7 problems

Let $r \geq 3$ and let $\mathscr{S} \subset \Pi(n,d,r)$ denote the collection of set partitions that can be made noncrossing by applying at most $r-3$ adjacent transpositions. For…

Character–web conjecture. If $\operatorname{ch}(T)$ is a cluster monomial, then

Let $\mathbb{C}[\operatorname{Gr}^{\sim}(3,n)]$ be the coordinate ring of the affine cone over the Grassmannian. A web invariant is called indecomposable, non-elliptic, or arboriza…

Forest-presentation conjecture. The web invariant $z$ is a cluster monomial if and only if $z=[D]$ for some forest diagram $D$.

Three-term non-cluster conjecture. Then $z$ is not a cluster variable.

Three-term cluster conjecture. Then $z$ and $z'$ are cluster variables, and the displayed relation is an exchange relation.

Cluster-monomial web-basis conjecture. All cluster monomials are web invariants.