155 problems

Potential-density conjecture. Integral points are potentially Zariski-dense in the $K$-scheme $X_{G,\underline{C}}(Y)_K$: there exists a finite extension $L/K$ such that the Zarisk…

For every punctured surface $\Sigma_{g,p}$ with $g\geq 2$, the mapping class group $\operatorname{Mod}(\Sigma_{g,p})$ acts ergodically on every non-Teichmüller component of the rel…

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 $\frak{S}$ 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…

Topological mirror symmetry in degree zero. For every $\kappa\in\hat{\Gamma}$, with $\gamma=w(\kappa)$,

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 $K$ be a knot in $bS^3$, let $bbSigmad_2K$ be its 2-fold branched covering, and let $X(bbSigmad_2K)$ be the $bSL_2(bC)$ character variety of $bpi_1(bbSigmad_2K)$. Write…

Let $\mathbb H_{u_1,u_2,\ldots,u_g}[X_1,X_2,\ldots,X_n;q,t,T]$ be the logarithmic HLV kernel, and let $\mathbb H_{u_1,u_2,\ldots,u_g,\lambda}(q,t)$ denote the coefficient of…

Let $\mathcal{M}_\B$ be the relevant character variety, let $\dim=\dim(\mathcal{M}_\B)$, and let $\alpha\in W_4H^2(\mathcal{M}_\B)$, where $W_\bullet$ is the weight filtration. Cur…

Let $\mathcal H_\lambda(z,w)$ be the partition function appearing in the source, let $\phi(0):=0$, and for every nonzero partition $\lambda$ define … where the sum is over the boxe…

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…