26 problems

Let $G_t^{KZ}$ be the Kontsevich–Zorich cocycle over the Teichmüller flow, and let $\mu_\kappa^{(1)}$ be its canonical absolutely continuous invariant measure. Zorich–Kontsevich co…

Let $\mathcal{Q}$ be a connected component of a stratum of quadratic differentials, and let $\mathcal{M}_{\mathcal{Q}}\subseteq\mathcal{Q}$ be the subset of quadratic differentials…

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

Kontsevich–Zorich conjecture. The Lyapunov exponents are (a) all non-zero, expressing non-uniform hyperbolicity, and (b) all distinct, expressing simplicity.

Zorich's conjectures. If $\nu_g>0$, then there exists a Lagrangian subspace $L_g$ such that the cycles remain at bounded distance from $L_g$. If, moreover, the exponents $\nu_j$, f…

Let $S$ be a generic flat surface in a connected component of a stratum, and let $\nu_1,\dots,\nu_g$ be the Lyapunov exponents associated with the Teichmüller geodesic flow. The as…

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…

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…

Let $X_n$ be the translation surface obtained from the regular $n$-gon, and let $l_0$ be the length of a side of the $n$-gon. For $n\geq 10$ with $n\equiv 2\pmod 4$, and for closed…

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 $\Lambda^+_{g,n}$ be the sum of the $g$ Lyapunov exponents of the Hodge bundle along the Teichmüller flow on the moduli space of area-one quadratic differentials…

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…

Let $\mathcal N$ be an affine invariant submanifold in a stratum of Abelian differentials, and call it higher rank when its rank, defined by … is greater than one. Here…

Let a Teichmüller curve be an algebraic curve generated by the Teichmüller geodesic flow, with its Lyapunov spectrum and Harder–Narasimhan polygon defined from the corresponding va…

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.

Let ${\mathcal{Q}}(S)$ be the space of quadratic differentials on the once-punctured torus, and let $\mu$ be the $\Phi^t_{\mathcal{T}}$-invariant Borel probability measure on…

Let ${\mathcal Q}$ be a non-hyperelliptic component of a stratum of abelian differentials with at least two zeros, and let $S$ be the underlying surface. Let the stabilizer be the…

Let ${\mathcal C}$ be an affine invariant manifold of rank $\ell\geq 1$. An affine invariant manifold is a submanifold of a stratum of abelian differentials locally defined by homo…

Let $\mathcal{C}$ be a connected component of a stratum of unit-area Abelian differentials, let $\mu_{\mathcal{C}}$ be its Masur–Veech measure, and let $\rho_{\mu_{\mathcal{C}}}$ b…

Let $\mu$ be an $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian differentials. Call $\mu$ algebraic when it is supported on an affine suborbifold an…

Let $\mathcal{H}(\mu)$ be a stratum of Abelian differentials, and consider Teichmüller curves contained in it. Kontsevich–Zorich conjecture. For many low-genus strata…

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…

Let translation surfaces be acted on by $\operatorname{GL}^+(2,\mathbb{R})$ through postcomposition in the charts of their translation structures. For a translation surface…