The main reference for the course will be the following book:
Zimmer, Robert J. Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984.