73 problems

Let $\mathfrak{S}$ be a triangulable pb surface and let $\lambda$ be an ideal triangulation. Let $\overline{\mathscr S}_\omega(\mathfrak{S})$ be the reduced stated ${\rm SL}_n$-ske…

Let $\mathfrak{S}$ be a surface, let $\mathfrak{X}_{{\rm SL}_2(\mathbb{C})}(\mathfrak{S})$ be its ${\rm SL}_2(\mathbb{C})$ character variety, and let $r$ be a classical shadow asso…

Thurston's conjecture. The bracelets bases of ordinary skein algebras are positive bases; equivalently, their structure constants lie in

Roger–Yang representation-theoretic conjecture. The representation theory of the Roger–Yang skein algebra should encode hyperbolic geometric information about the punctured surface…

Refined skein recursion conjecture. The equation obtained for the refined Chern–Simons model is the correct refinement of the skein recursion relations for the unknot.

Lê–Yu's conjecture. The Azumaya locus of $\mathcal{S}_A(\Sigma_g)$ should be the smooth locus of $\mathcal{X}_{\operatorname{SL}_2}(\Sigma_g)$. This conjecture is disproved by the…

Let $\Sigma_{g,n}$ be a punctured surface, let $\mathcal{C}(\Sigma_{g,n})_{\mathbb{C}}$ be its complexified curve algebra, and let $\mathcal{T}^{d}(\Sigma_{g,n})$ be its decorated…

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

Generic uniqueness conjecture. When $r$ belongs to a Zariski dense open subset of $\mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$, the representation $\rho_r$ is the only irreducible re…

Let the skein algebra of a marked surface have frozen variables, and consider its localization at those frozen variables. The skein algebra carries a (quantized) band basis and a c…

Let the skein algebra of a marked surface be equipped with its (quantized) band basis and dual canonical basis. The band basis is a basis constructed from topological objects such…

Reduced skein–cluster conjecture. Under these assumptions,

Localized skein–cluster conjecture. Under these assumptions,

Le–Yu injectivity conjecture. The projected stated skein algebra equals the stated skein algebra:

Skein–cluster equality conjecture. Under the same assumption as Theorem,

Refined torus skein conjecture. The refinement of the skein algebra of the torus is the quantum toroidal $cmathfrak{gl}_1$ algebra.

Let $\mathfrak{S}$ be a triangulable pb surface, let $\mathcal V$ be the vertex set of the associated quantum seed, and let $\mathcal V_{\mathrm{mut}}$ be its mutable vertices. Wri…

Let $\Gamma=H_1(\widetilde{\Sigma};\mathbb{Z})$ have its standard intersection pairing, and let … where $\sigma$ is the $\mathbb{Z}/2$-deck transformation action. Let $Q_{2\Gamma}$…

Let $\mathfrak{S}$ be a punctured surface with at least one puncture. The projected ${\rm SL}_n$-skein algebra is the quotient … where $\mathcal K_{\hat\omega,\mathfrak{S}}$ is the…

Let $S$ be a compact oriented surface of genus one with boundary, and let $\widetilde{G}=G\times F$ and $N$ be the associated group and representation. The quantized Coulomb branch…

Let $\Sigma=\Sigma_{g,p}$, let $Sk^{ta}(\Sigma)$ be the tagged skein algebra, let $U(\Sigma)$ be the upper cluster algebra, and let $\nu:Sk^{ta}(\Sigma)\to U(\Sigma)$ be the natura…

Let $\Sigma$ be a surface, let $\mathcal{S}_q^{\mathrm{MRY}}(\Sigma)$ be its MRY skein algebra, and replace the ground ring $\mathbb{C}$ by $\mathbb{Z}[q^{\pm1/2}]$. Let $v_i$ deno…

Let $H_g$ be the genus $g$ handlebody, let $\rho\in \mathfrak{X}_{{\rm SL}_3(\mathbb{C})}(H_g)$ be represented by an irreducible representation, and let…

Let $\mathfrak{S}$ be a surface, let $\bar\omega$ be the quantum parameter, and let $\mathscr S_{\bar\omega}(\mathfrak{S})$ be the ${\rm SL}_3$-skein algebra with center…

Half-index conjecture. These inner products can be computed by a careful deformation and a chain of dualities that maps them to Schur half-indices, namely twisted partition functio…