30 problems

Let $\Gamma_{L^\ast}$ be a graph in the class of loop BCFW graphs $\mathbf{\Gamma}^{\mathtt{BCFW}}_{k,n;L^\ast}$, and let … be the corresponding branch of the loop BCFW recursion.…

For ${\mathcal{N}} \in Q_{n,2}$, let $A_{\mathcal{N}}:=A_{n,k,2}\cap {\mathring{\Pi}}_{\mathcal{N}}$, where $A_{n,k,2}$ is the $m=2$ amplituhedron and…

Let $Z:{\mathbb R}^n\to{\mathbb R}^{k+m}$ be a linear map with $n\geq k+m$, and let $P=Z({\mathrm{Gr}}(k,n)_{\geq 0})$ be the resulting Grassmann polytope when $Z$ is well-defined…

Cluster-adjacency conjecture. Each element of $\operatorname{Facet}(Z_{\pi})$ is a cluster variable for $\operatorname{Gr}_{m,n}$, and these cluster variables are pairwise compatib…

Let $Z_{\hat{G}(\mathcal{T})}$ be a positroid tile of the amplituhedron $\mathcal{A}_{n,k,2}(Z)$. A facet is a codimension-one boundary piece, and each facet lies on a hypersurface…

Let $J=\{j_1,\ldots,j_k\}$ be a $k$-subset of $[n]$, and let $I_{j_\alpha}$ be the $j_\alpha$th cyclic $k$-interval. Let $\xi_J$ be the quantity defined in the source's geometric c…

Let $S_M$ be a $km$-dimensional positroid cell of ${\rm Gr}_+(k,n)$. Let $\pi_+$ be the positive amplituhedron map and let $\pi$ denote its complexified rational map on the closure…

For integers satisfying $k e 0$ and $k+m e n$, let $Z$ be a totally positive matrix and let $\mathcal{A}{n,k,m}(Z)$ be the image of $mathop{rm Gr}+(k,n)$ under the induced map. A po…

Let $m$ be even and let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+m/2,n}$. Generalized T-duality conjecture. The collection gives a triangulation, respectively dissection,…

Consider regular positroidal subdivisions $\{\Gamma_{\pi}\}$ and $\{\Gamma_{\pi'}\}$ of $\Delta_{k+1,n}$ and corresponding subdivisions $\{Z_{\hat{\pi}}\}$ and $\{Z_{\hat{\pi'}}\}$…

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…

A regular positroidal subdivision of $\mathcal{A}_{n,k,2}$ is a positroidal dissection of the form $\{S_{\hat{\pi}}\}$, where $\{S_{\pi}\}$ is a regular positroidal subdivision of…

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection. Good T-duality correspondence conjecture. The collection…

Let $S_{\pi^{(1)}}$ and $S_{\pi^{(2)}}$ be positroid cells in $Gr^+_{k,n}$ corresponding to coloopless permutations $\pi^{(1)}$ and $\pi^{(2)}$, with … and…

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection in $Gr^+_{k,n}$. T-duality conjecture. The collection…

Let $S_{\pi}$ be a loopless $(n-c)$-dimensional cell of $Gr^+_{k+1,n}$ with $c$ connected components, where $c$ is a positive integer. Then $S_{\hat{\pi}}$ is a coloopless…

For $m=4$, let $\mathcal{T}_{k,\ell}=\{f_1,\dots,f_N\}$ be the BCFW collection of positroid cells, with $\mathcal{T}_{\ell,k}=\{f_1^{-1},\dots,f_N^{-1}\}$ up to cyclic shift. BCFW…

A flip replaces one of the two triangulations associated with a nearly $(n,k,m)$-admissible affine permutation by the other. Flip-connectivity conjecture. Any two triangulations of…

Let $g\in\tilde{S_n}(-k,n-k)$ be nearly $(n,k,m)$-admissible, meaning that $\operatorname{inv}(g)=k\ell-1$, $\operatorname{dim}(Z(\Pi^{>0}_{k+g}))=km$ for every…

Polynomial numerator conjecture. There exists a polynomial $H(S,A,B)$ in the entries of $S,A,B$ such that

Let $\mathcal{T}$ be a degree-one $(n,k,m)$-triangulation. The universal amplituhedron form $\omega^{(\mathcal{T})}_{\mathcal{A}_{n,k,\geq m}}$ is the signed sum of the pushforward…

Amplituhedron form conjecture. The form $\omega^{(\mathcal{T})}_{\mathcal{A}_{n,k,m}(Z)}$ does not depend on the choice of triangulation $\mathcal{T}$. The form is intended to be a…

Let $f\in\tilde{S_n}(-k,n-k)$ be an affine permutation with $\operatorname{inv}(f)=k\ell$. A cell $f$ is $(n,k,m)$-admissible when it satisfies the source's admissibility condition…

Let $M(a,b,c)$ denote the number of plane partitions fitting in an $a\times b\times c$ box, namely … Let $Z\in\operatorname{Gr}_{>0}(k+m,n)$ and let…

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…