41 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 $n$ 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 $n$ be an odd integer, and consider the two orbits of primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$, denoted orbit A and orbit B. For a prime $p$, write…

Completeness conjecture. The known list of obtuse rational lattice triangles is complete: every obtuse rational lattice triangle is either a member of one of the two infinite famil…

Let $n$ be an integer greater than $8$, and consider primitive $n$-squared origamis in $\mathcal{H}(4)$, with $\mathcal{H}^{\mathrm{odd}}(4)$ denoting the odd component and…

Let $W_{d^2}[n]$ be the locus under consideration, and for odd $n$ let $W_{d^2}^{\epsilon}[n]$, with $\epsilon\in\{0,1\}$, denote its two parity-labelled subloci. Parity conjecture…

Let $(\mathcal{G}_{n})_{n}$ be the family of $\operatorname{SL}(2,\mathbb{Z})$-orbit graphs of primitive $n$-squared origamis in the stratum $\mathcal{H}(2)$. McMullen's expander c…

Interaction-strength conjecture. For every pair of closed curves $\alpha,\beta$ on $S_{m,n}$,

Let two interval exchange systems be isogenous, and suppose one is minimal and self-induced. Then the other is also minimal and self-induced. Associate to such systems their transl…

SL(2,Z)-orbit conjecture. For odd genus $g\geq 5$, the origamis constructed by Aougab-Menasco-Nieland lie in exactly two $\operatorname{SL}(2,\mathbb{Z})$-orbits inside the odd com…

Alternating-group monodromy conjecture. All of the $[1,1]$-origamis in Theorem have monodromy group isomorphic to $\operatorname{Alt}_{2g-1}$.

Double-heptagon connection-point conjecture. The point $P$ is a connection point if and only if $N$ is even and $[\overline{x}:\overline{y}]$ belongs to the orbit of…

Modulo-two orbit conjecture. An element $s=\frac{x}{y}\in\mathbb{Q}[\lambda_7]$ belongs to $H_7\cdot\infty$ if and only if $[\overline{x}:\overline{y}]$ belongs to the orbit of…

Non-periodic connection-point conjecture. For $n\geq 9$ odd, there are no non-periodic connection points on the double regular $n$-gon.

Heptagon connection-point conjecture. There exist non-periodic connection points on the double regular heptagon.

Let $X$ be a translation surface. Write $\mathop{\mathrm{KVol}}(X)$ for its KVol and $\mathop{\mathrm{SysVol}}(X)$ for its systolic volume, defined as the volume divided by the squ…

Directional weak-mixing conjecture. For every translation surface that does not factor over the circle, the directionally typical translation flow is weakly mixing.

Kontsevich–Zorich conjecture. For the hyperelliptic components $\mathcal{M}^{\mathcal{H}}_{2g-2}$ and $\mathcal{M}^{\mathcal{H}}_{(g-1,g-1)}$,

Let $g\geq 1$, let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a partition of $2g-2$, let $\mathcal H_1(\alpha)$ be the space of unit-area surfaces in the corresponding stratum of abelia…

For $g\in\{3,4,5\}$, let $J_g(\lambda,1)$ denote the corresponding family of translation surfaces of genus $g$. A compact Riemann surface is hyperelliptic if it admits a degree-two…

Let $J_2(\lambda,1)$ be the genus-two family of translation surfaces in the stratum $\mathcal{H}(1,1)$, and let $a,b,c$ be complex parameters. A genus-two hyperelliptic curve is gi…

Let $0<\alpha<1$ be irrational, let $T$ be the rotation by angle $\alpha$, viewed as a two-interval exchange with discontinuity $1-\alpha$, and let $T_f$ be the Veech extension fro…

Let $X_N$ be the Veech $N$-example surface, with directional flow $\phi_t^{\theta}$, and let $T_f$ and $T_{1-f}$ be the associated interval exchange transformations. Write the slop…

Let $\mathcal{M}$ be an orbit closure of surfaces of genus $g$. Its rank is denoted by $\operatorname{rank}(\mathcal{M})$. Almost High Rank Conjecture. If … then $\mathcal{M}$ is a…

Let $\mathcal{M}$ be a geminal orbit closure, and let the optimal covering map associated with $\mathcal{M}$ be the covering map of minimal degree arising in the classification of…