31 problems

Let $g$ be the genus and let $m_1,\dots,m_r$ be positive integers satisfying … Consider the even and odd connected components $\mathcal H^{\mathrm{even}}(2m_1,\dots,2m_r)$ and…

Smooth Weierstrass cubic period-test conjecture. For a generic smooth real Weierstrass cubic, the compact oval is nonseparating and the exterior Cauchy transform of the domain boun…

Separating/nonseparating algebraicity principle. If $\gamma$ is separating, then the exterior Cauchy transform is algebraic. If $\gamma$ is nonseparating, then the exterior Cauchy…

Let $k$ be an integer for which the indicated trigonal stratum ${\mathbf P}{\mathcal H}^{tri}_{3k+1}(4k,2k)$ is defined, and let…

Let $(M,\omega)$ be a horizontally periodic Veech surface, let $\mathcal{M}=\overline{\mathrm{SL}_2(\mathbb{R})\cdot\mathbb{T}(\omega)}$, and let $\mu_\mathbb{T}$ be a fully suppor…

Let $\Omega\mathcal{M}_g(\kappa)$ be a stratum of abelian differentials, and let $S\subset\Omega\mathcal{M}_g(\kappa)$ be an irreducible algebraic subvariety. Call a subvariety…

Let $\Omega\mathcal{M}_g$ denote the moduli space of abelian differentials, and call a subvariety invariant if it is invariant in the sense used for the stratum dynamics. Mirzakhan…

Let $\textnormal{S}_{\alpha}$ be a stratum of abelian differentials up to scaling, let $\widetilde{\textnormal{S}_{\alpha}^{\mathrm{an}}}$ be the universal cover of its analytifica…

Let $\textnormal{S}_{\alpha}$ be a stratum of abelian differentials up to scaling. An arithmetic point has a degree, and points of degree at least $3$ are those whose arithmetic de…

Let $\textnormal{S}_{\alpha}$ be a stratum of abelian differentials up to scaling, equipped with its bi-algebraic structure. A subvariety is linear if it is defined by linear equat…

Fix $n>1$. For each $i\in\{1,\ldots,n\}$, let $\mathcal{S}_i$ be a component of a stratum of Abelian or quadratic differentials, and let $\mathcal{M}\subset\prod_i\mathcal{S}_i$ be…

The Strong Diamond Conjecture. Under these assumptions, $\mathcal{M}$ is a full locus of covers. This would extend the full-locus conclusion beyond the cases established by the pre…

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

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

Let ${\mathcal H}(m)$ be a stratum of Abelian differentials and let $c_{\mathit{area}}({\mathcal H}(m))$ be its area Siegel–Veech constant. Configurations contributing to this cons…

Let ${\mathcal H}(m)$ be a stratum of Abelian differentials, and let $c_{\mathit{area}}({\mathcal H}(m))$ denote its area Siegel–Veech constant. The principal boundary strata arise…

For an integer partition $\underline{\kappa}=(k_1,\ldots,k_n)$ of $2g-2$, let $\mathcal{HT}(\underline{\kappa})$ be the stratum of abelian differentials on marked genus-$g$ Riemann…

Let ${\mathcal H}^{\mathrm{comp}}(m)$ be a nonhyperelliptic connected component of a stratum of Abelian differentials in genus $g$, and let…

Consider the Siegel–Veech constants obtained by counting saddle connections on strata of Abelian differentials, and distinguish saddle connections of multiplicity one from configur…

Let $m=(m_1,\dots,m_n)$ be an unordered partition of $2g-2$, and consider the Siegel–Veech constants associated with saddle connections on the stratum ${\mathcal H}(m_1,\dots,m_n)$…

Maximal systolic ratio conjecture. The supremum of the systolic ratio over genus two surfaces $(X,d_{\omega})$ equals

Let $\mathcal{M}$ be a connected component of a stratum of abelian differentials, let $\mathcal{M}(q)$ be its level-$q$ lift for $q\geq 2$, and let $\nu_{\mathcal{M}(q)}$ be the co…

Uniform error-bound conjecture. There exists a universal constant $C$ such that, for all $g$ and all $m\in\Pi_g$,

Asymptotic volume conjecture. For any $m\in\Pi_{2g-2}$,

Let a Teichmüller curve be a curve in a stratum of the moduli space of Abelian differentials, and call it algebraic primitive when its variation of Hodge structures has the decompo…