49 problems

Let $N(d,t)$ denote the smallest cardinality of a spherical $t$-design on the $(d-1)$-dimensional sphere, and let $C_d$ be a sufficiently large positive constant depending only on…

A spherical $t$-design on $\mathbb{S}^{d}$ is a finite set of points on the sphere that exactly integrates every polynomial of degree at most $t$ in the usual spherical-design sens…

Let $d$ and $t$ be positive integers, and let $N(d,t)$ denote the minimal number of points in a spherical $t$-design on the unit sphere … A spherical $t$-design is a finite point s…

Classification conjecture for 24-point spherical 5-designs. There are no 24-point spherical 5-designs in $S^3$ other than those in the three-dimensional family of deformations of…

Nonexistence conjecture for universally optimal 24-point codes. There is no universally optimal spherical code of 24 points in $S^3$.

Exponent reduction conjecture. The exponent $\frac{n(n+1)}{2}-1$ can be reduced to $\frac{n(n-1)}{2}$.

For $n\geq 3$, let $k_{\operatorname{max}}(n)$ denote the maximal strength of a spherical design in $\mathbb S^{n-1}$ that is an orbit of a finite subgroup of $O(n)$. Bounded-orbit…

Let $\tau(N)$ be the largest integer $t$ for which an $N$-point three-dimensional spherical $t$-design exists. Since every $t$-design is also a $t'$-design for $t'\leq t$, an $N$-p…

A spherical $t$-design is a finite set of points on the unit sphere whose averages agree with spherical averages for polynomials of degree at most $t$. Let $N$ denote the number of…

A spherical $t$-design is a finite set of points on the unit sphere whose averages agree with spherical averages for polynomials of degree at most $t$. Let $N$ denote the number of…

Let ${\cal C}_m$ be the Clifford group acting on the sphere associated with the code and invariants discussed above. For $m=2$, there are real points on $S^3$ at which the harmonic…

Shortest 3-design curve conjecture. The curve $\gamma$ is the shortest $3$-design curve in $S^{2k-1}$, up to isometry and reparameterization. This conjecture is suggested by numeri…

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…

Asymptotic sharpness conjecture. This lower bound is asymptotically sharp up to constants for each $d$. This is known to be true for $d\leq 3$, while the corresponding assertion in…

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…

Let an $m$-stiff configuration be a spherical $(2m-1)$-design in the unit sphere $S^{d-1}$ whose points lie on $m$ parallel hyperplanes. Let $d>2$ be an integer. Classification con…

Let an $m$-stiff configuration be a spherical $(2m-1)$-design in the unit sphere $S^{d-1}$ whose points lie on $m$ parallel hyperplanes. Non-existence conjecture. There is no $m$-s…

Consider the potentials $t^{2k}+5alpha C_{2k}^2(t)$ and $|t|^p$ on the sphere, and let $\mu_{\alpha \to 0}$ denote a minimizer of the first potential in the limit $\alpha \to 0$. L…

Let $d$ be odd, and let a spherical $3$-design on $S^d$ mean a finite subset of the $d$-sphere whose averages agree with spherical averages for all polynomials of degree at most…

Let $t_1,t\in\mathbb{N}$ with $t_1<t$. A spherical $t_1$-design is a finite multiset on $\mathbb{S}^d$ reproducing spherical integration for polynomials of degree at most $t_1$, an…

Let $t_1,t\in\mathbb{N}$ with $t_1<t$. A spherical $t_1$-design is a finite multiset on $\mathbb{S}^d$ reproducing spherical integration for polynomials of degree at most $t_1$; a…

Exceptional-index conjecture. The exceptional indices are determined by the projective indices according to

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.

QMC strength conjecture. It is conjectured that the strengths of Fekete points, equal area points, and Coulomb energy points are, respectively, $1.5$, $2$, and $2$.

Let $s_{2n}$ denote the best covering radius of a configuration of $2n$ points on $S^{n-1}$, and let $s_{2,2n}$ and $s_{3,2n}$ denote the corresponding covering-radius quantities a…