Consider a billiard ball bouncing around on a polygonal table. This dynamical system, which serves as a simplified model of basic processes in physics, is surprisingly complex. When the angles of the table are rational, the billiard table can be "unfolded" to get a closed surface with a natural flat geometry, a translation surface.
I will describe a concrete result that I proved concerning the equidistribution of corner-to-corner billiard trajectories. This problem about dynamics on an individual surface is intimately connected to a fancier dynamical system: the action of SL(2,R) on a stratum of translation surfaces. Here tools from Teichmuller theory, algebraic geometry, dynamics on homogeneous spaces, etc can be brought to bear. I will discuss some of the results that I've proved in this setting, including on strong regularity of invariant measures.