4 problems

Eight-face conjecture. The facet of $\mathbf{N}_{3,n}$ minimizing $\gamma_{j_1,j_2,j_3}$ has exactly eight codimension-$3$ faces combinatorially isomorphic to

Let $\mathcal A$ be a cluster algebra of finite type, let $\mathcal S(\mathcal A)$ be its associated cluster complex, and let $C$ denote the relevant cluster-variable set whose car…

Let $\mathcal S_{k,n}$ denote the possibly infinite cluster complex of the Grassmannian $\operatorname{Gr}(k,n)$. Let $\operatorname{TGr}_0^+(k,n)$ denote its positive tropicalizat…

Let $\operatorname{Trop}^+Gr_{k+1,n}$ be the positive tropical Grassmannian, whose cones carry their natural partial order. Positive tropical Grassmannian fan conjecture. The regul…