20 problems

For $a\geq 0$, let $W_a\subset\mathcal{M}_{g,n}^{comb}$ be the locus of metric fatgraphs having a vertex of valence at least $2a+3$. For a generic perimeter slice, regard $W_a$ as…

Let $g>0$, let $n$ be a positive integer, and let $k_1,\ldots,k_n$ be nonnegative integers such that $k_1+\cdots+k_n=2g+n-3$. Write $\overline{\mathcal M}_{g,n}$ for the moduli spa…

Let ${\mathcal{M}}_{g,n}$ be the moduli space of smooth curves of genus $g$ with $n$ marked points, and let $\psi_1,\dots,\psi_n$ denote the cotangent-line classes at the marked po…

Let $\mathcal{M}_{g,P}$ be the moduli space of smooth genus-$g$ curves with marked points $P$, and let $W_{m_*,P}$ denote the combinatorial class associated with ribbon graphs havi…

Let $k_1,k_2,\ldots,k_r$ be positive integers. Define $W_{k_1,k_2,\ldots,k_r}$ to be the set of fat graphs with vertices $v_1,\ldots,v_r$ of valences $2k_1+3,2k_2+3,\ldots,2k_r+3$,…

Let $M_g^s$ be the mapping class group of genus-$g$ surfaces with $s\geq 1$ boundary components, modeled by a space of fat graphs. Let $\kappa_k\in H^{2k}(M_g;\mathbb{Z})$ denote t…

Let $\overline{\mathcal M}_{g,n}$ be the moduli space of stable curves of genus $g$ with $n$ marked points, and let $\psi_i$ denote the first Chern class of the cotangent line at t…

Kontsevich's conjecture. This functional equals $\widetilde{\mathcal M}_{m_*}^{n}$ for a class $x$ of the form

Witten's conjecture. The combinatorial cycles can be expressed in terms of Mumford-Morita-Miller classes.

Let $M_N$ be a KSBA moduli space of surfaces of general type, and consider the Miller–Morita–Mumford classes, also called characteristic classes, of the diffeomorphism group of gen…

Let $Q=\mathbb{Q}[a_1,\dots,a_n]$, and let $\Omega^m_{g,n}\in R^*({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q$ be the class obtained by summing the tree contributions de…

For all vectors of positive integers $Z=(z_1,\ldots,z_n)$ of size greater than $2g-2$, let $A^g_{z_1,\ldots,z_n}$ and $B^g_{z_1,\ldots,z_n}$ be the classes in…

Let $M_{d,\chi}$ be the moduli space under consideration, and let the tautological generators in $A^*(M_{d,\chi})$ be those defined in. A relation among these generators has degree…

Let $C$ be an irreducible nonsingular projective curve of genus $g\geq 2$, let ${\mathcal M}_{\mathrm{Dol}}$ be its twisted Dolbeault moduli space with Hitchin fibration $h$, and l…

Let $k\geq 1$ and let $\boldsymbol{\mu}=k\boldsymbol{\mu}'$, where $\boldsymbol{\mu}'$ is a nonnegative partition of $2g-2$. Let…

Let $k\geq 1$ and let $\boldsymbol{mu}=k\boldsymbol{mu}'$, where $\boldsymbol{mu}'$ is a nonnegative partition of $2g-2$. The locus $\overline{\mathcal{H}}_g^1(\boldsymbol{mu}')$ i…

Let $g$ and $k$ be positive integers, let $\boldsymbol{\mu}$ be a partition of $k(2g-2)$, and let $H_{g,\boldsymbol{\mu}}^k$ be the weighted fundamental class of the compactified t…

Let $(a_1,\ldots,a_n)$ be a partition of $2g-2$ with non-negative parts. For sufficiently large $r$, let $\mathcal{W}_{g; a_1, \dots, a_n}(r)=r^{g-1}W^r_{g,n}(a_1, \dots, a_n)$ be…

Let $G$ be a complex reductive group, let $X$ be a topological space such as a closed orientable surface $\Sigma$, let $\xi\in \mathrm{H}^{2k}(BG,\mathbb{Q})$, and let $\int_C\xi$…

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