Related papers: Counting saddle connections in a homology class mo…

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that…

Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such…

We show that for almost every translation surface the number of pairs of saddle connections with bounded magnitude of the cross product has asymptotic growth like $c R^2$ where the constant $c$ depends only on the area and the connected…

We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A…

We prove an effective version of a celebrated result of Eskin and Masur: for any affine invariant manifold of translation surfaces, almost every translation surface has quadratic growth for the saddle connection holonomy vectors, with an…

We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface…

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal…

Let $\mathcal{H}$ denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on $\mathbb{R}^2$ is in $L^2(\mathcal{H}, \mu)$, where $\mu$ is Lebesgue…

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the…

We classify $\text{GL}(2,\mathbb{R})$ orbit closures in the product of strata of translation surfaces. Applications exist to joinings of certain Masur-Veech measures.

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat…

For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.

We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of…

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and…

The slope gap distribution of a translation surface is a measure of how random the directions of the saddle connections on the surface are. It is known that Veech surfaces, a highly symmetric type of translation surface, have gap…

In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus…

We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit…

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat…

We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also…