9 problems

Oriented-multicut adjacency conjecture. (i) An oriented multicut $\delta'(A)$ is not adjacent to every oriented multicut $\delta'(B)$ such that $B<A^T$. (ii) The orbit represented…

Oriented-multicut facet conjecture. The following inequalities define facets of $OMCUT_n$: (i) zero-extensions of every facet of $OMCUT_{n-1}$, including oriented triangle and non-…

Oriented-multicut skeleton conjecture. (i) All oriented multicuts are extreme rays of $QMET_n$. (ii) The oriented cuts form a dominating clique in $G_{OMCUT_n}$; consequently, the…

Adjacency and diameter conjecture. (i) A triangle facet is adjacent to a facet if and only if they are non-conflicting. (ii) The non-negativity facets $NN_{ij}$ and $NN_{i'j'}$ are…

Let $(M,g)$ be a gravitational instanton, and let an asymptotic cone mean a pointed-Gromov–Hausdorff limit obtained by rescaling $(M,g)$ along a sequence tending to infinity. Metri…

Let $X$ be an essentially non-branching metric-measure space, and let $C(X)$ denote its Euclidean $N$-cone. Euclidean cone curvature-dimension conjecture. The Euclidean $N$-cone…

Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}\ge 0$ and $E(M,p)\not=1/2$. Suppose that $\pi_1(M)$ contains a torsion-free nilpotent subgroup of nilpotency length $l$. An a…

Ridge-graph conjecture. (i) ${{\operatorname{Ri}}}(\text{\sl PMET}_n)$ has diameter $2$, with all non-adjacencies given by $L_{ii}\nsim M_{ik}$;…

Skeleton conjecture. (i) ${{\operatorname{Sk}}}(\text{\sl OCUT}_n)=K_{2S(n,2)}$ and is a subgraph of ${{\operatorname{Sk}}}(\text{\sl WQMET}_n)$. (ii) The complement of…