14 problems

Let $\mathfrak{M}_{\infty,1}$ be the homotopy colimit of the genus-stabilisation maps … For each $i\geq 1$, let $\kappa_i$ denote a class of degree $2i$ in the rational cohomology…

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…

Let $F_n$ be the free group of rank $n$, and let $\operatorname{Out}F_n$ be its outer automorphism group. Stable cohomology conjecture. The stable rational cohomology of…

Fix a positive integer $k$ and let $\lambda=(a_1,\dots,a_r)$ be a partition of $k$, so that $k=a_1+\dots+a_r$. Let $d_0(\lambda),d_1(\lambda),\dots$ be the stable multiplicities as…

Fix a positive integer $k$ and let $\lambda=(a_1,\dots,a_r)$ be a partition of $k$, so that $k=a_1+\dots+a_r$. For each cohomological degree $i$, let $d_i(\lambda)$ denote the stab…

Let $\mathcal{I}_g$, $\mathcal{I}_{g,1}$ and $\mathcal{I}_g^1$ be the Torelli groups of a closed surface, a surface with one boundary component, and a surface with one marked point…

Let $\mathcal{I}_g$ be the Torelli group of a closed genus-$g$ surface, and let $e_{2i}$ denote the Mumford–Morita–Miller class of degree $4i$. Kawazumi–Morita's conjecture. Stably…

For $p,q\ge0$, let $H^{p,q}=H^{\otimes p}\otimes(H^*)^{\otimes q}$, and let $\mathcal{H}$ and $\mathcal{E}$ be the wheeled PROPs defined from stable twisted cohomology and the corr…

Let $\mathcal{E}$ and $\mathcal{H}$ be the wheeled PROPs in the paper, and let … be the morphism of wheeled PROPs whose image is isomorphic to $\mathcal{K}$. Djament's conjecture.…

For a partition ${\boldsymbol\mu}$ with at most $g$ parts, let $V_{\boldsymbol\mu}$ be the corresponding irreducible ${\mathrm{Sp}}(H)$-module, and let … be the homomorphism induce…

Morita–Mumford conjecture. The stable cohomology ring is freely generated by the tautological classes:

Stable-cohomology top-degree conjecture. There exists a $G$-module $A$ with trivial action such that

Let $\bold{M}_g$ be the moduli space of smooth genus-$g$ Riemann surfaces, and let $\kappa_i\in H^{2i}(\bold{M}_g;\mathbb{Z})$ be the Mumford–Morita–Miller classes defined by … whe…

Quantum Mumford conjecture. The classes $qc_k$ generate