22 problems

Classification conjecture. Then $d=2,3,6,7,23$, and $\Omega$ is isometric to the corresponding example found in the examples section.

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

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

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…

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…

For $\alpha\in(0,1)$, let $\lambda=(1-\alpha)/(2\alpha)$, and let $k(\lambda)$ denote the minimum number of vertices of a graph whose largest eigenvalue is exactly $\lambda$, or…

Asymptotic linear bound conjecture. The optimal objective satisfies

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…

Let $d\geq 2$. An equiangular line system in $\mathbb{C}^d$ is a set of lines through the origin whose unit-norm representatives have constant squared pairwise inner-product magnit…

Let $N_{\alpha}(r)$ be the maximum number of a system of equiangular lines in $\mathbb{R}^r$ with common angle $\arccos\alpha$. The Lemmens–Seidel conjecture concerns the case…

Jiang–Polyanskii conjecture. If $k(\lambda)=\infty$, then

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

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…

Let $N_\alpha(n)$ denote the maximum number of equiangular lines in $\mathbb{R}^n$ with common angle $\arccos\alpha$. Bukh's conjecture. If $1/\alpha$ is an odd natural number, the…

Let $N_{1/(2r-1)}(d)$ denote the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos \frac{1}{2r-1}$, where $r\geq 2$ is an integer. Asymptotic equiang…