14 problems

Shortest 2-design curve conjecture. The curve $\gamma$ is the shortest $2$-design curve in $S^2$, up to isometry and reparameterization. The conjecture is motivated by the precedin…

Let $d>3$. An equiangular tight frame (ETF) with parameters $(r,n)$ is a set of $n$ unit vectors in $\mathbb{S}^{r-1}$ whose coherence attains the Welch bound. XXY's ETF existence…

Projective-index classification conjecture. Every finite group with a linear action on $\mathbb{C}^2$ has one of these five projective-index sets and associated canonical groups.

Folklore conjecture. Let $C\subset\mathbb{S}^{n-1}$, with $n\geq 2$, be a spherical 4-distance 7-design. Then $C$ is a tight spherical 7-design. In particular,

Monotonicity conjecture. For each fixed positive integer $p$, the sequence $R_p(n)$ is non-increasing in $n$.

Let $d\geq 5$. An equiangular tight frame (ETF) with parameters $(d,n)$ is a real ETF for $\mathbb{R}^d$ of size $n$. ETF extension conjecture. There exists an ETF with parameters…

Bannai et al.'s classification conjecture. One of the following holds: (1) $n=2$ and $C$ is a regular hexagon or a regular heptagon; (2) $C$ is a tight spherical 5-design; or (3)…

Let $p>0$ with $p\notin 2\mathbb N$, and let $\mu$ be a probability measure on $\mathbb S^{d-1}$. Its $p$-frame energy is … A finite discrete measure is a measure supported on fini…

Worst-case error monotonicity conjecture. For fixed $t$, if $N'>N$, then

Let $g(n)$ denote the maximum cardinality of a spherical two-distance set in $\mathbb{R}^n$. Maximum spherical two-distance set conjecture. … The conjecture reflects the observed p…

Let $\Lambda$ be a lattice, and let $M_1(\Lambda)$ and $M_2(\Lambda)$ denote its first and second layers. A finite subset of a sphere is a spherical $2$-design when it satisfies th…

The discrepancy asymptotic conjecture. If the fundamental conjecture for optimal Riesz energy on spheres holds, then

Convergence conjecture. The squared worst-case error in the Sobolev space $\mathcal{H}^{3/2}$ satisfies

Shell design conjecture. The following assertions hold: