orders of the zeroes of . If is any partition of then the stratum is the subset of of translation surfaces which have a holomorphic form whose zeroes match the partition. The stratum is naturally a complex orbifold of complex dimension (note that is the moduli space of tori, which is well-known to be an orbifold; in higher genus, the failure to be a manifold is even more dramatic). Local coordinates are given by where and is as above a symplectic basis of this space. Masur-Veech volumes The stratum admits a -action and thus a real and complex projectivization . The real projectivization admits a natural section if we define it as the space of translation surfaces of area 1. The existence of the above period coordinates allows to endow the stratum with an integral affine structure and thus a natural volume form . We also get a volume form on by disintegration of . The Masur-Veech volume is the total volume of for . This volume was proved to be finite independently by William A. Veech and Howard Masur. In the 90's Maxim Kontsevich and Anton Zorich evaluated these volumes numerically by counting the lattice points of . They observed that should be of the form times a rational number. From this observation they expected the existence of a formula expressing the volumes in terms of intersection numbers on moduli spaces of curves. Alex Eskin and Andrei Okounkov gave the first algorithm to compute these volumes. They showed that the generating series of these numbers are q-expansions of computable quasi-modular forms. Using this algorithm they could confirm the numerical observation of Kontsevich and Zorich. More recently Chen, Möller, Sauvaget, and don Zagier showed that the volumes can be computed as intersection numbers on an algebraic compactification of . Currently the problem is still open to extend this formula to strata of half-translation surfaces. The SL2(R)-action If is a translation surface obtained by identifying the faces of a polygon and then the translation surface is that associated to the polygon . This defined a continuous action of on the moduli space which preserves the strata . This action descends to an action on that is ergodic with respect to . Half-translation surfaces Definitions A half-translation surface is defined similarly to a translation surface but allowing the gluing maps to have a nontrivial linear part which is a half turn. Formally,