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Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel–Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemann surfaces. This space carries actions of the foliated Laplacian derived from the \mathrm{SL}_{2}(\mathbb{R})-action as well as various differential operators related to relative period translations.In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum, exhibiting all Siegel–Veech transforms requires novel configurations of saddle connections. We also show that the continuous spectrum of the foliated Laplacian is much larger than the space of Siegel–Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations. 

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in W = [0,N]^d are almost equilateral having all sides almost equal to √dN/√6, and the sine of the typical angle between rays from the visual spectra from the origin of W is, in the limit, equal to √7/4, as d and N/d tend to infinity. We also show that there exists an interesting number theoretic constant Λd,K, which is the limit probability of the chance that a K-polytope with vertices in the lattice W has all vertices visible from each other. 

Motivated by the study of the growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one gener- alization, we give a result regarding the upper bound on the asymptotics of the number of classes of irreducible special Lagrangians in K3 surfaces with bounded period integrals. In another generalization, we give the exact leading term in the asymptotics of the number of Mukai vectors of semistable coherent sheaves on algebraic K3 surfaces with bounded central charges, with respect to generic Bridgeland stability conditions. 

Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics.