43 problems

Let $\mathcal{Q}$ be a connected component of a stratum of quadratic differentials, and let $\mathcal{M}_{\mathcal{Q}}\subseteq\mathcal{Q}$ be the subset of quadratic differentials…

Let $\mathcal H$ and $\mathcal Q_g$ denote the moduli spaces of Abelian and quadratic differentials, respectively, and consider a connected component of a stratum of … A group…

Spin-structure classification conjecture. The parity of the spin structure classifies the non-hyperelliptic connected components of the strata of quadratic differentials with presc…

Fix integers ${\bf d}=(d_1,\dots,d_n)$ with each $d_i$ positive or equal to $-1$, satisfying $\sum_i d_i=4g-4$, and let ${\cal Q}_{\bf d}$ be the moduli space of non-square meromor…

A Joyce structure on a moduli space of quadratic differentials with simple poles is a geometric structure of the type considered in the paper. The examples above admit a Killing ve…

Let $M\in{\mathbb Z}_+$ be fixed. For every $0\leq p\leq M$ and every odd positive partition ${\underline{k}}$ of $4g-4+p$, write its parts as $\kappa_1,\dots,\kappa_n$. Adapted vo…

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 $\boldsymbol{\theta}=(\theta_1,\theta_2,\theta_3)$ 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…

Let $\boldsymbol{z}$ be a generic configuration on the unit circle, and let $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ be the master function. Involution-symmetry conjecture. Ev…

Let $\mathcal{QD}(\boldsymbol{z})$ be the specified space of quadratic differentials with $m$ poles and $n$ zeros, and let $\Gamma(Q)$ denote the horizontal trajectories of…

Let $G(\mathbb{S},n)$ be the relative graded Brauer graph algebra associated with the weighted surface data $(\mathbb{S},n)$, and let…

Non-varying-stratum characterization. The stratum $\mathcal{Q}_g(d_1,\ldots,d_n)$ is non-varying if and only if $L^+$ equals the sum of the $g$ smallest numbers in $S$. This is pre…

Let $(\mathbb{S},\mathbb{M})$ be a marked bordered surface, with the genus-zero assumption $|\mathbb{M}|\geq 3$, and let … be the mapping-class-group-equivariant continuous map fro…

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…

Yang–Zagier–Zhang conjecture. The error term in this asymptotic admits an expansion in $g^{-1}$ whose leading term is $-\frac{\pi^2}{144}\frac{1}{g}$, with explicit coefficients fo…

For fixed $n\geq 0$, let $C_{\rm area}(\mathcal{Q}_{g,n})$ be the area Siegel–Veech constant associated with the principal stratum $\mathcal{Q}_{g,n}$, and set $M=-\pi^2/144$. Area…

For a fixed integer $n\geq 0$, let $\mathcal{Q}_{g,n}$ denote the principal stratum of quadratic differentials and let ${\rm Vol}\,\mathcal{Q}_{g,n}$ be its Masur–Veech volume. Def…

Let $\hat\Pi_{4g-4}$ be the set of partitions of $4g-4$ with parts in $\{-1,0,1,2,\dots\}$ and at most $\log(g)$ entries equal to $-1$. For a stratum ${\mathcal Q}(d_1,\dots,d_n)$…

A stratum of meromorphic quadratic differentials is a moduli space of quadratic differentials with prescribed orders of zeros and at most simple poles. Geometric-properties conject…

Let $\boldsymbol{d}\in\hat\Pi_{4g-4}$, and let ${\mathcal Q}(\boldsymbol{d})$ be a non-hyperelliptic connected component of a stratum of meromorphic quadratic differentials with at…

Let $\boldsymbol{d}=(d_1,\dots,d_n)$ be an unordered partition of $4g-4$ with parts $d_i\in\{-1,0,1,2,\dots\}$, and let $\hat\Pi_{4g-4}$ consist of those partitions having at most…

Let $(\mathbb{S},\mathbb{M})$ be a marked bordered surface, and let … be the mapping class group equivariant continuous monodromy map from the preimage of the dense open set of pro…

The Siegel–Veech and Lyapunov intersection formulas. The area Siegel–Veech constant and the sum of Lyapunov exponents are