29 problems

Let $\mathcal{P}$ be an integral circle packing, let $\mathcal{K}\subset\mathbb{Z}$ be its set of curvatures, and let $\mathcal{K}_a$ be the set of integers passing all local obstr…

Let a simply-connected, bounded Jordan domain be approximated by a sequence of electrical networks, for instance networks arising from a sequence of disk packings. Choose conductan…

A circle packing is a packing of congruent circles in the plane, and a packing is diffusively dominant if its mass distribution diffusively dominates that of every other packing. C…

Let $E$) be a moiety of a bounded primitive Eisenstein circle packing. Let $N$ be a positive real number, and let $c_E$ denote a constant depending on $E$. Let…

Let $\mathcal P$ be an Apollonian circle packing with symmetry group $\Gamma<\operatorname{PSL}_2(\mathbb C)$. For any circle $C\in \mathcal C_\Lambda^*$, consider its orbit…

Equivalence conjecture for infinite ideal polyhedra. The following statements are equivalent, with the Euclidean alternatives applying to IIP and the hyperbolic alternatives applyi…

Finite-subcomplex relaxation conjecture. Condition (3) can be relaxed so that it is required to hold only outside a finite sub-complex of $\mathcal{D}$.

Uniformization of the CRF. The following statements are equivalent: the combinatorial Ricci flow on $\mathcal{T}$ converges for every initial value in hyperbolic background geometr…

Let ${\mathcal{P}^{\operatorname{full}}}$ be an ambient primitive integral Apollonian packing. A prime component ${\mathcal{P}^{\operatorname{pr}}}$ is a component consisting of th…

Let $G$ be a CP hyperbolic triangulation, without necessarily assuming bounded degree or nonamenability. Extension conjecture. The statements of Theorems $\mathrm{thm:main0}$ and…

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.

Let $G$ and $\hat{G}$ be a pair of simple, 3-connected plane graphs corresponding to a cellular decomposition of $\mathbb{C}$ and its dual. A circle configuration is a configuratio…

Connelly's conjecture. There exists no circle packing of the triangulated plane with $r$ greater than the ratio attained by the reference packing shown in the figure.

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,…

Let $M\geq1$, and let $\mathcal{S}$ be a packing of closed elliptical disks whose aspect ratios are at most $M$. Let $s_0\in\mathcal{S}$. Ellipse-packing critical and exponential-d…

Let $E_{\mathcal{S},p}(s_0,r)$ be the event that $s_0$ is connected to distance $r$ by open disks in site percolation with parameter $p$. Uniform exponential decay conjecture. Ther…

Let $4\mathcal{S}4$ be a circle packing and let $s_0\in\mathcal{S}$. For $0<p<1$, let $E_{\mathcal{S},p}(s_0,r)$ denote the event that $s_0$ is connected to distance $r$ by open di…

Converse quantile conjecture. If $[\boldsymbol{G},\boldsymbol{o}]$ has a point-stationary CP whose CP-cocycle is $\boldsymbol{c}$, then the quantile condition $$ holds along some s…

Let $S$ be an oriented closed surface of genus $g\geq 2$, let $\tau$ be a polygonal cell decomposition of $S$, and let $c$ be a complex structure on $S$. A complex projective struc…

For each $n\in\mathbb{N}$, let $\Pi_n$ denote the set of compact packings of the plane with $n$ distinct circle radii, represented by radius ratios $(r_{n-1},r_{n-2},\ldots,r_1)$.…

Let $f\in C_{\hat{\alpha},\hat{\eta}}$ be a flat label, where $C_{\hat{\alpha},\hat{\eta}}$ denotes the space of labels with the prescribed data $\hat{\alpha}$ and $\hat{\eta}$. A…

Local-global conjecture. Every sufficiently large admissible integer from $\mathcal{P}$ is a curvature. Equivalently,

Nine-point packing conjecture. There is one unique, up to isometry, optimal arrangement of 9 points in ${\mathbb T}^2$, shown in the paper's Figure 9, with

Collectively jammed density conjecture. If such a packing is collectively jammed and is not a triangular packing, then its density is at most

Density Gap Conjecture. If $n+1$ is a triangular lattice number but $n$ is not, then the most dense packing of $n$ equal disks in the triangular torus has density