First lecture by Alessandro Carderi. 

Title: A short introduction to amenable groups.

Abstract: In this first lecture we will discuss different definitions of amenability as long as examples and counter-examples. The aim of the lecture will be to introduce and motivate the main theorem. 

The second lecture by Thi Ndang.

Title: The top Lyapunov exponent.

Abstract: The top Lyapunov exponent of a smooth transformation of a compact manifold, with respect to a probability measure measures the exponential rate of expansion of typical tangent vectors. I'll begin by giving examples of cocycles. The derivative of a smooth transformation of a compact manifolds is such an example. Then I'll define the top Lyapunov exponent in all previous examples of cocycles, using Kingman's subadditive ergodic theorem. I'll move on to the main part of the lecture: proving that a cocycle has uniform subexponential growth if and only if all top Lyapunov exponents vanish.