27 problems

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…

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…

Let $K$ be a prime alternating knot, and let $K'$ be a 2-twin of $K$, meaning a knot sharing the same two-fold branched covering as $K$. Greene's conjecture. The knot $K'$ must be…

Let $d$ be a positive integer, and let $\Lambda$ be a collection of $k\geq 3$ nontrivial partitions of $d$. The collection satisfies the Riemann-Hurwitz condition when it satisfies…

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 $f:X\to Y$ be a finite, degree $d$, morphism from a reduced pure $n$-dimensional projective variety $X$ onto an $n$-dimensional projective manifold $Y$, with $X$ having at wors…

Naive $k$-ampleness conjecture. The vector bundle ${\cal E}$ is $k_X$-ample.

Let $\mathcal K$ be the set of branching knots in $\mathbf{S}^3$ involved in the preceding cyclic-branched-covering construction. Conjecture. The set $\mathcal K$ contains all toru…

Takahashi manifold cyclic branched-covering conjecture. For all $n>1$, the manifolds $T_n$ are $n$-fold cyclic coverings of $\mathbf S^3$, branched over a knot that does not depend…

Let $\mathcal K$ be the set of knots in $\bf S^3$ that occur as branch knots for the family of Dunwoody manifolds considered in the paper. Conjecture. The set $\mathcal K$ contains…

Let $M(\sigma)$ be a Dunwoody manifold associated with a 6-tuple $\sigma$, and let $p_{\sigma}$ denote the parameter associated with $\sigma$. Consider the subclass of manifolds sa…

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…

A branch datum records the prescribed branching information of a branched covering of the Riemann sphere, and its degree is denoted by $n$. Prime-degree conjecture. Any branch datu…

Let $\alpha$ be a relatively uniformly discrete multiset in $\mathbb R^n$, meaning that it is a finite sum of uniformly discrete multisets. The associated construction gives a bran…

B. and M. Shapiro conjecture. If all critical points of $R$ are real, then $R$ can be transformed, by post-composition with a linear fractional transformation, into a quotient of t…

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…

A generalized $n$-manifold is a generalized manifold of dimension $n$, and a map is Lipschitz BLD if it is both Lipschitz and bounded length distortion. Let…