8 problems

A Petrie scheme of an abstract polytope is the shortest cycle or bi-infinite flag sequence generated by a product of the flag-adjacency involutions in a permutation of all ranks; i…

Finiteness conjecture. In this case, ${\mit \Gamma}$ is finite if and only if $(s,t)=(1,2),(1,3),(1,4)$ or $(2,3)$, up to interchanging $s$ and $t$. This is a concrete instance of…

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 $p\geq 3$ be prime. The regular polytope $\mathcal{C}_{p,p,3}$ has facets $\mathcal{M}_{p,p}$ and vertex-figures $\mathcal{M}_{p,3}$. Universality conjecture for…

Let $p\geq 3$ be prime. The regular polytope $\mathcal{C}_{p,3,3}$ has facets $\mathcal{M}_{p,3}$ and vertex-figures $\{3,3\}$. Universality conjecture for $\mathcal{C}_{p,3,3}.$ F…

Let $A_n$ denote the alternating group of degree $n$. A self-dual string C-group is a string C-group representation whose associated polytope is self-dual. Existence conjecture. Fo…

A string C-group of alternating type is a string C-group whose underlying group is an alternating group; for $n\geq 12$, the known constructions give alternating groups a rank at l…

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