288 problems

Let $S$ be a closed oriented surface, and let $\operatorname{Mod}(S)$ be the pure extended mapping class group acting on the character variety $\mathfrak{X}(S)$. The action preserv…

Let $K$ be a knot. Its colored Jones polynomial is the sequence of quantum knot invariants indexed by colors, and its A-polynomial is the algebraic curve encoding the boundary char…

Let $X$ be the variety under consideration. A representation $\rho:\pi_1(X)\to \mathrm{GL}_r(\mathbb{C})$ is integral if it is conjugate to a representation…

Let $F_2$ be the rank-two free group, and consider a type-preserving representation $\rho:F_2\to\mathrm{PSL}(2,\mathbb{R})$, meaning a non-elementary representation for which the p…

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…

Let $\mathcal{C}$ be a generic tuple of conjugacy classes, let $\bs\mu$ encode the eigenvalue multiplicities, and let $\mathcal{M}_{\mathcal{C}}$ be the associated character variet…

Let the graphs arising from the other surfaces in the paper's family be the analogous graphs obtained from the corresponding surface actions. Bourgain–Gamburd–Sarnak's expander con…

Let $\mathcal{M}_{\mathcal{C}}$ be a character variety with its mixed Hodge structure, and suppose its mixed Hodge polynomial has the curious Poincare-duality symmetry described in…

Let $(u,v)$ be a generator pair for $F_2$, and let $\rho\colon F_2\to SL(2,\mathbb C)$ be an irreducible representation satisfying the $BQ$-conditions: for every primitive element…

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…

For a character $[\rho]\in\mathcal{X}$, an end invariant is a lamination $\lambda\in\mathscr{PL}$ for which there exists a sequence of distinct $X_k\in\mathscr{C}$ converging to…

Let $S$ be a closed surface, let $\operatorname{MCG}(S)$ act by precomposition on $\mathfrak{X}(\pi_1(S),\operatorname{PSL}_2(\mathbb{C}))$, and let $\mathfrak{X}_{QF}$ denote the…

Let $S_{g,n}$ be an orientable punctured surface, and let $\mathfrak{X}_{tp}(S_{g,n})$ be the space of type-preserving representations…

Motivic and $e$-polynomial mirror symmetry conjecture. The motives and $e$-polynomials of the character varieties $\mathcal{X}_G(\Gamma)$ and $\mathcal{X}_{^L G}(\Gamma)$ are equal…

Kapustin–Witten mirror symmetry conjecture. For Langlands-dual groups, certain topological and geometrical invariants of the corresponding character varieties should coincide or mi…

Let $M$ be a 3-manifold with torus boundary, let $\rho$ range over representations into $\operatorname{SL}_2(\mathbb{C})$, and let…

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…

Let $\mathcal{M}_B$ be a smooth Betti moduli space, or character variety, of complex dimension $d$ associated with a punctured Riemann surface, and let…

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 $G$ be an algebraic group, and let $\mathrm{Char}_G$ be the character variety parametrizing $G$-local systems. A local system is motivic when it is of geometric origin. Esnault…

Let $\frak{S}$ be a triangulable punctured surface. A congruent ${\rm SL}_3$-lamination is an element $\ell \in \mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^t)$, and…

Let $\Sigma_{g,n}$ be a punctured connected oriented surface, let $C$ be a collection of parabolic conjugacy classes, and consider the relative representation space … The mapping c…

Strong AJ conjecture. Suppose $K$ is a knot in $S^3$. Then

Let $M$ be an oriented compact $3$-manifold with torus boundary, and let $X^\mathrm{irr}(M)$ be the character variety of irreducible $\mathrm{SL}_{2}(\mathbb{C})$-representations.…

Let $r\geqslant 1$ and let $G$ be a connected reductive complex affine algebraic group. Write $\mathcal{X}_r(G)$ for the smooth locus of the $G$-character variety of the rank-$r$ f…