36 problems

Reduced-coordinate conjecture. The variables $\mathbf{w}$ are cluster coordinates on $\mathcal{X}_{\mathrm{red}}$: every mutation in the $\mathbf{w}$ variables can be lifted to a s…

Let $\mathbb{I}^{\rm br}$ be the conjectural ${A}_2$-bracelets basis map for $\mathscr{O}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$, and let $\mathbb{I}$ denote the paper's correspondin…

Goncharov–Shen mirror-duality conjecture. This basis is in bijection with

Let $V_2$ be the two-dimensional complex Cartan subalgebra for the Lie algebra of type $G_2$, and let $V_2^{\rm reg}$ be the complement of the six root-kernel lines. Let $W_{G_2}$…

P=W and PI=WI conjectures. There exists a real Lagrangian fibration $h$ such that

Let $cmathcal{T}$ be a triangulation of a general $n$-triangulated $m$-gon of type $cmathcal{P}(m_1,n_1,\ldots,m_k,n_k)$, and let $cGamma_{n+1}$ denote the index set used for the v…

Zigzag-mutation invariance conjecture. Zigzag mutation is an isomorphism

Monomial-reduction conjecture. There exists a mutation sequence

Decorated-reduction conjecture. Under certain conditions for the decorated Newton polygon, there exists an integrable system such that:

Reduced Goncharov–Kenyon integrability conjecture. The Hamiltonians $\mathcal{Z}_{a,b}(\mathbf{x})$ for $(a,b)$ in the interior of $N$ define an integrable system on the subvariety…

Let $\beta\in\mathrm{Br}^{+}_{n}$ be a positive braid word, let $\mathrm{BS}(\beta)$ be the associated double Bott–Samelson variety, and let $\mathcal{D}(\mathrm{BS}(\beta))$ denot…

Let $A$ be a locally acyclic cluster algebra and let $\mathcal{A}$ be its corresponding cluster variety. Let $\mathcal{D}(\mathcal{A})$ denote the deep locus and…

Let $A$ be a skew-symmetric, locally acyclic cluster algebra in which all frozen variables are invertible, and let $\mathcal{A}=\operatorname{Spec}A$. Let…

A versal family of a planar singularity gives a family of Lagrangian curves in the complex symplectic plane $(\mathbb{C}^2,dp\,dx)$, and the equations of these curves can be transf…

Gross–Hacking–Keel–Kontsevich's order-of-vanishing conjecture. The theta functions on $\mathcal{A}_{\mathrm{prin}}$ respect the order of vanishing.

Let $X$ be a generic Fano variety with klt singularities. A polarized cluster variety is a cluster variety equipped with a polarization; write it as $U=Y\setminus D$ with polarizat…

Let $Y$ be a cluster variety that is locally acyclic and of full rank. Write $Y_{>0}$ for its positive part and $Omega(Y_{>0})$ for its natural top form. The simplex-like compactif…

Fix cluster data $Gamma$ and let $cA$ and ${}^LcX$ denote the cluster variety of $cA$-type and the Langlands-dual cluster variety of $cX$-type constructed from the Langlands-dual d…

There are two types of cluster varieties, the $\mathcal{A}$ and $\mathcal{X}$ varieties. The Langlands dual of a cluster variety is obtained from the Langlands dual fixed data, so…

Let $\mathbb{I}^{\rm GHKK}$ be the Gross–Hacking–Keel–Kontsevich duality map for the cluster varieties associated with the mutation class of the $3$-triangulation quiver, and let…

Let $\mathbb{I}^{\rm br}$ and $\mathbb{I}$ denote the conjectural bracelets and bangles basis maps, respectively, and let their structure constants be defined by multiplication in…

Let $\mathscr{A}_{\rm L}(\frak{S};\mathbb{Z})$ be the set of ${\rm SL}_3$ laminations and let $\mathbb{I}^{\rm br}_{{\rm SL}_3}$ be the map obtained by replacing loop components wi…

Let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ be the set of tropical-integer points of the ${\rm SL}_3$ cluster $\mathscr{A}$-space of a generalized marked surface…

Gross–Hacking–Keel–Kontsevich's conjecture. The function $f^\mathrm{trop}$ is min-convex. This conjecture concerns the convexity properties of tropicalizations of regular functions…

P=W conjecture. For any diffeomorphism $f:X\to Y$, the $P=W$ identity holds under $f^*$.