11 problems

Let $n$ satisfy $2\leq n\leq 8$, and let $2^k$ be the largest power of $2$ strictly less than $n$. A sphere packing is weakly recurrent and dense when it has the corresponding recu…

Layered extremal word conjecture. If $\Pi$ is a set of layered patterns, then

Let $a<b$ be coprime positive integers, and let … For a finite set $S\subset\mathbb{Z}$, write $d_p(S)$ for its packing density and $d_c(S)$ for its covering density. The packing a…

Connelly's uniformity conjecture. The ternary triangulated packing numbered $53$ is the most uniform triangulated packing other than HCP. Its proposed deformation yields the highes…

Density-maximizing conjecture. For the ratios $r_b$ and $r_c$, the periodic nontriangulated binary packing depicted in Fig. 7 maximizes density among all binary packings with the s…

Kertész's density conjecture. If ${\bf Q}$ contains a TS-packing of $N$ unit balls, then

Let $B^d$ be the unit ball in $d$-dimensional Euclidean space, let $\delta_\rho(B^d,n)$ be the optimal parametric density for a finite packing of $n$ balls with parameter $\rho>0$,…

Consider packings by $n$ different given discs. A packing is saturated if no further disc can be added. Saturation conjecture. If the densest compact packing is saturated, then the…

Let $d\geqslant 2$ be fixed. For a $d$-dimensional convex body $C$, let $\delta(C)$ and $\theta(C)$ denote its packing and covering densities by congruent copies. Kuperberg's conje…

Isostatic conjecture. The density $\rho$ of every such packing satisfies

Square-torus Markov-bound conjecture. Every such packing satisfies $M\leq 6.25$. Equivalently,