15 problems

Uniqueness conjecture. Every metric space $X$ with a closed and discrete set $\operatorname{dist} X$ has the property of uniqueness in the class $\operatorname{\mathcal{G\!H}}$.

Let $\mathbb{S}_{\mathrm{E}}^m$ denote the $m$-dimensional Euclidean sphere equipped with its standard $\mathbb{Z}_2$-action, and let $d_{\mathrm{GH}}^{\mathbb{Z}_2}$ be the…

For each positive integer $k$, let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, and let $\mathrm{dis}(\gamma_{2k+1})$ denote it…

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…

In spherical coordinates on $\mathbb{S}^2$, define … and … where $(x(\phi,\theta),y(\phi,\theta),z(\phi,\theta))=(\cos(\phi)\sin(\theta),\sin(\phi)\sin(\theta),\cos(\theta))$. Let…

Let $\alpha:\mathbb{S}^1\to\mathbb{S}^2$ be the embedding … and let $R_\alpha$ be its induced correspondence. Variant correspondence conjecture. … This correspondence arose from co…

Let $\psi_{2k+1}:\mathbb{S}^{2k+1}\to\mathbb{S}^1$ and $\psi_{2k}:\mathbb{S}^{2k}\to\mathbb{S}^1$ be the projections associated with the trigonometric moment curve embeddings, and…

Let $k\in\mathbb{N}$, and let $\gamma_{2k}:\mathbb{S}^1\to\mathbb{S}^{2k}$ and $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the trigonometric moment curve embeddings defined…

Let $\mathcal{M}$ denote the collection of compact metric spaces, and for $i,j\in\{1,2,3\}$ let $e_{ij}:X_1\times X_2\times X_3\to X_i\times X_j$ be the coordinate projection. A co…

Let $\mathcal{M}$ denote the collection of compact metric spaces. Given three compact metric spaces $X_1,X_2,X_3\in\mathcal{M}$, an isometric embedding…

Let $D54A^m$ denote the unit $m$-sphere with its intrinsic spherical metric, let $d_{\operatorname{GH}}$ be the Gromov–Hausdorff distance, and let $D701_m$ be the quantity defined…

Circle distance conjecture. One has

Let $4\mathcal{M}^14$ denote the collection of all $1$-connected compact metric spaces, and let $4\mathfrak{S}_p4$ be the operator whose kernel consists of the spaces mapped to the…

Let $S^1$ be the circle of unit radius in $\mathbb{R}^2$ centered at the origin, and let $P_n$ be an inscribed regular polygon with $n$ sides; equip both with the Euclidean metric.…

Let $M$ be a closed Riemannian manifold, and let $d_{\mathrm{GH}}$ denote the Gromov–Hausdorff distance. For sufficiently small $r$, the metric Vietoris–Rips thickening…