7 problems

Let $E_2$ denote the second relaxation, or four-point bound, in the semidefinite hierarchy for energy minimization. For five particles on $S^2$, consider all completely monotonic p…

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

Chen–Gonzales–Goodman–Kang–Okoudjou's conjecture. If $p\in(p_{k-1},p_k]$ for $k=1,2,\ldots,d$, then $L_k^d$ minimizes the $p$-frame potential when $N=d+1$.

Let $N=d+1$, let $p\in(0,2)$, and consider systems of $N$ unit vectors in $\mathbb{R}^d$. For an integer $k$ with $0\leq k\leq N-1$, a regular $k$-simplex is a set of $k+1$ unit ve…

Let $N$ particles lie on $S^{n-1}$, and let a $k$-point bound be a lower bound for their energy obtained from the corresponding moment method. Suppose there exists a completely mon…

Let $E_s(\omega_N)$ denote the Riesz $s$-energy of a configuration $\omega_N\subset\mathbb{S}^2$, let $E_{\log}(\omega_N)$ denote its logarithmic energy, and let $\omega_N^{\log}$…

Let $\mathcal{E}_s(N)$ be the minimal Riesz $s$-energy of an $N$-point configuration on $\mathbb{S}^2$, and let $\sigma$ be normalized surface area measure. For $-2<s<2$, $s\neq 0$…