I will give a number of standard examples of actions of higher-rank abelian groups on manifold and will discuss a number of geometric objects occurring in algebraic actions and non-linear actions, namely Lyapunov exponents and (coarse) Lyapunov manifolds.

Using such objects and the notion of metric entropy, I'll explain the proof of the following theorem: For any action of SL(n,Z), n>= 3, on a manifold of dimension at most n-2, there always exists an invariant probability measure.
Speaker(s): Aaron Brown (University of Chicago)