10 problems

Let $(X,\tilde g)$ be the universal cover of a $d$-dimensional smooth Riemannian manifold $(M,g)$. A mediatrix is the locus equidistant from the reference point and one of its nont…

Let $X$ be a closed, connected, irreducible, orientable 3-manifold with infinite fundamental group. Its universal covering space is denoted by $\widetilde{X}$. Universal Covering C…

Campana's torus-submersion conjecture. Up to a finite étale cover of $X$, the manifold $X$ admits a torus submersion over a projective manifold $Y$ such that $K_Y$ is ample and the…

Let $X$ be a complex projective manifold with infinite fundamental group, and let $\widetilde{X}$ be its universal cover. Suppose that $\widetilde{X}$ is quasi-projective. Claudon–…

Product-cover conjecture. There exist possibly noncompact Kähler manifolds $\mathsf{Y}$ and $\mathsf{Z}$, with $\omega_\mathsf{Y}$ trivial, and a covering map

Let $X$ be a smooth complex compact Kähler variety, and let $\widetilde{X}$ denote its universal cover. Shafarevich's conjecture. The universal cover $\widetilde{X}$ is holomorphic…

Let $B$ be a fixed graph, let $\widehat B$ be its universal cover, and let $A_{\widehat B}$ be the adjacency operator on $\widehat B$. Let $\mathcal{C}_n(B)$ denote the model of de…

Compactifiable universal cover conjecture. After passing to a finite étale cover, there exists a locally trivial fibration $X\rightarrow A$ with simply connected fibre $F$ onto a c…

Semialgebraic universal-cover classification conjecture. The following are equivalent:

Let $D$ be a multipartite degree matrix, let $T_D$ be its universal cover, and let $T_{D,r}(s)$ denote the radius-$r$ ball around $s\in V(T_D)$; write $\rho(\mathord\cdot)$ for spe…