23 problems

Let $n$ point charges be placed in Euclidean space, and consider their electrostatic field or potential away from the charges. Maxwell's conjecture. The field of $n$ point charges…

Consider the random tiling model on a plane bipartite lattice, with the lattice embedded in the plane in a suitable way. Universality conjecture. The validity of the model does not…

Let monomers be holes in the square lattice, with two-point correlations defined by the random tiling model. Krauth–Moessner conjecture. The two-point correlations of monomers beha…

Let a finite configuration of positive charges be given, and let $\alpha>0$. Consider its nondegenerate equilibrium points and its effective Voronoi cells. Effective Voronoi-cell b…

Let an interior lattice structure be extended periodically to an infinite periodic arrangement, and consider the electrostatic potential associated with this extension. Lattice arr…

Linear slice conjecture. The restriction of $V$ to any straight line or flat plane in $\mathbb{R}^3$ has at most $O(n)$ equilibria.

Gabrielov–Novikov–Shapiro conjecture. For all $p\geq 1$, the number of equilibria of the restriction of $V_p$ to any straight line in $\mathbb{R}^3$ is at most $2n-1$.

Gabrielov–Novikov–Shapiro conjecture. For $n$ unit point charges in generic position, and all $p\geq 1$, the number of index-$j$ equilibria of $V_p$ is at most $\#_j$.

Consider ${\rm S}^{1}$-symmetric black holes whose fixed-point set, called the axis, is compact, and suppose the black hole has ends at which its charge is nonzero. Compact-axis bl…

The Myers–Korotkin–Nicolai (MKN) black holes provide the topologies under consideration, and a solution of the type discussed immediately before is an ${\rm S}^{1}$-symmetric charg…

Consider an ${\rm S}^{1}$-symmetric charged electrostatic solution, and call the fixed-point set of the ${\rm S}^{1}$-symmetry the axis. Compact-axis conjecture. ${\rm S}^{1}$-symm…

Let $N$ positive electric charges be fixed in $\mathbb{R}^d$. Finiteness conjecture. The configuration has only finitely many points of equilibrium. The source calls this a folklor…

Electric-zero bound conjecture. The quantity $Z(K)$ satisfies

Let $zi e zj$ be point locations in $mathbb{R}^2$, let $miinmathbb{R}$ be charges, and consider the electrostatic potential … Let the associated Voronoi complex encode the arrangem…

The planar zero-set conjecture. The zero set of a two-dimensional finite electrical field $F$ consists of a finite set of points.

Let a hypersphere and a plane stereographically connected to it be given, and consider solutions of the Poisson equation on each of these spaces. Stereographic correspondence conje…

Let $p=(m,n,k)$ range over the sites of the simple cubic lattice $\mathbb{Z}^3$, and assign charges of equal absolute magnitude to these sites, subject to a prescribed total charge…

Same-sign equilibrium finiteness conjecture. For any such set of charges, the set of equilibrium points is finite.

Ciucu's electrostatic conjecture. Taking the exponential of the negative of this electrostatic energy recovers, up to a multiplicative constant, asymptotic expressions for the dime…

Let $\Gamma$ be the set consisting of the boundary of the union of the two unit disks centered at $-1$ and $0$, together with the line segment connecting their intersection points.…

Convex-curve conjecture. The zeros of $G(d;z)$ in $A_2$ are located on a curve which is convex from the left.

Alternating-location conjecture. If $d$ is even, all the zeros of $G(d;z)$ in $A_1$ are the zeros on the unit circle $\mathcal C_0$ given in that proposition. If $d$ is odd, the ze…

Zero-separation conjecture. For every positive integer $d$, the zeros of $G(d;z)$ form three groups separated by the sets $A_1$, $A_2$, and $A_3$, except that when $6$ divides $d$…