154 problems

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p$. Consider $N_\Phi$ a Seifert fibered space with base orbifold $S^2(2,2,2,3)$ and gluing matrix $\Phi$ $=$ ……

Let $K$ be a knot. Let $P_K^0(q,E,Q)$ be the factor of the $A_q$-polynomial corresponding to the inhomogeneous recursive relation, let $A_K(l,\alpha)$ be the $A$-polynomial, let…

Let $M$ be an oriented compact $3$-manifold with a torus boundary, and let $\mathrm{R}^{\rm para}_{\mathrm{SL}_2(\mathbb{C})}(M)$ be the set of conjugacy classes of boundary-parabo…

Let $M$ be an oriented closed $3$-manifold, let $G$ be a simply connected complex simple Lie group, and let $\mathrm{R}^{\rm irr}_G(M)$ be the character variety of irreducible repr…

Let $\mathcal{M}_{\B}^d$ be the $\operatorname{GL}_n$ character variety with central monodromy $\zeta_n^d I_n$, and let $\hat{\mathcal{M}}_{\B}^e$ be the corresponding…

Let $\mathfrak{m}=\mathfrak{m}(a_1/b_1,a_2/b_2,\ldots,a_t/b_t)$ be a Montesinos knot with $t$ tangles. Consider its $\operatorname{PSL}_2(\mathbb C)$- and…

Genus-one product identity.

Polynomiality and positivity conjectures. The following two assertions are conjectured:

Hausel's mixed-Hodge polynomial conjecture.

Genus-one partition identity. The following combinatorial identity holds:

Let $K$ be a hyperbolic knot in $S^3$, let $R_0$ be the relevant irreducible component of its $SL_2(\mathbb{C})$-representation variety, and let $\mu$ and $\lambda$ denote the meri…

Let $K$ be a hyperbolic knot in $S^3$, let $R_0$ be the relevant component of the $SL_2(\mathbb{C})$-representation variety, and let $\mu$ and $\lambda$ denote the meridian and lon…

Let $\Sigma$ be a closed, orientable $3$-manifold. Write $\lambda_{\operatorname{PSL}_2(\mathbb C)}(\Sigma)$ and $\lambda_{\operatorname{SL}_2(\mathbb C)}(\Sigma)$ for the respecti…

Let ${\mathcal X}_{\kappa}$ be the character variety with parameter $\kappa$, and let ${{\mathcal E}(\rho)}$ be the set of end invariants of a character $\rho$. The ending invarian…

Let $\rho$ be an $\mathrm{SL}(2,\mathbb C)$ character and let ${{\mathcal E}(\rho)}$ denote its set of end invariants, viewed as a subset of the projective lamination space…

Free-group ergodicity conjecture. If $r\geq 3$, the action of $\mathsf{Out}(\pi)$ on $\mathsf{Hom}(\pi,G)$ is ergodic.

Andersen's scalar-operator conjecture. If

Invariant-forms conjecture. The symplectic structures $\omega_\mathbb{B}$ generate the subalgebra of $\Omega^*(\mathsf{Hom}(\pi,G)/G)$ consisting of $\mathsf{Mod}_{\Sigma}$-invaria…

Let $\rho$ be an $\mathrm{SL}(2,\mathbb C)$ character, let $\mathcal E(\rho)$ be its set of end invariants, and let $\mathscr{PL}$ denote the projective lamination space. Cantor-se…

Dynamics conjecture. The action of $\Gamma$ on $\Omega^K$ is wandering, and for each $c>2$ its action on

Let $K$ be a knot in $S^3$. Its deformation variety $D(K)$ is the image of the character variety of $\mathrm{SL}_2(\mathbb C)$ representations of the complement…

Let $K$ be a knot, let $\Sigma_2(K)$ be its double branched cover, and let $HC_0^{\operatorname{ab}}(K)$ denote the abelianized degree-zero contact homology, identified with the co…

Let $G$ be a complex connected reductive group, and let $M_{Dol}^{vc}(G)$ be the moduli space of $G$-Higgs bundles with vanishing Chern classes over a compact Riemann surface of ge…

Generic spectrum conjecture. For a generic $\rho: \Sigma_g \longrightarrow \mathrm{SL}(2,\mathbb{R})$ which is not simple Fuchsian, the spectrum of $\rho$ is closed and has empty i…

Let $M$ be a three-manifold and let $G$ and $G^{\vee}$ be Langlands dual groups. Write $\mathfrak{M}_G(M)$ and $\mathfrak{M}_{G^{\vee}}(M)$ for the corresponding stacks of local sy…