19 problems

Let $\Omega$ and $k$ satisfy the assumptions in Section 1. Suppose also that $\Omega$ is convex and $\rho_->0$. An optimal-density uniqueness and convexity conjecture asserts that the…

Let $\Lambda_2$, $\Lambda_8$, and $\Lambda_{24}$ be the hexagonal lattice in $\mathbb{R}^2$, the $E_8$ root lattice in $\mathbb{R}^8$, and the Leech lattice in $\mathbb{R}^{24}$, r…

K-contact energy-minimization conjecture. The curl eigenform $\eta$ defined by a $K$-contact structure is always energy-minimizing on $\Xi_\eta$.

Let $\mathbb{S}^2$ be the unit sphere and let $\mathcal{E}_2(\mathbb{S}^2;N)$ denote the minimal Riesz $2$-energy of $N$ points on $\mathbb{S}^2$. Let $\gamma_n(\alpha)$ be the gen…

Integral Fejes Tóth conjecture. The maximum of $I_1(\mu)$ over all Borel probability measures on $\mathbb S^d$ is achieved by $\mu_{ONB}$. The discrete conjecture remains open for…

Fejes Tóth's conjecture. This energy is maximal when $x_1,\dots,x_{d+1}$ are mutually orthogonal and $x_k=x_{k-(d+1)}$ for every $k$ with $d+1<k\le N$. Equivalently, periodically r…

Let $\Gamma_n$ denote the set of convex $n$-gons with each edge of length $1$, and let $E_n^\alpha$ be the energy associated with the increasing continuous function…

Minimality of the rock-salt structure. There exist $\delta_0,\delta_1>0$, depending only on $d$, such that the global minimizer of $E_{f_1,f_2}$ is of the form

Let $p>0$ with $p\notin 2\mathbb N$, and let $\mu$ be a probability measure on $\mathbb S^{d-1}$. Its $p$-frame energy is … A finite discrete measure is a measure supported on fini…

Let $N=m+kd$ points be given in $\mathbb{S}^{d-1}$, where $1\leq m<d$ and $d\geq 2$, and let $\mathbf{A}\in\mathbb{R}^{N\times N}$ be their Gram matrix. The repeated orthonormal se…

Let $\mathbb{A}$ and $\mathbb{B}$ be the annular domains appearing in the class $\mathcal{F}(\mathbb{A},\mathbb{B})$ of admissible mappings, and let…

Let $E_2$ denote the second relaxation, or four-point bound, in the semidefinite hierarchy for energy minimization. For five particles on $S^2$, consider all completely monotonic p…

Let $\alpha>0$ and $t>0$, and let $\tilde{E}_t(y;\alpha)$ be the reduced theta energy for the body-centred-orthorhombic family, with $t_0(\alpha)$ defined by the preceding lemma. C…

Let $f(t)=e^{-\alpha t^2}$, with $0<\alpha<\pi e$. For configurations of density $\rho$ in $\mathbb{R}^n$, let the minimal $f$-energy be the infimum of the corresponding Gaussian p…

Let $S^2$ be the unit sphere, and let a completely monotonic potential function be a potential function whose derivatives alternate in sign on the relevant domain. For five points…

Let $\varphi(x)=e^{-c|x|^2}$ in $\mathbb R^n$, where $n\in\{8,24\}$, and let $h$ be the limiting auxiliary function from the sharp energy-bound convergence conjecture. Let…

Square-antiprism conjecture. For every completely monotonic potential function $f$ on $(0,4]$, some square antiprism minimizes the energy $E_f$ among all eight-point codes in $S^2$…

Let $\Omega$ and $\Omega^*$ be bounded $k$-connected domains in $\mathbb{C}$, where $k\geqslant 2$. Suppose that $\Omega\subset\Omega^*$ and that the inclusion is a homotopy equiva…

Cohn–Kumar conjecture. For $n=2$, $8$, or $24$, the linear programming bounds for potential energy minimization in $\mathbb{R}^n$ are sharp for every completely monotonic potential…