A celebrated result of Avila and Viana proves simplicity of the Lyapunov spectrum for the Kontsevich Zorich cocycle over strata of abelian differentials, equipped with the Masur Veech measure.

We use ideas from random walks reminiscent of the work of Eskin and Mirzakhani and amenable actions to extend this result to all strata of quadratic differentials, with a different proof. I will focus in the lecture on explaining the connections to the work of Mirzakhani,