10 problems

Billiard-flow deviation conjecture. For all rational-angle Euclidean polygons, the deviation of ergodic averages for the billiard flow is the same for almost all directions, and de…

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…

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 $g$ and $n$ be fixed such that the moduli space of dilation surfaces $\mathcal{MD}_{g,n}$ is non-empty. For $k\geq 3$, consider $k$-affine interval exchange transformations and…

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…

Nonarithmetic rank-one sparsity conjecture. Outside of genus two, a sequence of nonarithmetic rank-one orbit closures cannot equidistribute in a component of a stratum.

Mirzakhani's covering conjecture. Higher-rank arithmetic affine invariant submanifolds are either connected components of strata or branched covering constructions.

Mirzakhani's arithmeticity conjecture. Higher-rank affine invariant submanifolds are arithmetic.

Let ${\mathcal L}_1$ be a suborbifold in the moduli space of unit area Abelian differentials, or in the moduli space of unit area meromorphic quadratic differentials with at most s…