22 problems

Let $V=\{x\in\mathbb R^{d+1}:\sum_{i=0}^d x_i=0\}$, and for vectors $v_i$ spanning $\mathbb R^d$ let … A convex set is a universal cover if every subset of $\mathbb R^d$ of diamete…

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–…

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…

Let $M$ be a closed, connected, orientable, irreducible $3$-manifold with infinite fundamental group $\pi_1(M)$. Its universal covering space is denoted by $\widetilde{M}$. Univers…

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…

A universal cover is a convex set that contains every subset of Euclidean space of diameter $1$ after a suitable rotation, reflection, and translation. Consider the regular cross p…

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

Kollár–Pardon conjecture. Under these assumptions,

Let $X$ be a normal projective irreducible complex variety whose universal cover $\widetilde{X^{\mathrm{an}}}$ is biholomorphic to a semialgebraic open subset of an algebraic varie…

Let $\mathcal{T}$ be an arbitrary universal cover, and let $A_{\mathcal{T}}$ be the associated Jacobi operator. Suppose the cumulative distribution function of the density of state…

Let $X$ be a compact Kähler manifold with nef anticanonical bundle, and let $\widetilde{X}$ be its universal covering. Universal-cover splitting conjecture. The universal covering…

Let $f:S\to B$ be a fibration without multiple fibres over a curve $B$ of genus $b\geq 1$, and assume that for a general fibre $F$, the homomorphism $\pi_1(F)\to\pi_1(S)$ has infin…

Let $M$ be a closed irreducible $3$-manifold with infinite fundamental group $cpi_1(M)$. The universal-cover conjecture asserts that $M$ has universal cover homeomorphic to…

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:

Makeev's conjecture. The body $D_n$ is a universal cover for unit diameter sets.

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…

Affine universal-cover conjecture. Up to some finite étale cover, $X$ is a torus.

Let $H$ be a finite graph and let $\widetilde H$ be its universal cover. Consider a large random lift of $H$, and separate its non-backtracking spectrum into the part arising from…

Let $H$ be a finite graph, called the base graph, and let $\pi\colon G\to H$ be a random lift of degree $n$. Eigenpairs of $A(H)$ pull back to eigenpairs of $A(G)$; call these old…