10 problems

Spurious-minima growth conjecture. Based on numerical experiments, the number of spurious local minima in $S^2$ increases “dramatically” with the number of points.

Hesthaven's conjecture.

Fekete–Pommerenke energy conjecture.

Let $S_4$ be the real four-dimensional simplex, and let $F_5$ be the set of $126=\binom{9}{4}$ points constructed by placing the specified boundary points on every face and the rem…

Let $S_3$ be the real three-dimensional simplex, and let $F_5$ be the set obtained by taking the boundary points described in the paper together with the four interior points … max…

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N_4=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_i$ be its fundamental Lagrange polynomials. The degre…

Let $(L_1,\phi_1),\ldots,(L_m,\phi_m)$ be a collection of pairs as above, with normalized volumes as defined in the surrounding text. For each pair, let $\mu_{(L_j,\phi_j)}$ denote…

Let $K$ be a non-pluripolar compact subset of $\C^{n}$, and let $\mu_K:=MA(\psi_K)$ be its pluripotential equilibrium measure. The measure $\mu_K$ is supported on $K$ and is a prob…

Let $E\subset\mathbb{C}^n$ with $n>1$. Consider Vandermonde determinants computed at Fekete points, together with the associated sequences of probability measures. Leja's conjectur…

Abrikosov's conjecture. Fekete points should be organized as nodes of a honeycomb lattice relative to the conformal metric, perhaps with slight disturbances near the boundary.