29 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…

Simple-connectivity conjecture. Filling in these $8$-cycles is enough to make the complex simply connected.

Let $[n]$ be the set of integers from $1$ to $n$, and let ${[n] \choose k}$ denote the collection of all $k$-element subsets of $[n]$. For nonnegative integers $k<n$, a subset…

Let $[1\dots n]$ be labeled cyclically, and let a non-crossing family be a family of pairwise non-crossing $k$-subsets, where non-crossing means that no chord with endpoints in…

Move-connectivity conjecture. If

Connectivity conjecture. If

A normal plabic graph is a plabic graph whose boundary vertices are black of degree $1$, whose internal black vertices have degree $3$, and in which no two vertices of the same col…

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 $(G,\mathbf{D})$ be a plabic tangle of dimension at most $km$ admitting a brushing. Let $F_{\widehat{(G,\mathbf{D})}}$ be its associated space. Higher-intersection positivity c…

Fix a positive integer $m$, and let $G$ be a plabic graph of type $(k,n)$ with $\dim\Pi_G=km$. Let $T_G$ denote the associated family of $k$-planes. Intersection-number-zero criter…

Let $(G,\mathbf{D})$ be a dominant plabic tangle, let a brushing be the additional structure used to define brushed promotion, and let $\boldsymbol{\sigma}$ be a choice of signs. B…

Let $(G,\mathbf{D})$ be a dominant solvable plabic tangle, where $G$ has $n$ outer boundary vertices and inner disks $D^{(1)},\dots,D^{(\ell)}$, and let $m$-VRCs denote the associa…

Let a promotion map be a rational map between products of Grassmannians arising from a plabic tangle. Promotion-map conjecture. Promotion maps are quasi-cluster homomorphisms. The…

Let $G$ be a plabic graph, and let $rG$ denote any rotation of $G$. Its associated Newton–Okounkov body is denoted by $\Delta_G$. Rotation invariance conjecture. All rotations of a…

Let $I$, $N$, $k$, and the word set $(\pm I)^m$ be as in the affine plabic-fence construction. For a word $\mathbf w$, let $\mathbf w_1$ and $\mathbf w_2$ be the associated positiv…

Let $G(\mathbf w,f)$ and $G(\mathbf w',f')$ be affine plabic fences, with restricted move equivalence defined using the relevant affine plabic-fence moves. Affine plabic-fence clas…

Let $G$ and $G'$ be connected simple plabic graphs whose quivers $Q_G$ and $Q_{G'}$ are mutation equivalent. Let $L^{\operatorname{plab}}_G$ and $L^{\operatorname{plab}}_{G'}$ be t…

Let $G$ be a plabic graph with $\operatorname{c}(G)$ connected components and $n$ interior faces. Let $L^{\operatorname{plab}}_G$ be its associated plabic graph link, and let…

Let $G$ be a simple plabic graph with $\operatorname{c}(G)$ connected components. Let $Q_G$ be its planar dual quiver, let $L^{\operatorname{plab}}_G$ be the associated plabic grap…

Let $G$ be a planar bicoloured PBDTP graph, and let its edges carry the geometric signature. Form the associated sign matrix by assigning the corresponding geometric-signature sign…

Let $G$ be a reduced plabic graph, and let $\Sigma_G^S$ and $\Sigma_G^T$ denote respectively its source and target seeds. A quasi-cluster transformation is the relevant equivalence…

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…

Let $\mathcal{A}$ be the set of vertices of a generic convex $n$-gon, and let $\Upsigma_{\mathcal{A},k}$ be the associated higher associahedron. The generic polygon invariance conj…

Let $G$ be a reduced plabic graph, meaning a reduced planar bicolored graph, corresponding to a Schubert variety or, more generally, a positroid variety. Let the target labeling of…

Conjecture on smaller non-realizable plabic graphs. Much smaller non-realizable plabic graphs exist.