548 problems

For every integer $g\ge 2$ and every prime $p$, let $\mathcal{S}_g$ denote the supersingular locus in the moduli space of principally polarized abelian varieties of dimension $g$ o…

Let $C$ be a smooth projective curve, let $S\subset C$ be a finite subset, and fix an integer $q\geq 2$. Let $\mathbb T$ denote the parameterizing scheme for non-isotrivial familie…

Aluffi–Chen–Marcolli conjecture. The polynomial $P_{\overline{\mathcal{M}}_{0,n+1}}(t)$ is real-rooted.

Let $g\in\frac12\mathbb{Z}_{\geq0}$ and $n\in\mathbb{Z}_{\geq1}$ satisfy $2g-2+n\geq1$, and let $\omega_{g,n}$ be the multidifferential defined by Definition RTR. Refined topologic…

The Betti moduli space $M_B$ parametrizes irreducible local systems on a punctured curve with prescribed central monodromy, while $M_{\mathrm{Dol}}$ is the corresponding Dolbeault…

Let $Q$ be a connected component of the coarse moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ such that $X$ is a connected smooth projective complex variety,…

Let $(M,\mathcal P_M)$ be as in Theorem 61, let $\Sigma$ be the cylindrical end of $M$, and let $R(M;\mathcal P_M)$ and $R(\Sigma;\mathcal P_\Sigma)$ denote the corresponding repre…

Let $L$ be the line bundle and $\phi$ the vortex data on ${\mathbb P}^1$; let $\tau$ satisfy the inequality in equation, and let $\alpha>0$. The datum $\phi$ is polystable. Uniquen…

Let $Y$ be a nodal curve with $b4$ nodes, let $\Gamma$ be its dual graph, and let $\mathcal B$ be the set of bonds of $\Gamma$. Define the vector-weight collection…

Let $n$ be a positive integer, let $d,e$ be integers satisfying $(d,n)=(e,n)=1$, and let $\mathcal{M}_{\Dol}^d$ and $\hat{\mathcal{M}}_{\Dol}^e$ denote the corresponding Dolbeault…

Let $\mathcal{M}^{G}$ be the moduli space associated with a simple group $G$, and let the star product be the product defined by equation on the chord-diagram algebra. Descent conj…

Let $X$ be the bordered holomorphic curve and let $\tilde M(f)$ be the compactified moduli space associated with a fixed homotopy class $[f]$. A simple path in $X$ has no self-inte…

Let $M$ be a manifold with symplectic form $\omega$ and compatible almost complex structure $J$, let $L_1(j),\dots,L_{p_j}(j)$ be transversely meeting Lagrangian submanifolds, and…

Let ${\cal M}$ be Hitchin's moduli space of rank-$2$ Higgs bundles with fixed determinant of degree $1$ over a Riemann surface $\Sigma$ of genus $g>1$. It is a simply connected, no…

Dynamical model conjecture. There exists an endomorphism $\Phi_C$ such that:

An $N=2$ super-Riemann surface with genus-zero compact body is an $N=2$ super-Riemann surface whose underlying compact body has genus zero. The $N=2$ super-Riemann sphere is denote…

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 $\M_g$ be moduli space with the Teichmüller metric, and let ${\cal C}_g$ be the curve complex of a closed genus-$g$ surface: its vertices are nontrivial, nonperipheral isotopy…

For $g\geq 2$, let $\M_g$ be the moduli orbifold of genus-$g$ Riemann surfaces. Nonpositive-curvature conjecture. The orbifold $\M_g$ admits no complete, finite-volume Riemannian m…

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…

Generic injectivity conjecture. The map $\sigma$ is generically injective.

Let $X$ be a proper rigid analytic space over $K$. A formal versal deformation of $X$ is the formal deformation supplied by the local deformation theory of $X$. Global versal-famil…

Uniform boundedness conjecture. For every type $t$, there is a bounded region

Let $V$ be an irreducible variety of dimension $2g+2n-5$, with $4\leq 2n-2<g$, and let $\mathcal{F}\rightarrow V$ be a family of smooth $n$-gonal curves of genus $g$ with maximal v…

Let $X$ be a closed oriented $3$-manifold and let $G=\operatorname{SU}(2)$. Let $\mathcal M_j$ be the union of the connected components of the moduli space of flat $G$-connections…