8 problems

Let $(M,V,varepsilon)$ be a closed connected weighted marked surface, where $varepsilon:V\rightarrow\{0,1\}$. Consider an initial Euclidean or hyperbolic polyhedral metric induced…

Let a Jordan domain be approximated by inversive distance circle packings, and let the associated discrete conformal maps be the maps induced by those packings. Bowers–Stephenson's…

Let $(M,V,\varepsilon)$ be a marked weighted connected closed surface with $\varepsilon:V\rightarrow\{0,1\}$, and suppose there exists a PL or PH metric generated by a discrete con…

Let $(M,V,\varepsilon)$ be a closed connected weighted marked surface, where $\varepsilon:V\rightarrow\{0,1\}$, $\chi(M)=0$, and $d$ is a piecewise linear metric on…

Hexagonal square-tiling rigidity conjecture. All squares $S_i$ have the same size.

Strong discrete uniformization convergence conjecture. The discrete uniformization maps $f_n$ associated to these data converge uniformly to the Riemann mapping associated to…

Let $4\mathcal{T}4$ be a plane triangulation graph, with EQ-type, VEL-type, and CP-type denoting the discrete types associated respectively with equilateral triangulation geometry,…

LQG scaling-limit conjecture for random planar maps. Liouville quantum gravity describes the scaling limit of random planar maps under discrete conformal embeddings.