8 problems

Let two flat surfaces be equipped with their Steklov spectra. Flat inverse Steklov spectral conjecture. Two flat surfaces are Steklov isospectral if and only if they are isometric.…

Let $\mathcal F^\alpha_\rho=\big({H}^{\alpha}_{g,n}\big)^{-1}(\rho)$ be a level set of Veech's holonomy map in Teichmüller space, where $\rho\in\mathbb U^{2g}$. The lifted holonomy…

Let $\mathscr M_{1,2}$ be the moduli space of genus-one curves with two marked points, and let $\Omega^{\alpha_1}$ be Veech's volume form associated with the parameter $\alpha_1$.…

Let $p_1,p_2,p_3,p_4$ be the variables used for the basic invariants of $G_{33}$, and define $C_6$, $C_9$, $C_{12}$, and $C_{18}$ by … … … … Let $F_{12}$, $F_{18}$, $F_{24}$, and…

Let $G_{29}$, $G_{32}$, and $G_{33}$ be complex reflection groups. In the construction under consideration, impose the flatness conditions at selected special points to determine u…

Characterization conjecture. A hexagonally triangulated flat plane is conformal to the regular one if and only if the PL conformal factor $w$ is linear.

Rigidity conjecture. This two-parameter family characterizes all hexagonal triangulations of the flat plane conformal to the regular one.

Let $K$ be a compact nonpositively curved triangle-square complex, let $\widetilde K$ be its universal cover, and let $F$ be an intrinsically aperiodic flat. Intrinsically aperiodi…