The finite blocking problem asks - given a polygonal billiard table and two points when can all billiard shots from one point to the other be blocked by a finite collection of points? Resolving this problem naturally leads to studying the dynamics of a $GL(2,{\mathbb R})$ action on the moduli space of Abelian differentials.

By work of Eskin, Mirzakhani, and Mohammadi, every $GL(2, {\mathbb R})$ orbit closure in the moduli space of Abelian differentials is a linear manifold. Recent work of Mirzakhani and Wright constructed a “dynamically interesting” boundary of an orbit closure that places even stronger constraints on the structure of an orbit closure.

Applying these two results we will establish finiteness results for the finite blocking problem. As a separate application we will study holomorphic sections of the universal curve defined over “closed families of Riemann surfaces containing Teichmuller disks”.

We will end by describing work in genus two that promotes finiteness to explicit classification and we will use this information to completely solve some explicit blocking problems. Part of this work is joint with Alex Wright.