Siegel-Veech transforms are in L^2

We describe a counting problem which can be viewed as a higher-genus version of the Gauss circle problem: how many corner-to-corner trajectories of (at most) a fixed length exist on a surface obtained from a Euclidean polygon with sides identified by translations (a *translation* surface)? We show how if you consider this problem as the underlying surface varies, the count is in L^2 of the natural measure on the appropriate moduli space. This builds on work of William Veech, and is joint work with Yitwah Cheung and Howie Masur. All relevant definitions will be introduced, and there will be lots of pictures and examples.