19 problems

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…

Quantum modularity conjecture. This generating function is a quantum modular form.

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 $\textnormal{S}_{\alpha}$ 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…

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 $g\to\infty$,

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…

Siegel–Veech universality conjecture. For all such nonhyperelliptic connected components,

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 ${\mathcal M}_1$ be a regular $\operatorname{SL}(2,\mathbb R)$-invariant suborbifold in a stratum of Abelian differentials, and let $c_{\operatorname{area}}({\mathcal M}_1)$ be…

Rauzy-class decomposition conjecture. The extended Rauzy class $cmathfrak{R}_{ex}$ is a disjoint union of exactly $k$ nonempty Rauzy classes.