Fix a translation surface X, and consider the measures on X coming from averaging the uniform measures on all the saddle connections of length at most R. Then as R → ∞, the weak limit of these measures exists and is equal to the area measure on X coming from the flat metric. This implies that, on a rational-angled billiard table, the billiard trajectories that start and end at a corner of the table are equidistributed on the table. We also show that any weak limit of a subsequence of the counting measures on S 1 given by the angles of all saddle connections of length at most Rn, as Rn → ∞, is in the Lebesgue measure class. The proof of the equidistribution result uses the angle result, together with the theorem of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.

Strata of exact differentials are moduli spaces for differentials on Riemann surfaces with vanishing absolute periods. Our main result is that classes of closures of strata of exact differentials inside the moduli space of multi-scale differentials lie in the divisorial tautological ring. By relating exact differentials to rational functions we obtain a new proof that classes of Hurwitz spaces are tautological and a new method for computations.

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We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-Mirzakhani-Mohammadi, this yields the (previously conjectured) convergence of sequences of Siegel-Veech constants associated to Teichmüller curves in genus two.The proof uses a recurrence result closely related to techniques developed by Eskin-Masur. We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniform constant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface.

We investigate the closure M of a linear subvariety M of a stratum of meromorphic differentials in the multi-scale compactification constructed in [BCG + 19]. Given the existence of a boundary point of M of a given combinatorial type, we deduce that certain periods of the differential are pairwise proportional on M , and deduce further explicit linear defining relations. These restrictions on linear defining equations of M allow us to rewrite them as explicit analytic equations in plumbing coordinates near the boundary, which turn out to be binomial. This in particular shows that locally near the boundary M is a toric variety, and allows us to prove existence of certain smoothings of boundary points and to construct a smooth compactification of the Hurwitz space of covers of P 1 . As applications of our techniques, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright [Wri15] to the case of real linear subvarieties of meromorphic strata.