12 problems

Let ${\mit \Gamma}$ be the group with generators $\rho_0,\ldots,\rho_5$ and the standard Coxeter relations for the Schl"afli type $\{3,3,3,4,3\}$, together with the extra relation…

Let $d\geq 3$ and $n\geq 3$ be integers. Consider a regular $d$-polytope of type $\{n,n,\dots,n\}$ with generating involutions $\rho_0,\rho_1,\dots,\rho_{d-1}$ and automorphism gro…

Let $\beta_f$ denote the Schl"afli symbol corresponding to a regular polytope represented by the polynomial $f$. Define $g(\beta_f)$ to be the least integer $s$ such that every int…

Let $\beta_f$ denote the Schl"afli symbol corresponding to a regular polytope represented by the polynomial $f$. Define $g(\beta_f)$ to be the least integer $s$ such that every int…

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…

All-net conjecture. The 24-cell, the 120-cell, and the 600-cell fail to be all-net.

Let $n$ be sufficiently large, and let $k$ satisfy … Consider regular polytopes of rank $n-k$ in the setting of the classification results discussed in the paper. Extension conject…

Let $n\geq 12$. An abstract regular polytope with automorphism group $\mathrm{A}_n$ has rank $d$, where rank means the number of involutory generators in its string C-group represe…

Polar-duality conjecture. If the first sum attains its maximum at a point $N$, then the second sum attains its minimum at $N$, and conversely. If the first sum is independent of th…

Core semisparsity conjecture. If $N$ is semisparse in $W$, then $\operatorname{Core}(W,N)$ is also semisparse.

Let $q$ be an odd square prime power. Let $L_2(q)$ be the simple projective special linear group, and let $b$ and $c$ be the outer automorphisms described above, with…

The 2-skeleton conjecture. The complex $\mathcal K$ coincides with the $2$-skeleton of $\mathcal P$.