An Introduction to Counting Closed Geodesics on Translation Surfaces

A translation surface is a collection of polygons in the plane with parallel sides identified by translation to form a surface with a singular Euclidean structure. Closed geodesics on a translation surface are straight lines which start and end at a vertex of one of the polygons. Through some concrete examples we will give probabilistic results on counting closed geodesics for certain families of translation surfaces. These translation surfaces live in SL(2,R) orbit closures that are in fact algebraic varieties in the moduli space of translation surfaces, and are equipped with a natural SL(2,R) invariant probability measure.