Billiards in quadrilaterals, Hurwitz spaces, and real multiplication of Hecke type

After a brief introduction to the dynamics of the $\mathrm{GL}(2,\mathbb R)$ action on the Hodge bundle (the space of translations surfaces), we will give a construction of six new orbit closures and explain why they are interesting. Joint work with Alex Eskin, Curtis McMullen, and Ronen Mukamel.