13 problems

Tropical fan conjecture. If the lineality space of $\operatorname{Trop}^+ \operatorname{Spec} \mathcal{A}$ has dimension $|C|$, then the quotient of…

Let $\widetilde{\varphi}:\operatorname{Gr}(4,n)\to\operatorname{Gr}(4,n')$ be a promotion map. Promotion-map copositivity conjecture. The map $\widetilde{\varphi}$ is copositive, n…

Let $(G,\mathbf{D})$ be a tangle admitting a brushing, with $\dim\Pi_G=km$ and positive intersection number $\operatorname{IN}_m(G)>0$. Let $F_{\widehat{(G,\mathbf{D})}}$ be the as…

Let $k\geq 1$ and $n\geq k+4$. Let $\mathcal{A}_{n,k,4}$ be the amplituhedron, and let $\mathcal{BCFW}_{n,k}$ denote the collection of $4k$-dimensional BCFW positroid cells in the…

Let $\mathcal{M}_{n,k}$ be the momentum amplituhedron. For a Grassmannian graph $G$, let $\Phi^\circ_G$ be the associated stratum, and let $\dim_{\mathcal{M}}(F)$ denote the dimens…

Let $\mathcal{M}_{n,k}$ be the momentum amplituhedron, and let $\mathcal{S}_{n,k}$ consist of the decorated permutations whose strata are boundary strata, together with the decorat…

Let $\Delta_{k,n}$ be the $k$th hypersimplex. For a $k$-element subset $J\subset\{1,\dots,n\}$, let $\Pi_J$ be the associated polytope obtained by eliminating the block containing…

Positroid stratification conjecture. The positroid stratification of $Gr^{\geq 0}(k,n)$ is a regular CW-complex. In particular, the closure $\overline{\Pi}_{\mathcal{M}}$ of each p…

Fix $Z ioperatorname{Gr}{0}(k+m,n)$. Let $f,f1,\dots,fN$ be affine permutations in $tilde{Sn}(-k,n-k)$, and let $g$ be another such affine permutation. The notions of $Z$-admissibil…

Let $\mathcal{W}(k,n)$ be the cell complex associated to admissible Wilson loop diagrams with $k$ propagators on $n$ vertices. Let $B$ be a cell of dimension $3k-1$, let $W$ and…

Let $Z\in\operatorname{Mat}_{k+4,n}^{>0}$, and let $\mathcal{C}_{n,k,4}$ be the collection of BCFW cells in $\operatorname{Gr}_{k,n}^{\geq 0}$. Arkani-Hamed–Trnka triangulation con…

Let $\mathcal{J}\subset {[n]\choose k}$ be an arrangement of largest minors, and let $W\in {[n]\choose k}$. Say that $W$ is $(\geq t,\mathcal{J})$-largest if, for every arrangement…

Pairwise smallest-minors conjecture. There exists an arrangement of smallest minors $S\subset\binom{[n]}{k}$ containing $I$ and $J$ if and only if either the pair is 1-interlaced,…