16 problems

Let $\mathcal{W}$ be the set of rank $2$ webs $W(R,S,T,V)$ on the Grassmannian $\operatorname{Gr}(k,n)$, where $R,S,T$ are cyclically separated subsets and $V$ is disjoint from the…

Positivity characterization conjecture. Traces of all $2$-webs in $H$ are positive if and only if, up to gauge equivalence, one of the following conditions holds:

Let $\hat{S}$ be a compact oriented surface with marked boundary points and punctures. Let $\mathcal{P}_{{\rm PGL}_3,\hat{S}}^+(\mathbb{Z}^t)$ denote the integer tropical points sa…

Let $\tilde{G}$ and $\tilde{G}'$ be tangled tree diagrams related by crossing changes, and let $G\in\mathsf{Tree}_{\mathfrak{sp}_4,\Sigma}$ be a tree-type elementary web. Crossing-…

Let $\mathbbm{k}$ be a field, and let \text{\boldmatha} \in \mathbbm{Z}^{r} and \text{\boldmathb} \in \mathbbm{Z}^{s} be the parameters indexing the relevant objects, with…

Rank-inequality conjecture. One has

Assume there is a bijection between generalized oscillating tableaux of length $k$ with $n$ parts, whose final component is $(m,\ldots,m)$ for some $m\in\mathbb{Z}$, and…

Let a generalized oscillating tableau have length $k$ and $n$ parts, with final component $(m,\ldots,m)$ for some $m\in\mathbb{Z}$. Consider $\operatorname{sl}(n)$ webs with $k$ bo…

Let $K$ be a web in $\R^{3}$, meaning an embedded trivalent graph, and let $\Jsharp(K)$ be the associated finite-dimensional vector space over the field $\F$ of two elements. A Tai…

Let $R_{p,r}(k)$ denote the generalized Catalan number associated with parameters $p,r,k$. For $n \geq 3$, let an $sl_n$ web have a boundary string consisting of one boundary edge…

Let $R_{p,r}(k)$ denote the generalized Catalan number associated with parameters $p,r,k$. An $sl_n$ web is a diagrammatic object for the representation theory of $U_q(sl_n)$; a we…

Bivariant web conjecture. In any such cluster algebra: all cluster variables are bi-web invariants; a bi-web invariant is a cluster monomial if and only if it can also be written a…

Web compatibility conjecture. Two cluster or coefficient variables are compatible if and only if their product is a web invariant.

Arborization conjecture. The forest form of a cluster monomial can be found from the web form via the arborization algorithm.

Let $w$ be a minuscule web with boundary $\vec{\lambda}$ and dual diskoid $D$. Let $F(\vec{\lambda})$ be the Satake fibre, let $Q(D)$ be the diskoid-configuration variety, and let…

Let $w$ be a closed web, meaning a web with no boundary, and let $D$ be its dual diskoid. Let $Q(D)$ be the variety of diskoid configurations associated with $D$, let $\Psi(w)$ den…