16 problems

Fano ampleness conjecture. If $Y$ is Fano with $\rho(Y)=1$, then ${\cal E}$ is ample, and is spanned in most cases. This is presented as the guideline for the paper’s section on co…

Let $M$ be a closed hyperbolic manifold of dimension $n\geq 4$, and let $X$ be a nontrivial finite covering of $M$ branched along a closed totally geodesic submanifold of codimensi…

Let $d=d'd”$, where $d'$ is the minimal prime factor of $d$ and $d”>2$, so that $d$ is neither prime nor equal to $4$. Let $\mathcal{D}$ be branch data with $k$ partitions, and sup…

Let $d$ be a positive integer, and let $\mathcal{D}$ be a collection of nontrivial partitions of $d$. A collection satisfying the Riemann–Hurwitz formula is called a candidate bran…

Suppose $f\colon (S^2,A)\righttoleftarrow$ is a post-critically finite hyperbolic-type branched covering without a Levy cycle. Let $\Gamma$ be a completely $f$-invariant multicurve…

Let $f\colon (S^2,A)\righttoleftarrow$ be a post-critically finite hyperbolic-type branched covering, and let $\overline{E}^p(f)$ denote its asymptotic $p$-conformal energy for…

Let $d$ and $n$ be positive integers, let $\mathscr M_d$ be the space of degree-$d$ branched self-coverings of the sphere $S^2$, and let … After giving $S^2$ its complex structure,…

Two-branch-point covering conjecture. On every finite Riemann surface $U$, there is a branched covering

Admissible cone metric count conjecture. There are at least $g$ admissible cone metrics of the form $a(d_1,\ldots,d_n)$.

Chart simplification conjecture.

Let $(F,\Gamma)$ be a branched covering surface-knot of degree $N$. Let $u_w(F,\Gamma)$ and $u(F,\Gamma)$ denote the weak simplification number and simplification number, respectiv…

Let $(F,\Gamma)$ be a branched covering surface-knot. The quantities $u_w(F,\Gamma)$ and $u(F,\Gamma)$ denote, respectively, the weak simplification number and simplification numbe…

Enumerative interpretation conjecture. The functions $\Phi_g(c)$ are generating functions for some family of branched coverings of the sphere, and the coefficients $\theta_g(k)$ en…

Mirzakhani's covering conjecture. Higher-rank arithmetic affine invariant submanifolds are either connected components of strata or branched covering constructions.

Let $M^4$ be a closed, oriented, connected $4$-manifold. A simple branched covering is a branched covering whose local monodromies around the branch set are transpositions; let…

Chisini's conjecture. Given two generic coverings $f:S\longrightarrow\mathbb{P}^2_{\mathbb{C}}$ and $f':S'\longrightarrow\mathbb{P}^2_{\mathbb{C}}$, both of degree at least $5$, wi…