33 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…

Let $E_f(L)$ denote the lattice energy of a two-dimensional lattice $L$, and consider the potential $f(r)=e^{-\beta\pi r}-e^{-\alpha\pi r}$, where $\alpha$ and $\beta$ are paramete…

Let $e_{\rm UEG}$ denote the energy per unit volume of the three-dimensional uniform electron gas at unit density, and let $\zeta_{\rm BCC}(1)$ be the Epstein zeta value at $1$ for…

Crystallization conjecture. Interacting particles spontaneously arrange in periodic structures at low temperatures.

A ground state is a configuration of interacting particles that minimizes the system's energy, and a periodic lattice is a particle arrangement invariant under translations by the…

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

Minimality for $\textnormal{E}_{\textnormal{BPD}}$ in general. For all $N\in\mathbb N\backslash\{0,1\}$, we have

Macroscopic-scaling conjecture. In this case, there should exist $s>d$, a nontrivial measure $\mu$ on $\mathbb{R}^d$, and a sequence of minimizers $(X_N)_{N\geq 2}$ of $E_N$ such t…

Let $L$ be a two-dimensional lattice of area $A$, and let $E_f(L)$ denote its lattice energy for a potential $f$. A potential is completely monotone if it satisfies…

In Euclidean space of dimension $d\in\{2,3\}$, consider systems of interacting particles and their ground state energy in the thermodynamic limit. Crystallization conjecture. The g…

Let $\mathcal{X}_{s,\mu}$ denote the Riesz equilibrium infinite configurations. For $s>d$, let $r=\mu^{-1/s}$; for $s<d$, let $r=\rho_b^{-1/d}$, where $\rho_b>0$ is the background…

Let $s\in(-2,\infty)\setminus\\{d\\}$, and let $e(s)$ denote the minimal energy per unit volume of the Riesz gas. For a Bravais lattice $\mathscr{L}$ with unit-volume fundamental c…

For a Riesz interaction in the Coulomb case $s=d-2$, or for the logarithmic interaction $f(r)=-\log r$ in dimension $d=2$, consider electrons embedded in a uniform background of po…

Let $\mathsf{F}_N$ denote the energy functional of the two-dimensional one-component plasma for $N$ particles, and call its minimizers Fekete points. The lattice property concerns…

A periodic lattice is a discrete subgroup of Euclidean space whose quotient is compact. The crystallization conjecture. Most ground states of physical interacting systems, especial…

Wigner crystallization conjecture. The minimum of the jellium energy should be achieved by a crystal, called a Wigner crystal.

Let $\mathsf A_2$ denote the triangular lattice in two dimensions, and let $V$ be a pair potential. Consider the corresponding two-body energy and its asymptotic energy per point f…

Let $r_0>0$. For a Bravais lattice $L$ in dimension $3$, let $E_{\alpha,r_0}[L]$ denote the Morse lattice energy, let $\mathcal{L}_3$ be the class of Bravais lattices, and let…

Crystallization conjecture. As $n\to\infty$, the minimizers of $F_n$ over all possible points $y_1,\ldots,y_n\in\Omega$ arrange themselves in a periodic lattice.

Let $f$ satisfy $f(r)=F(r^2)$, where $F$ is a completely monotone function, meaning the Laplace transform of a nonnegative Borel measure. Let $e^\rho$ denote the equidistant altern…

Let $\mathbb{W}$ denote the renormalized energy of an infinite configuration of points, and consider its restriction to lattice configurations in low dimensions. Crystallization co…

Let $C_{\mathrm{Jel}}(s,d)$ be the Jellium next-order constant obtained by minimizing over general configurations, and let $C_{\mathrm{Jel}}^{\mathrm{lattice}}(s,d)$ be the corresp…

In two dimensions, consider the renormalized energy for the Riesz gases with power-law exponents in the range under consideration. Abrikosov's conjecture. The renormalized energy i…

Let $\overline{\mathcal{A}}_1$ be the class of admissible electric fields with unit background density, and let … be the minimum value of the renormalized energy. The infimum is at…

Let $bc\rm M\floor$ be a submanifold of $c2^d$ of dimension $d'$, and consider the hypersingular interaction $V(x)=|x|^{-s}$ with $s>d'$. For $N$ particles constrained to…