22 problems

Let $S$ be a surface equipped with a metric of constant curvature, and let $T$ be a triangulation of $S$. Write $\mathcal{GT}(S,T)$ for the space of geodesic triangulations associa…

Let $P$ be a strictly convex polygon in $\mathbb{RP}^d$. Suppose that $P$ intersects a hyperplane with multiplicity $k$, where $k\geq d+2$. Call a vertex configuration a flattening…

Let $P$ be an embedded closed polygon in $S^2$ that bisects the area, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. Discre…

Let $P$ be an embedded non-contractible closed polygon in $\mathbb{RP}^2$, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. D…

Let ${\rm L}$ be the discrete logarithmic function on a quasicrystallic quad-graph ${\mathcal D}$, obtained by restricting the discrete logarithmic function on the quad-surface to…

Let $z^{2a}$ be the discrete power function on a quad-surface $\Omega_{\mathcal D}$ associated with a quasicrystallic quad-graph ${\mathcal D}$ whose edge slopes are…

Let $W$ denote the conformal energy of a simplicial Willmore sphere. Quantization conjecture. The conformal energy is quantized: … This conjecture is motivated by numerical experim…

Let a cubic graph be a graph whose edge operators satisfy the cubic-graph relations displayed in the source, and let the edge operators and their adjoints obey the anticommutation…

Non-candidate crease-rule pattern conjecture. Crease-rule patterns which are not candidate crease-rule patterns do not admit a rigid-ruling folding motion.

Let $f(u,v)$ be a smooth conjugate net, and let $(u_0,v_0)$ be a generic point. A reparametrization replaces $f(u,v)$ by $F(U,V)$ with $F(U(u),V(v))=f(u,v)$, where $U$ and $V$ have…

Doyle's conjecture. There are no locally univalent circle packings of the hexagonal triangulation $\mathcal{T}_H$ other than regular hexagonal packings and Doyle spirals.

Nonexistence conjecture. The analogue of Lemma for Euclidean flexible nets is not true.

TCD–cluster correspondence. The $\tau$-variables of $T_F$ are gauge equivalent to the $\mathcal A$-cluster variables of the dual configuration $F^\star$ as defined by Fock and Gonc…

Let $Q^{\mathbb{C}}(\operatorname{cr})$ and $Q^{\mathbb{R}}(\operatorname{cr})$ denote the complex and real spaces of discrete holomorphic quadratic differentials associated with a…

Let $N\ge 6$. Recall that $\mathcal{M}_N$ is the configuration space of $N$-Kaleidocycles, $\pi_c:\mathcal{M}_N\to\mathbb{R}$ is the projection onto the $c$-axis, where $c=\cos\nu$…

Let an abstract triangulated sphere be realized in $\mathbb{R}^3$, and let $\operatorname{D}$ denote its discrete Dirac operator. A triangulated-sphere realization conjecture. Ever…

Let a simply connected triangulated surface in space have discrete Dirac operator $\operatorname{D}$. A triangulated-surface Dirac nullity conjecture. The nullity … is invariant un…

Let a pentagon be iteratively transformed by the $cP$-evolute operation, where the $cP$-evolute is the polygon whose sides are the perpendicular bisectors of the original polygon's…

Let a conformal polyhedral approximation of the sphere be a conformal polyhedral surface obtained by approximating the sphere with a triangulated polyhedral surface, and let its di…

Let $(x_{\mathbf{i}},\nu_{\mathbf{i}})$ and $(y_{\mathbf{i}},\mu_{\mathbf{i}})$ be independent principal contact elements obtained as trajectory surfaces of a discrete rotating mot…

Weak convergence conjecture. If $\delta_2(P_h,P_h^\star)\to0$ as $h\to0$, then

Let $f^\epsilon$ be the discrete conjugate net approximating a smooth conjugate net $f:\Omega\to\mathbb{R}^3$, with points $f_{ij}^\epsilon$ on $f(\Omega)$ and sufficiently small…