14 problems

Let $n$ be an odd integer with $n\geqslant 5$. Let $a_n$ and $b_n$ denote the cardinalities of the two discrete orbits of primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$,…

Let ${\mathcal{H}}(2)$ be the stratum of abelian differentials on genus-$2$ surfaces with one double zero. For each invariant of the primitive square-tiled surfaces in this stratum…

Let $n$ be prime, and consider the elliptic points associated with primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$. Elliptic-point count conjecture. For prime $n$, the nu…

Let $M\in{\mathbb Z}_+$ be fixed, and let $\tilde{\underline{k}}=({\underline{k}},-1^p)$, where $p\leq M$ and ${\underline{k}}$ is a positive partition of $4g-4+p$. Let…

For each cylinder type, let … where $F$ counts the relevant integer solutions for the saddle-connection lengths and the sums arise in counting square-tiled surfaces in a stratum of…

Let $(\mu,k)$ be a spherical profile, meaning profile data for square-tiled surfaces on the sphere. Let $\operatorname{ST}(\mu,k)$ denote the corresponding set of square-tiled surf…

Let $\boldsymbol{\theta}=(\theta_1,\theta_2,\theta_3)$ be the relevant profile data for a square-tiled surface, and let $\theta_4$ be a non-negative integer such that … For an inte…

A square-tiled surface is a combinatorial quadrangulation equipped with the cylinder-shear reconfiguration operation. The associated quadratic differential determines a point in a…

Fix a stratum of Abelian or quadratic differentials and let $k$ be a fixed number of cylinders. Consider the contribution of square-tiled surfaces having $k$ cylinders to the Masur…

Abelian-stratum mod-Poisson conjecture. For every $x>0$, uniformly for $k\in\{0,1,\ldots,\lfloor x\log(g)\rfloor\}$,

Let $\mathcal{C}$ be a non-hyperelliptic connected component of a stratum of holomorphic quadratic differentials, and let $p_{\mathcal{C}}(k)$ be the probability that a random quad…

Abelian mod-Poisson conjecture. For every $x>0$, uniformly for $k\in\{0,1,\ldots,\lfloor x\log(g)\rfloor\}$, as $g\to\infty$,

Consider square-tiled-surface $T,R$-orbits in the connected components of strata in genera $3$, $4$, and $5$ belonging to the authors' specified list. For an orbit, let its Siegel–…

Let $g\ge 2$, and let a Teichmüller curve lie in a stratum of Abelian differentials in genus $g$. Write its Lyapunov exponents as $\lambda_1\ge\cdots\ge\lambda_g$. The spectrum is…