Dynamics of non-ergodic foliations, Poincare sections and best approximations

The study of the dynamics of Teichmuller flows has in large part been inspired by the analogy with homogeneous flows, e.g an analog of Ratner's theorems for unipotent flows attained for the $SL(2,\mathbb R)$-action on the moduli space of holomorphic differentials in the celebrated work of Eskin-Mirzakhani-Mohammadi. Interestingly, some results about homogeneous flows and the behavior of their trajectories were inspired by investigations of dynamical properties of foliations of surfaces and Teichmuller geodesic rays. In this talk, I will describe some contributions in this direction pertaining to various concepts in Diophantine approximation such as singular vectors and Khintchine-Levy constant.