8 problems

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…

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 $\hat\Pi_{4g-4}$ 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.

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.