5 problems

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

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…