46 problems

Let $G$ be a split real semisimple Lie group, let $X$ be its associated symmetric space, and let $\rho$ lie in a Hitchin component of representations of $\pi_1(\Sigma_g)$ into $G$.…

Let $\mathcal{X}(\Gamma,\mathsf{SO}(p,p+1))$ be the character variety of representations of the surface group $\Gamma$ into $\mathsf{SO}(p,p+1)$. In addition to the components defo…

Let $G^\textbf{R}$ be a real form of a complex simple Lie group, and let $\hat{\rho}$ be a magical $\mathfrak{sl}_2$-triple of $G^\textbf{R}$. The generalized Cayley correspondence…

Let $\hat{S}$ be a compact oriented surface with marked boundary points and punctures, and let $\mathcal{X}_{{\rm G},\hat{S}}$ and $\mathcal{A}_{{\rm G}^L,\hat{S}}$ be the moduli s…

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…

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

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…

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 $S$ be a punctured surface, and let $G=SL_m$. The cluster ensemble $({\cal A}_{G,S},{\cal X}_{G,S})$ yields a family of unitary projective representations of the mapping class…

Let $\Flag(\mathbb R^d)$ denote the full flag variety and let $\Gr(k,\mathbb R^d)$ denote the Grassmannian of $k$-planes. If $\Lambda\subset\Flag(\mathbb R^d)$ is the image of a po…

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 $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a compatible hermitian structure $\rho$. Let $d_A$ be the unique unitary, $\sigma$-invariant connection satisfying…

The geometric Hitchin space is denoted by $\hat{\mathcal{T}}^n(S)$, and the Hitchin component is the corresponding component of the character variety for a split real semisimple Li…

Let $S$ be an oriented surface of negative Euler characteristic, let $G$ be a simple Lie group with a $\Theta$-positive structure, and let $\mathcal{P}_\Theta(S,G)$ denote the set…

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…

Higher Teichmüller spaces are spaces of representations of the fundamental group of a closed surface $S$ into higher-rank simple Lie groups. Let $\mathcal{T}(S)$ denote Teichmüller…

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…