27 problems

Let $S$ be a surface with holes, let $S'$ be obtained by cutting $S$ along a simple loop, and let $\gamma_\pm$ be the two new boundary components. Let ${\bf V}_{G,S;\lambda}$ be th…

Let $X \in \mathcal{T}^{\pm}(\mathbb{S}, \mathbb{M})$ and let $\hat{X}$ be as before. Write $X_n={\mathrm{PSL}_n(\mathbb{C})}/{\text{PSU}(n)}$ for the symmetric space and let…

Let $S$ be a closed surface with fundamental group $\Gamma$, let $d\geq 2$, and let $\lambda$ be a maximal geodesic lamination on $S$. Denote by $\mathfrak R(\lambda,d)$ the space…

The ordered-quiver fruitfulness conjecture. The quivers $Q_n$ provide fruitful seeds, and any quiver obtained by gluing them according to an ideal triangulation of a bordered surfa…

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 $(P,\Phi)$ be a $G$-Fock bundle over $S$ with hermitian structure $\rho$ such that $\Phi\in\mathcal{P}$ is positive. Let $\mu$ be the induced $\mathfrak{g}$-complex structure o…

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle. A hermitian structure $\rho$ is compatible when it commutes with $\sigma$, and positive when the associated positivity condition for the…

Let $\Sigma$ be a closed oriented surface, let $\mathcal{T}^n$ be the moduli space of higher complex structures on $\Sigma$, and let … be the formal map whose image is an open subs…

Let $\Sigma$ be a closed oriented surface and let $(\mathrm{G},\mathrm{P}^\mathbb{R}_\Theta)$ admit a $\Theta$-positive structure. A representation…

Let $X$ be a compact Riemann surface, let $\Sigma$ be its underlying surface, and let $\mathrm{G}^\mathbb{R}$ be a simple Lie group. Write ${\mathcal M}(X,\mathrm{G}^\mathbb{R})$ for th…

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 $\Delta$ be an ideal triangulation of a triangulable generalized marked surface $\frak{S}$, let $\mathbb{I}^q_\Delta$ be the proposed quantum duality map, and let…

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…

Let $\tilde{\Sigma}$ be the spectral curve associated with a point of T^*\bm\bm\bm\bm\bm\bm\bm\bm\hat{\mathcal{T}}^n, and let $\alpha$ be the Liouville form on…

Let $\Sigma$ be a surface, and let $\mathcal{A}(\lambda)$ be a family of $\operatorname{PSL}_n(\mathbb{C})$-connections satisfying … with…

The cotangent bundle T^*\bm\hat{\mathcal{T}}^n is obtained by Hamiltonian reduction from the hyperkähler punctual Hilbert scheme, with its given symplectic structure near the zer…

Guichard–Wienhard conjecture. The extra components in the moduli space of $SO(p+q,q)$-Higgs bundles conjectured to contain positive representations are those connected components c…

Let $\mathcal{C}_{\mathfrak{g},\Sigma}$ be the mutation class associated to an admissible pair $(\Sigma,\mathfrak{g})$. Let $\mathbf{w}_0=(w_0,\ldots,w_0)\in W(\mathfrak{g})^p$ be…

Boundedness conjecture. The collection

Non-decoupling conjecture. The Hitchin equation never decouples:

Negative curvature conjecture. The sectional curvature $k_\sigma^N$ is negative at every immersed point. Consequently, the induced curvature $\kappa$ is negative at every immersed…

Labourie's conjecture. Does there exist a unique $\rho$-equivariant minimal surface in $G/K$?

Guichard-Labourie-Wienhard conjecture. The set of $\Theta$-positive representations $\rho:\pi_1(\Sigma_g)\to G$ is open and closed in $\operatorname{Hom}(\pi_1(S),G)/G$. In particu…

Labourie's conjecture. The quotient of $\mathcal{T}_H(S,\operatorname{PSL}(n,\mathbf{R}))$ by the mapping class group is a holomorphic vector bundle over $\mathcal{M}(S)$, with fib…