May 19th, 2020

First lecture Thi Ndang.
Title: MargulisRuelle inequality (1)
Abstract: The aim of this lecture and the next one is to understand the title's inequality.
Given an ergodic dynamical system defined on a smooth compact Riemanian manifold, the sum of positive Lyapunov exponents with multiplicities is an upper bound of the measure theoretical entropy. As topological entropy measures the exponential growth rate of all distinguishable orbits of the dynamical system, measure theoretic entropy will measure it for typical orbits. This lecture will focus on the Lyapunov exponents' side of the inequality. The next lecture will focus on measure theoretical entropy.
In the first part of my talk, I'll define topological entropy and prove that for Cat maps acting on the two torus, it is equal to the logarithm of the top eigenvalue. In the second part, I'll state and sketch a proof of Oseledec's Theorem which will allow me to introduce all the Lyapunov exponents and measurable subbundles of the smooth manifold. In the last part, as in the Cat map example, we'll focus on the unstable subbundles and introduce Pesin manifolds.
Second lecture Alessandro Carderi.Title: An introduction to cost and graphings
Abstract: In this lecture we will explore the theory of cost for p.m.p. equivalence relations, prove some basic facts and compute some examples, as long as presenting some of the important theorems of the area. We will state the strategy of the proof of the GaboriauLyons Theorem. We will also start working with graphs, or to be precise, fields of graphs, sometime called graphings, or unimodular (pointed and colored) random graphs, which will be the main topic of the next two lectures. In particular some of the proofs we will give, will be obtained by playing with the graphs.