11 problems

Global minimizer conjecture. The hexagonal lattice $\mathsf{A}_2$ is the global minimizer of these energies among point configurations at fixed density. The paper proves only local…

Hexagonal crystallization conjecture. The minimum over all two-dimensional lattices of unit density always exists and is always achieved at the hexagonal lattice.

Signed-tiling conjecture. If $n=3$ or $n=6$, then the contribution of $L$ is $0$ unless there exists a signed tiling of $L$ by stones, bones, and snakes; if such a signed tiling ex…

Hexagonal-lattice random-current conjecture. All phase transitions of the random current and random-cluster model on the hexagonal lattice coincide:

Let $f:\mathbf{R}^2\to\mathbf{R}$ be a radial Schwartz function, and write $r=|x|$. Cohn–Elkies conjecture. There exists such an $f$ satisfying … … … For the hexagonal lattice, the…

Let $r_1,r_2,\ldots$ be the positive real numbers of the form … where $j,k$ are integers. A radial Schwartz function is a function $f:\mathbf{R}^2\to\mathbf{R}$ invariant under rot…

Let $\mathcal{C}\subset\mathbb{R}^2$ be an infinite point configuration with unit density, and let $f$ be completely monotone with $|f(r)|=O(r^{-1-\varepsilon})$ as $r\to\infty$ fo…

Let $\mathcal{C}\subset\mathbb{R}^2$ be an infinite point configuration with density … where $B_R$ is the ball of radius $R$ centered at $0$. Let $f$ be completely monotone and sat…

In two dimensions, consider the linear programming bounds for energy minimization and the hexagonal lattice. Cohn–Kumar sharpness conjecture. The linear programming bounds should b…

Let $r_1,r_2,\dots$ be the positive real numbers of the form … where $j$ and $k$ are integers. Radial Schwartz functions $f\colon{\mathbb R}^2\to{\mathbb R}$ are considered through…

Universal-cover connective-constant conjecture. There exists $k\in\mathbb N\cup\{0\}$ for which