9 problems

Haag–Kertzer–Rickards–Stange conjecture. The sporadic set $S({\mathcal P})$ is finite.

Let ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral Apollonian packing, and let $C_{{\mathcal{P}^{\operatorname{full}}}}(X)$ be the number of circles of curvature at…

Let $C_{{\mathcal{P}^{\operatorname{pr}}}}(X)$ be the number of circles of curvature at most $X$ in a prime component ${\mathcal{P}^{\operatorname{pr}}}$, and let…

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 ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral packing, and let $\Sigma$ be the union of the residue classes modulo $24$ that have representatives in…

Consecutive-curvature sequence conjecture. There is a sequence of consecutive tangent spheres

Curvature realization conjecture. The set of curvatures of $\Omega(\mathcal{B}_{\mathcal{R}^4})$ is $\mathbb N$.

Local-global conjecture for $K$-Apollonian packings. There exists an $M$ dividing $24$ such that all sufficiently large integers whose residues modulo $M$ lie in $S_M$ occur as red…

Graham–Lagarias–Mallows–Wilks–Yan conjecture. All sufficiently large integers with residues in $S$ occur as curvatures in $\mathcal P$.