32 problems

Lemmens–Seidel's conjecture.

For $\alpha\in(0,1)$, let $N_{\alpha}(d)$ be the maximum number of equiangular lines in $\mathbb{R}^d$ with fixed angle $\arccos\alpha$. Bukh's conjecture. For each relevant positi…

Jiang–Polyanskii conjecture.

A set of $m$ lines in ${\mathbb C}^d$ spanned by unit vectors $z_1,\ldots,z_m$ is equiangular if there is a constant $a$ such that … The absolute bound for such a set is…

Balla's conjecture for $1/\alpha$ is odd. For every integer $k\geq 1$ and every $d\geq 1$,

Balla's conjecture. For any $d\geq 1$,

Gerzon's equality conjecture. The only dimensions in which equality in Gerzon's bound is attained are

Let $\alpha\in(0,1)$, define … and let $\kappa=\kappa(\lambda)$ be the spectral radius order of $\lambda$, namely the least number of vertices of a graph whose largest adjacency-ma…

The six equiangular lines are represented by vectors in quaternionic two-space. Regular-720-cell conjecture. The $720$ vectors in $\mathbb{H}^2$ giving the six equiangular lines ar…

For a lattice in dimension $n$, consider the equiangular lines produced by its minimal vectors. Minimal-vector line-count conjecture. For $n=16$, $18$, $19$ and $20$, the largest n…

Let $t'_n$ denote the largest number of lines in an equiangular family obtained from the minimal vectors of a lattice in dimension $n$. Dimension $8$–$13$ conjecture. For…

Representation truncation conjecture. The optimal value of $\mathrm{las}_2$ can be obtained using only the representations

Cubic growth conjecture. Based on the computed data, the corresponding expressions for $D_3(a)$ and $D_4(a)$ are cubic rather than quadratic.

Continuous non-Archimedean functional Zauner conjecture. For every locally compact group $G$ and every $d\in\mathbb{N}$, there exist such collections satisfying: the maps…

Let $d$ be a positive integer, let $q$ be a prime power, and let $\mathbb{F}_{q^2}$ be the quadratic extension of the finite field with $q$ elements. A unitary geometry on…

Existence conjecture. These optimal measurement systems exist for every natural number $n$.

Bukh–Balla–Dräxler–Keevash–Sudakov conjecture. For sufficiently large $n$,

Let $N_\alpha(d)$ be the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos \alpha$. For a positive integer $r\geq 2$, set $\alpha=\frac{1}{2r-1}$. Ba…

Let $M_a(n)$ denote the maximum number of equiangular lines in $\boldsymbol{R}^n$ with common angle $a$. Lemmens and Seidel's conjecture. … Lemmens and Seidel obtained results lead…

Let $\Omega$ be a set of $d(d+1)/2$ equiangular lines in $\mathbf{R}^d$. Here $\operatorname{Inc}(\Omega)$ denotes the maximum size of an incoherent subset of $\Omega$. Absolute-bo…

Let an equiangular set have angle $\alpha=\frac{1}{5}$ and base size $K=4$, and consider a $(4,2)$ pillar and its Seidel graph. Finite-family conjecture for $(4,2)$ pillars. In the…

Let an equiangular set have angle $\alpha$ and base size $K$, and let a pillar be one of the structures in the paper's pillar decomposition. Large-pillar conjecture. There is a con…

The $q$-moment conjecture. If $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, then

Maximal equiangular line sizes conjecture. The maximal numbers of equiangular lines in the specified dimensions are given by the table above.

Let $N_\alpha(n)$ denote the maximum number of equiangular lines in $\mathbb{R}^n$ with angle $\arccos\alpha$. For $\alpha\in(0,1)$, define … and let $k(\lambda)$ be the smallest o…