11 problems

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 $\#_j$.

Maxwell's conjecture. The electrostatic potential defined by $n$ point charges in $\mathbb{R}^3$ has at most $(n-1)^2$ equilibria.

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…

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…

Consider a configuration of $N$ fixed electric charges in $\mathbb{R}^3$ having finitely many points of equilibrium. Maxwell's conjecture. The number of equilibrium points is at mo…

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

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.…

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 $d$…