4 problems

Let $V_{\alpha}(\bar x)$ be the potential of a generic configuration of point charges of the same sign, and let $a_{\alpha}^j$ denote the number of its critical points of index $j$…

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 $\#_j$.

Voronoi degree conjecture. The Voronoi degree of a generic hypersurface of degree $d$ in $\mathbb{R}^n$ equals

Let the unit disc be equipped with the uniform probability distribution. For an optimal set of $n$-means, consider the associated Voronoi regions. Conjecture on optimal quantizers…