1,169 problems

A stratum of translation surfaces is a quasi-projective variety, and its orbifold fundamental group is finitely presented. A connected component of a stratum is a classifying space…

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…

Kontsevich's volume formula. The Masur–Veech volume is

Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a degree-$d$ map over $\overline{\mathbb{Q}}$, and let $c$ range over its critical points, counted with multiplicity. Define the critical hei…

Let $X$ be a smooth algebraic curve of genus $g\ge 2$, let $L\in\operatorname{Pic}(X)$ have degree $1$, and let $\Gamma=\operatorname{Pic}^0(X)[n]$. Denote by … the moduli space of…

Let $f_U:U\to V$ be a smooth projective family, and suppose that $V$ has a morphism $\mu:V\to P_h$, where $P_h$ is the coarse moduli scheme associated with the moduli functor of po…

Let $X$ be a projective complex $3$-manifold, let $beta eq 0$ be an algebraic curve class, and let $tau{ki}(etai)$ denote descendent insertions with $etaito H^{li}(X,mathbb{Q})$. W…

Homological stability conjecture. For every $m>1$, there is an isomorphism

Let $\mathcal{M}_g$ be the coarse moduli space of genus $g$ Riemann surfaces, viewed as a quasiprojective complex variety. Looijenga's conjecture. For every $g\geq 2$, the variety…

Zograf's conjecture. Based on numerical data, Zograf conjectured that

For each integer $n\geq 1$, let $\mathrm{Per}_n(0)\subset \mathrm{M}_2$ be the PCF-special curve parametrizing quadratic rational maps for which one critical point has exact period…

Let $X$ be a smooth projective curve of genus $g\geq 2$, let $L\in\operatorname{Pic}^{d}(X)$ with $(2,d)=1$ and $0<d<2$, and let $\mathrm{M}(2,L)$ be the moduli space of stable ran…

Let $G$ be a reductive group, let $C$ be a curve, and let $E$ be a principal $G$-bundle over $C$ with canonical parabolic reduction. Behrend's conjecture. The canonical parabolic r…

Let a polarized variety mean a projective variety equipped with a polarization, and consider the moduli space of K-polystable polarized varieties. K-moduli conjecture. A set of K-p…

Let $w^{\alpha}$ be the normal coordinates for the Dubrovin–Zhang hierarchy, let $\tilde u^{\alpha}$ be the normal coordinates for Buryak's double ramification hierarchy, let…

Kuznetsov's conjecture. There is a correspondence

Let $f:\mathfrak{X}\to S$ be an algebraic family of polarized algebraic manifolds over a quasi-projective manifold $S$, with period map $\Phi:S\to\Gamma\backslash D$ and lifted per…

A hyperelliptic curve with extra involutions is a smooth hyperelliptic curve possessing an automorphism group larger than the subgroup generated by its hyperelliptic involution. It…

Let $C$ be a smooth projective curve of genus $g\geq 1$, and let $r$ and $d$ be coprime integers. Consider the moduli space of stable vector bundles on $C$ of rank $r$ and degree…

Let $(M,J,\omega,\Omega)$ be an almost Calabi--Yau $m$-fold, let $X$ be a compact special Lagrangian $m$-fold in $M$ with conical singularities, and let…

Let $\mu=(m_1,\ldots,m_n)$ be a partition of $2g-2$, and let $\mathbb P\mathcal H(\mu)$ be a varying stratum of holomorphic differentials up to scaling. Let … where…

Let $V_{\mathbb{A}_{1,k}}$ denote the volume of the moduli space of $(1,k)$-annuli, viewed as a polynomial in $\pi^2$. Here $k$ is a positive integer. Leading-term conjecture for a…

Let $H_g((d),\boldsymbol{\nu})$ be the one-part double Hurwitz number, where $\boldsymbol{\nu}=(a_1,\ldots,a_n)$ is a partition of degree $d$, and let $r$ be the number of simple b…

Let $A$) be the Jacobian of a smooth genus-two curve, embedded by $|3\Theta|$ in $\check{\mathbb{P}}^8$, and let $\mathscr{C}_3=Z(F)$ be the unique cubic hypersurface singular alon…

Let $S$ be a closed orientable surface of genus $k\geq 2$. For a quasi-Fuchsian hyperbolic 3-manifold, let its convex core be the minimal compact totally convex subset, whose bound…