197 problems

Let $\Sigma$ be a closed oriented surface of genus $g$, let $s^i\Sigma$ denote its $i$-th symmetric product, and let $HF_g^*$ be the Floer cohomology ring associated with…

Let $X=\mathbb{C}P^2\# n\overline{\mathbb{C}P^2}$ carry a symplectic form $\omega$, and let $\Gamma(X,\omega)$ be its Lagrangian root system. Write $PR[\Gamma(X,\omega)]$ for the p…

Let $X$ be the $Z_n$ curve associated to the family in the source, with parameters $p$ and $q$, and let $g$ be its genus. A divisor of degree $g$ is non-special when it is not spec…

Let ${\cal M}_g$ be the mapping class group. Recall that a group ${\varGamma}$ is uniformly perfect if there is a natural number $N$ such that every element of ${\varGamma}$ is a p…

Let ${\cal M}_g$ be the mapping class group and let $H_b^2({\cal M}_g)$ denote its second bounded cohomology. Bounded-cohomology conjecture. For every genus $g$, the comparison map…

Let ${\cal K}_g$ be the Johnson kernel, let $d_i\in H^{4i-3}({\cal K}_g;{\mathbb Q})$ be the higher secondary classes, and let ${\cal M}_g$ act on $H^*({\cal K}_g;{\mathbb Q})$. Se…

Let ${\mathfrak h}^{\mathbb Q}_{g,1}$ be the rational graded Lie algebra of symplectic derivations, let $H_{\mathbb Q}$ be the rational symplectic module, and let…

Let $\operatorname{Map}_{g,1}$ be the mapping class group of a genus-$g$ surface with one marked point, and let $\operatorname{Map}_{g,1}(-)\subset\operatorname{Map}_{g,1}$ be the…

Let $S$ be a surface, let $\operatorname{Mod}(S)$ be its mapping class group, and let $\Lambda_G$ denote the limit set of a subgroup $G\leq \operatorname{Mod}(S)$ in the projective…

Let $C$ be a genus-$g$ Heegaard surface with fundamental group $\pi_1(C)$, let $X$ be the associated three-manifold, and let $L_+$ be one of the two lattices in the Heegaard splitt…

Mumford's conjecture. This homomorphism should be an isomorphism in the stable range of Harer's stability theorem; in the formulation stated earlier, it is an isomorphism in degree…

For $g\geq 3$ and $L\geq 2$, let ${\rm Mod}_g[L]$ be the level-$L$ congruence subgroup acting trivially on $H_1(\Sigma_g,\mathbb Z/L\mathbb Z)$. Picard-number-one conjecture for le…

Let $\T_g$ be the Torelli group of a closed genus-$g$ surface. Torelli finite-presentation conjecture. The group $\T_g$ is finitely presented for $g\geq 4$. This is presented as a…

For $g\geq 3$, let ${\rm Mod}_g$ be the mapping class group of a closed genus-$g$ surface. Kähler-group conjecture. The group ${\rm Mod}_g$ is a Kähler group, meaning that it is is…

For $g\geq 3$, let $\M_g$ be the smooth moduli orbifold of genus-$g$ Riemann surfaces. Maximality conjecture for $\M_g$. The orbifold $\M_g$ does not finitely orbifold-cover any ot…

For a fixed finite generating set of ${\rm Mod}_g$, define the density of a subset $S$ by … Let $\Tor_g$ denote the set of finite-order elements of ${\rm Mod}_g$. Torsion-density c…

Fix a finite generating set for ${\rm Mod}_g$, let $B(r)$ be the ball of radius $r$, and define the density of $S\subset {\rm Mod}_g$ by … Let ${\cal P}$ be the set of pseudo-Anoso…

AMU conjecture. For a mapping class $\phi\in\Gamma$, $\phi$ is Pseudo-Anosov or reducible with Pseudo-Anosov pieces if and only if $\rho_k(\phi)$ is infinite order for large enough…

Let $S_{g,p}$ be a genus-$g$ surface with $p$ punctures, let $\brun(S_{g,p})$ denote the set of Brunnian pseudo-Anosov mapping classes on $S_{g,p}$, and let $L(\brun(S_{g,p}))$ den…

Let $H$ be the handlebody used to construct tunnel number one 3-manifolds, let $\partial H$ be its boundary, and let $\mathcal{MCG}(\partial H)$ be its mapping class group. Choose…

Euler-component ergodicity conjecture. For each integer $1\leq k\leq 2g+b-2$, the $\mathsf{Mod}_{\Sigma}$-action on the component $e^{-1}(2-2g+b+k)$ of $\mathsf{Hom}(\pi,G)$ is erg…

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 $\mathcal H_{g,1}$ and $\overline{\mathcal H}_{g,1}$ be the homology-cobordism groups in the source, let $\sigma$ be the homomorphism to the inverse limit of automorphism group…

Let $\mathfrak h_{g,1}^{\mathbb Q}$ be the graded Lie algebra under consideration, and let $\operatorname{Sp}$ act on it through the symplectic action on $H_{\mathbb Q}$. Define…