21 problems

Let $X$ be a Riemann surface equipped with poles whose orders are all $4$, and prescribe the residues and leading order terms at those poles. Theorem provides a half-plane differen…

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…

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 $\boldsymbol{z}$ 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…

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 $\mathcal{Q}_{g,0}$ denote the moduli space of holomorphic quadratic differentials in genus $g$ without marked points. Large-genus asymptotics conjecture. As $g\to+\infty$, its…

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…

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 ${\mathcal Q}(d_1,\dots,d_n)$…

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…

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

Let $MV_{g,n}$ denote the Masur–Veech volume, and set … For each $g\geq 0$, consider polynomials $p_g,q_g\in\mathbb{Q}[n]$. Polynomial structure conjecture. For any $g\geq 0$, ther…

Let $H_n[d]$ be the fixed-degree coefficients considered in the generating series, and for each $d$ define … For each $d\geq 0$, there exists a polynomial $P_d$ of degree $d$ with…

Let $\log {\mathbf{S}}_\bigtriangleup$ be a topological log surface with its hat homology group $\widehat{\operatorname{\bf H}}_{}(\log {\mathbf{S}}_\bigtriangleup)$, and let…

Fix a genus $g$. Let $Q_n$ be a sequence of connected components of strata, let $p_n$ be the number of poles in the stratum containing $Q_n$, and suppose that $p_n$ tends to infini…

A translation surface is a surface equipped with a flat structure whose singularities may include poles, and its positive Lyapunov exponents are the exponents of the Kontsevich–Zor…

For a parameter $b$, let $\Theta$ be the variable in the quadratic differential … where $\Lambda$ is a complex parameter. Quadratic-differential support conjecture. The support of…

Let $b=u+iv$ with $v\ne 0$, and let $\Gamma_b$ be the real rational cubic with a node at the origin defined by … where $x=\operatorname{Re}z$ and $y=\operatorname{Im}z$. Its three…

Mirzakhani's conjecture. If an affine invariant submanifold $\cM$ is defined over $\bQ$, and $\cM$ is not a connected component of a stratum, then every translation surface in…

Boissy–Lanneau's equivalence conjecture. The sets $cmathfrak{R}_{ex}(cpi)$ and $cmathfrak{R}'_{ex}(cpi)$ coincide.