70 problems

Let $\mathbb W$ denote the renormalized Coulomb energy of unit-density point configurations in the plane. The triangular lattice conjecture. The triangular lattice is a global mini…

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 $N$ be a Fibonacci number, let the Fibonacci lattice be the corresponding lattice point configuration on the torus, and let $f$ be a potential in the broad class of potentials…

Consider the optimization problem for five points on a sphere under the potential $\operatorname{sign}(s)/x^s$, where $s$ is the parameter. Let the triangular bipyramid be the conf…

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…

A configuration $\mathcal P$ is universally optimal if it minimizes energy for every sufficiently rapidly decreasing potential $\varphi$ that is completely monotonic as a function…

Let the Jacobi theta function of a three-dimensional lattice be the theta function referred to in the source, and let FCC and BCC denote the face-centered-cubic and body-centered-c…

Let $f(r)=-\log r$ be the logarithmic potential, and consider configurations of $11$ points on $S^3$. A flat torus in $S^3$ is the product of two circles of radius $1/\sqrt{2}$ in…

View $\mathbb{R}^4$ as the orthogonal direct sum of two planes, and let $T\subset S^3$ be the flat two-dimensional torus given by the product of the circles of radius $1/\sqrt{2}$…

Let $\mathcal{C}_n\subset S^{n-1}$ be the spherical code consisting of one north pole and two dual $n$-point simplices, with $\alpha$ the unique root in $(0,1/n)$ of … The cosine o…

A lattice is universally optimal among lattices if it minimizes energy for every completely monotonic potential among lattices in the relevant Euclidean space, after the prescribed…

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…

For $n\geq 3$, a universal optimum in $\mathbb{R}^n$ is a point configuration minimizing energy for every completely monotonic potential function. Finiteness conjecture. For each…

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

Near-minimal local-minima conjecture. For large numbers of particles, there are many local minima of $V$ with energies very close to the lower bound.

Let $\phi$ be a completely monotonic potential of distance squared in dimension $2$, and consider point configurations at fixed density. The two-dimensional completely monotonic gr…

Let $r>0$, let $z_1,\ldots,z_n$ be points with $|z_i-z_j|\geq 2r$ whenever $i\ne j$, and let … Let $\mathbb{A}$ be the repulsion matrix associated with $E$, and let $\vec 1$ and…

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…

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…

Let $C_{s,d}$ be the limiting normalized minimal Riesz $s$-energy constant, and for a lattice $\Lambda$ let … be its Epstein zeta function. For the hexagonal lattice…

Let $\Omega=\mathbb{FP}^d$ be a projective space, and let $I_{F_s}$ denote the energy associated with the kernel $F_s$. Let $s^*\in[-2,-1]$ be the unique value such that $I_{F_s}$…

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…

Optimal boundary-condition conjecture. For every $lambda\band [0,infty]$, the function $Q_phi$ minimizes

Twelve-vortex branch conjecture. This branch is born stable and minimises $H$ for fixed small positive values of $\mu$.

Ten-vortex asymmetric-branch conjecture. The stable branch that minimises $H$ for small fixed values of $\mu$ is asymmetric.