421 problems

Let $(V,d)$ be a finite metric space with $n$ points. For distinct points $x,y,z\in V$, say that $z$ lies between $x$ and $y$ when … For distinct $x,y\in V$, define the line genera…

Let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, let $\Pi_{2k+1}:\mathbb{S}^{2k+1}\to\gamma_{2k+1}(\mathbb{S}^1)$ be the neares…

Busemann's conjecture. Every G-space is finite-dimensional, and every finite-dimensional G-space is a topological manifold.

Kleiner's conjecture. Approximately self-similar and combinatorially Loewner metric spaces attain their conformal dimension.

Let $\ell_1^n$ denote $n$-dimensional real space with its usual $\ell_1$ norm, and let $e(X)$ be the maximum cardinality of an equilateral set in a metric space $X$. The canonical…

De Palma–Trevisan conjecture. The divergences defined this way are genuine metrics on quantum state spaces. The conjecture has recently been justified under certain additional assu…

Metric cone conjecture. If the Kähler–Ricci shrinker has maximal volume growth, then $(Z_0,d^Z_0)$ is a metric cone and coincides with the metric completion of…

Let $\pi: (X,L)\to \mathbb{D}^*$ be a polarised maximal degeneration of compact Calabi-Yau manifolds, and let $(X_t,\omega_{CY,t},\Omega_t)$ denote its fibre with the Calabi-Yau st…

Austin's conjecture. If $\{f_1,\ldots,f_r\}$ is a pairwise commuting family of $\lambda$-contractions on $X$, then the maps have a common fixed point; that is, there exists…

Mondino's conjecture. $(X,\mathsf{d})$ is homeomorphic to an orbifold, possibly with boundary.

Let $f:M\to\Sigma$ be an elliptic bundle over a Riemann surface $\Sigma$ of genus at least $2$, and let $\omega(t)$ solve the Chern-Ricci flow on $M$ from an arbitrary Gauduchon me…

Nabutovsky's conjecture. There exist geodesic nets in the Euclidean plane with $4$ unbalanced vertices and an arbitrarily large number of balanced vertices. Moreover, this may alre…

Ballmann's higher rank rigidity conjecture. Any Hadamard space of rank at least two with a geometric group action admits an affine map which is not a dilation.

Edge-contraction and subdivision conjecture. For all $L,C\in\mathbb{N}$, there exists $C'\in\mathbb{N}$ such that if a graph $G$ is $(L,C)$-quasi-isometric to a graph in…

Let … be a non-collapsed Gromov–Hausdorff-convergent sequence of compact Alexandrov spaces of dimension $n$ with curvature bounded below by $-1$. Perelman's bi-Lipschitz stability…

Let $n$ be the ambient dimension, let $u,v\in\mathbb{R}^n$ be distinct points, and let $[u,v]$ denote the line segment joining them. For a parameter $c$, write…

Elementary molecule conjecture. Every extreme point of $B_{\mathcal{F}(M)}$ is an elementary molecule.

The conjectured Lipschitz bound. For all $x,y,a\in\mathbb{B}^n$,

Walter's conjecture. One has

A Riemannian metric is called area-extremal when the corresponding Llarull-type rigidity theorem remains valid with the condition that the map $h$ is area-contracting rather than…

Alexandrov gluing conjecture. The gluing produces an Alexandrov space $Y$ if and only if the gluing is by isometry and, for every $p\in Y$, $T_p(Y)$ is a metric cone over…

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle, let $\pi:X\rightarrow X_{can}$ be the associated holomorphic Calabi–Yau fibration, and let $S$…

Nonexistence conjecture. There is no such manifold for which

Generic means conjecture. For a generic set of points $x_1,\ldots,x_n$ in $\bar{X}$, $\bar{\bar{X}}$, $\bar{Z}$, or $\bar{\bar{Z}}$, means exist and are unique.

Discrete level-set conjecture. There is a result of a similar nature for Lipschitz quotient mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$.