We give a necessary and sufficient condition for the existence of infinitely many non-arithmetic Teichmuller curves in a stratum of abelian differentials. This is joint work with Simion Filip and Alex Wright.

A deep result of Furstenberg from 1967 states that if $\Gamma$ is a lattice in a semisimple Lie group $G$, then there exists a measure on $\Gamma$ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.