25 problems

Circle patterns are configurations of circles with prescribed combinatorial structures. A conformal mapping is a structure-preserving map between planar domains that preserves angl…

Kojima–Mizushima–Tan conjecture. The map

Let $S$ be a closed, oriented surface of genus at least $2$, let $\Gamma$ be the 1-skeleton of a triangulation of $S$, let $\mathcal C_\Gamma$ be the space of pairs $(\sigma,C)$ co…

Generalized Andreev conjecture. There exists a compact convex hyperbolic polyhedron $Q$ combinatorially equivalent to $P$ whose dihedral angles are given by $\Theta$, and $Q$ is un…

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 $Z^c$ be the square grid circle pattern with $0<c<2$, whose circles are labeled by … and which covers the infinite sector with angle $c\pi/2$, with the centers of the border ci…

Let $\Omega$ be a projectable cell complex, let $P:\Omega\to\mathbb{C}$ be its embedding into $\omega=P(\Omega)$, and let $z$ be a solution on $\mathbb{Z}^n_+$ of the cross-ratio e…

Uniqueness conjecture. There is only one initial value $p_0$ such that $p_n>0$ for every $n\in\mathbb{N}$.

Let $Z^{\gamma}_{n,m}$ be the discrete map considered in the paper, with parameters $\gamma$ and $\alpha$, and let $c(\gamma)$ be the constant obtained from the asymptotics of…

Embeddedness conjecture. The hexagonal circle pattern $z^c$ is embedded: the interiors of all elementary quadrilaterals…

Let $0<\alpha=\beta<\frac{1}{2}$ and set $\gamma=1-2\alpha$, so that $0<\gamma<1$. Consider the hexagonal circle patterns $z^{3\alpha}$ with an intersection point at the origin and…

Let $P(\Theta,R^{\dagger})$ be the space of circle patterns with prescribed intersection angles $\Theta$ and reference radii $R^{\dagger}$, and let $\mathcal{T}_{\mathrm{WP}}$ deno…

Let $Z_1$, $Z_2$, and $Z_4$ denote the conditions referenced in the theorem above, and let the theorem assert the existence result under the stronger condition $Z_4$. Weaker-condit…

Let $S$ be a closed, oriented surface of genus at least $2$. An admissible weighted graph $(\Gamma,\theta)$ in $S$ is a graph embedded in $S$ that is the 1-skeleton of a cell decom…

Ideal circle-pattern uniformization conjecture. The following statements are equivalent: the combinatorial Ricci flow on $\mathcal{T}$ converges for every initial value in hyperbol…

Circle-pattern symplectic non-degeneracy conjecture. The pullback is non-degenerate, and hence defines a symplectic structure on the space of circle patterns.

Circle-pattern homeomorphism conjecture. The space of circle patterns is homeomorphic to the Teichmüller space of the closed torus.

Let $P(\Theta)$ be the space of circle patterns with a fixed Delaunay angle structure $\Theta$, let $P(S_g)$ be the space of marked complex projective structures, and let…

Let $S_g$ be a closed surface of genus $g$, let $E$ be the edge set of a triangulation of $S_g$, and let $\Theta:E\to[0,\pi)$ be a Delaunay angle structure. Let $P(\Theta)$ be the…

Let a closed surface be equipped with a fixed triangulation, and prescribe intersection angles for its Delaunay circle patterns. Kojima–Tan–Mizushima conjecture. The space of such…

Holonomy–Teichmüller conjecture. For any Delaunay angle structure $\Theta$ on a closed triangulated surface with genus $g>1$, the holonomy map is an embedding of a manifold homeomo…

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…

A biperiodic circle pattern is a square-grid circle pattern invariant under two independent lattice periods. Miquel integrability conjecture. For biperiodic circle patterns, Miquel…

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…