62 problems

Wolf's conjecture. The metric $g$ is flat.

Let a group action on a metric space be locally elliptic if every cyclic subgroup has bounded orbits, and elliptic if the whole group has bounded orbits. Consider an action of a fi…

Let $X$ be a Hadamard space, and let its asymptotic rank be the largest dimension of a Euclidean flat that occurs asymptotically. An isoperimetric filling inequality is a bound on…

Let $A_S$ be a spherical Artin group, and let $\Delta_S$ be its spherical Deligne complex, equipped with its naturally defined metric. Charney–Davis conjecture. The complex…

Let $M$ be a tame, complete, finite-volume $n$-manifold of bounded nonpositive curvature. A manifold has geometric rank one when it satisfies the rank-one condition used in the sou…

Let $M\to E\to B$ be a smooth fiber bundle whose fiber admits a negatively curved Riemannian metric. Assume that $B$ is a paracompact, Hausdorff space with the homotopy type of a s…

A cubical complex is locally $5$--large when the link of every vertex is $5$--large, meaning that every simplicial cycle of length less than $5$ has a diagonal. The phrase “contain…

Local Euler characteristic conjecture. Each local Euler characteristic should satisfy

For $g\geq 2$, let $\M_g$ be the moduli orbifold of genus-$g$ Riemann surfaces. Nonpositive-curvature conjecture. The orbifold $\M_g$ admits no complete, finite-volume Riemannian m…

The FAn conjecture. The following are equivalent:

Conjecture. Then

Let $M$ be a manifold with an $F$-structure, and suppose the $F$-structure has positive rank. A local splitting structure is a geometric structure associated with the nonpositively…

Uniform boundedness conjecture. There is a constant $C$ depending only on the genus of $S$, the area of $S$, the minimum length of the geodesics in $S$, and the minimum of the curv…

Let $M$ be a closed, nonpositively curved, locally symmetric manifold with no local $\mathbb{R}$ factors. The Gromov norm of $M$, namely the simplicial volume, is positive. Gromov'…

Let $M$ be a closed $n$-manifold with nonpositive sectional curvature, negative Ricci curvature, and no local $\mathbb{R}$ factors. Let $N$ be any closed Riemannian manifold and le…

Let $M$ be a closed manifold which admits a locally symmetric Riemannian metric $g_{loc}$ with nonpositive sectional curvature. Assume that $(M,g_{loc})$ has no local factors isome…

Let $M^d$ be a compact, closed, $d$-dimensional manifold with a piecewise Euclidean structure, and suppose that it is non-positively curved in the sense of Gromov. Let $\chi(M^d)$…

Euler characteristic conjecture.

Zero-Liouville-measure conjecture. The set $\Lambda_0$ has zero Liouville measure.

Let $d$ be a positive integer, let $\textsc{Sym}_d$ be the symmetric group, let $T$ be its set of transpositions, and let $\delta=(1\ 2\ \cdots\ d)$. The interval complex…

Let $M$ be a closed nonpositively curved $4$-manifold, and let $\|M\|$ denote its simplicial volume and $\chi(M)$ its Euler characteristic. Connell–Ruan–Wang's equivalence conjectu…

Let $M$ be a closed, connected, oriented $n$-dimensional manifold. Its simplicial volume is … A Riemannian metric on $M$ has negative definite Ricci curvature when its Ricci curvat…

Let $M$ be a manifold. Suppose $M$ admits a Riemannian metric with nonpositive sectional curvature everywhere and negative definite Ricci curvature at some point. Connell–Wang's co…

Let ${{\normalfont\textsc{Poly}}}_d^{mt}$ be the space of all monic degree-$d$ complex polynomials up to translation with its stratified Euclidean metric, and let $K_d$ be the dual…

Let ${{\normalfont\textsc{Poly}}}_d^{mt}$ be the space of all monic degree-$d$ complex polynomials up to translation, equipped with its stratified Euclidean metric. CAT(0) conjectu…