We will present a geometric criterion for the ergodicity of the billiard flow in a polygon with non-rational angles and discuss its application to the Diophantine case.

We study cascades of bifurcations in a simple family of maps on the circle, and connect this behavior to the geometry of an absolute period leaf in genus $2$. The presentation includes pictures of an exotic foliation of the upper half plane, computed with the aid of the Möller-Zagier formula.

We discuss the current status of the problem of understanding the closures of the strata of curves together with a differential with a prescribed configuration of zeroes, in the Deligne-Mumford moduli space of stable curves.