15 problems

Let $D$ be a Postnikov diagram, let $Q_D$ be the associated ice quiver, and let $\mathsf{A_{MS}}$ be the cluster algebra defined by $Q_D$ after localising at the frozen cluster var…

Compatibly split positroid conjecture. The only compatibly split irreducible subvarieties of $\mathrm{Gr}(k,n)$ are the closed positroid varieties.

Williams's irreducibility conjecture. This set was conjectured to be irreducible. The claim is now a theorem, since the set equals the positroid variety…

Let $v otinquotmax$ and $wgeq v$, let $b1asigma=v^{-1}w$ be the resulting permutation decorated by declaring a fixed index $j$ black if $jin v^{-1}[k]$, and set $I=v^{-1}[k]$. Let…

Let $Gr(k,n)_{geq 0}$ be the totally nonnegative Grassmannian, stratified by the nonnegative parts of the positroid cells $(\Pi_u^w)_{\geq 0}^\circ$. Postnikov's conjecture. The po…

Galashin–Lam's conjecture. There exists a non-algebraic deformation retract

Vanishing conjecture. For every ${\bf m}\in{\mathbb Z}_{\geq0}^{{\mathbb Z}_n}$, the higher cohomologies

Let $\Pi_{\bf J}$ be the global positroid variety defined by the relations of Proposition, with fiber $\Pi_{\bf J}(\veps)$ over each value of $\veps$, and let…

Let $D$ be a Postnikov diagram with associated positroid $\mathcal{P}=\mathcal{P}_D$. The diagram gives two cluster algebra structures on the same algebra, namely the source-labell…

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…

Let $\widetilde \Pi^{\circ}_\pi$ be the relevant open positroid variety, and let $\mathbb{P}_\pi$ be the free abelian group of Laurent monomials in the target frozen variables. The…

The source concerns open positroid varieties and the cluster structures determined by relabeled plabic graphs. For a relabeled graph $G^\rho$ and an ordinary plabic graph $H$ with…

Postnikov's conjecture. There is an isomorphism of rings

Affine Grothendieck polynomial conjecture. The $K$-theory class of the structure sheaf of $\Pi_f$ is the image of $\tilde{G}_f$ under this identification:

Affine Grothendieck polynomial conjecture. The $K$-theory class of the structure sheaf $[\mathcal O_{\Pi_f}]$ is the image of $\tilde{G}_f$ under this identification: