10 problems

Fix the combinatorics $G$ of a periodic $2$-colored tiling of the plane, and consider the parameters of locally foldable tilings with those fixed combinatorics. Measure-zero conjec…

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 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 $M$ be a fixed surface, and let $H$ be a covector whose corresponding sections of the level surfaces associated with $M$ may exhibit chaotic behavior. Novikov–Maltsev conjectur…

$4n+2$ conjecture. Every periodic trajectory on a triangle tiling has period $4n+2$ for some positive integer $n$.

Rauzy-gasket escaping-trajectory conjecture. Every trajectory through the circumcenter is escaping, and trajectories passing increasingly close to the circumcenter, with…

Triangle no-spiraling conjecture. Trajectories on triangle tilings never spiral.

$4n+2$ period conjecture. In a triangle tiling, every periodic trajectory has a period of the form

Triangle periodic-length conjecture. There exist triangle tilings with periodic trajectories of arbitrarily large length.

Isosceles escaping-trajectory conjecture. An isosceles triangle tiling has an escaping trajectory if and only if its vertex angle is not of the form