19 problems

Kontsevich's volume formula. The Masur–Veech volume is

Let $\gamma$ be a periodic trajectory in a cyclic quadrilateral tiling, and let $\Omega_{\gamma}$ be the domain bounded by it. Tree conjecture for cyclic quadrilateral tiling billi…

Fractal flat surface conjecture. There exists a surface $\mathcal{S}(KS)$ that can be appropriately viewed as a suitable limit of the flat surfaces $\mathcal{S}(KS_n)$ and such tha…

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 $\theta_1,\dots,\theta_g$ be the Lyapunov exponents of the Rauzy–Veech cocycle, ordered so that … A complete Lagrangian flag is a nested sequence of Lagrangian subspaces whose…

Let $\theta_1>\theta_2>\dots>\theta_g$ be the Lyapunov exponents of the Rauzy–Veech cocycle. The asymptotic deviation theorem uses these exponents to determine the dimensions of th…

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…

Let $N_g$ denote the number of distinct $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2g-2)$ with $2g-1$ squares, such that each of their horizontal and vertical cylinder d…

Fix the combinatorics of a periodic $2$-colored tiling of a plane, defined by a bipartite graph $G$ on the $2$-torus. Suppose that a tiling with these combinatorics folds into a bo…

Let $\gamma$ be a singular periodic trajectory passing through a vertex $v$ of a tiling. Let $P$ and $P_e$ be tiles intersecting along an edge $e$. Bounded flower conjecture. The t…

Let $\hat\Pi_{4g-4}$ be the set of partitions of $4g-4$ with parts in $\{-1,0,1,2,\dots\}$ and at most $\log(g)$ entries equal to $-1$. For a stratum ${\mathcal Q}(d_1,\dots,d_n)$…

The affine-invariant-submanifold intersection formulas. The area Siegel–Veech constant and the sum of Lyapunov exponents are

Mixing conjecture. The vertical flow on $S(\mathcal{B},w^\pm, \leq_{r,s})$ is mixing.

Mixing conjecture. The specific surface has a mixing translation flow.

Veech dichotomy conjecture. Under this assumption, the Veech dichotomy will hold for the Koch snowflake billiard table $\Omega(KS)$.

Infinite-genus fractal flat surface conjecture. The surface $\mathcal{S}(KS)$ is a well-defined flat surface of infinite genus, has a fractal, self-similar structure, and has geode…

Self-similar geometry conjecture. The self-similarity of the Koch snowflake curve is somehow reflected in the geometry of $\mathcal{S}(KS)$.

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

Boissy–Lanneau's equivalence conjecture. The sets $cmathfrak{R}_{ex}(cpi)$ and $cmathfrak{R}'_{ex}(cpi)$ coincide.