Number theory in Teichmüller dynamics

On the moduli space of Riemann surfaces equipped with a holomorphic one-form, there is a natural action of SL(2,R). 
After work of McMullen, Eskin-Mirzakhani-Mohammadi, and Filip, we now know that every SL(2,R) orbit-closure
is in fact a smooth subvariety of moduli space.  While these orbit closures have a dynamical origin, they have many
interesting arithmetic properties, for example they are defined over number fields, they often lie on Shimura varieties,
they are often cut out by modular forms, etc.

In this talk, we give a survey of our current knowledge of the problem of classifying SL(2,R) orbit-closures focusing
on connections to number theory.