19 problems

Ooguri–Vafa integrality conjecture. The wavefunction has the product expansion

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 $\beta\in\mathrm{Br}^{+}_{n}$ 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…

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…

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 $\mathbb{I}^{\rm br}$ 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…

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 $\mathbb{I}$ be the classical duality map and let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ index its basic functions. Laurent coefficient positivity conjecture. For…

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…

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

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

Let $Y$ be a log Calabi–Yau variety and let $f$ be a regular function on $Y$. Its tropicalization $f^{\operatorname{trop}}$ is the associated tropical function. Gross–Hacking–Keel–…

Let $\varphi$ be a tropical function on the tropicalization of a cluster variety, or of a fiber of a cluster variety, let $\s{Y}$ denote that variety or fiber, and let $\f{w}$ be a…

Let $\varphi$ be a tropical function in a cluster-variety setting, and consider convexity along broken lines and convexity with respect to every seed. The convexity-equivalence con…