A collection of polygons with the property that to each side one can find another side parallel to it can be endowed with a translation surface structure by glueing along those edges.
This means that the closed surfaces obtained carries a flat metric outside finitely many conical singularities. Geodesics (which are straight lines) connecting such singularities are called saddle connections.
While the asymptotic number of saddle connections of length less then T growth roughly like T^2 (in the sense that there are lower and upper bounds of that order), one can say more for a generic surface with respect to the
moduli space of such structures thanks to the natural SL2-action it is equipped with. I shall present some results with polynomial error saving for counting saddle connections in the setting of
a) general loci (j/w Nevo,Weiss)
b) prescribed congruence restrictions in homology (j/w Magee, Guetierrez-Romo)
c) lattice-surfaces using Eisenstein series (j/w Burrin,Nevo,Weiss)