88 problems

A stratum of translation surfaces is a quasi-projective variety, and its orbifold fundamental group is finitely presented. A connected component of a stratum is a classifying space…

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

Mirzakhani's conjecture. All totally geodesic subvarieties of dimension at least $2$ are covering constructions.

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 $K$ be a connected component of a stratum of translation surfaces, and let $N_{cyl}(M,L)$ and $N_{sc}(M,L)$ denote the cylinder and saddle-connection counting functions for a s…

Eskin–Kontsevich–Zorich regularity conjecture. All ergodic $SL(2,\mathbb{R})$-invariant finite measures $\nu$ on $\mathcal{H}$ are regular. This regularity condition is known for t…

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 ${\mathcal{H}}(2)$,…

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…

Let an affine invariant manifold in a stratum of Abelian or quadratic differentials have rank at least two. Mirzakhani's conjecture. Its orbit closure should cover a stratum of Abe…

Let $\mathcal H(4)^{\mathrm{hyp}}$ denote the hyperelliptic connected component of the stratum of translation surfaces with one zero of order $4$. An algebraically primitive Teichm…

Let ${\mathcal{H}}(2)$ be the stratum of abelian differentials on genus-$2$ surfaces with one double zero. For each invariant of the primitive square-tiled surfaces in this stratum…

Boshernitzan's conjecture. Given a rational triangle, every nonsingular orbit in the invariant surface containing a perpendicular direction is periodic.

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 $X$ be a finite-area translation surface with uniquely ergodic vertical flow, and let $s$ be a straight non-vertical slit joining two nonsingular points $P,Q\in X$. Gluing…

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 $\mathcal{H}:=\mathcal{H}_{g}(k_{1},\dots,k_{s})$ be a stratum, and let $(\mathcal{G}_{n})_{n}$ be a family of $\operatorname{SL}(2,\mathbb{Z})$-orbit graphs of primitive $n$-s…

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 $n$ be a positive integer, and let $\mathcal{A}_n(1,1)$ and $\mathcal{S}_n(1,1)$ denote the cardinalities of the orbits of primitive $n$-squared origamis in $\mathcal{H}(1,1)$…

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…

A pillowcase-tiled surface of degree $n$ is a degree-$n$ cover of a quadratic differential in $\mathcal{Q}(-1^4)$ branched over at most the four poles. It is 1-cylinder if the clos…

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

A suspension over a staircase transformation is the flow obtained by suspending a staircase transformation with an appropriate roof function; a translation flow is the flow induced…

Let $\rho:S\rightarrow S$ be a pseudo-Anosov map with expansion factor $\beta$, and let $\phi_t^u$ denote the unit-speed flow in the unstable direction. If $\beta$ is Pisot of degr…

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…