21 problems

Let $\mathbb{Z}^n$ have the Manhattan metric, let $\{0,1\}^n\subseteq\mathbb{Z}^n$ have the induced metric, and let $\widetilde{H}_i(-;\mathbb{Z})$ denote reduced homology with int…

For fixed $m>0$, let $\mathcal{G}_m^n$, $\Delta_m^{n,r}$, and $\Gamma_n^{\alpha,r}$ be the complexes defined in the paper. Let $V(\Delta)$ denote the vertex set of an induced subco…

Vietoris–Rips ultrametric condition. The Vietoris–Rips clusters for some $\epsilon>0$ realise the sufficient ultrametric condition

Let $s(n;r)$ denote the size of the smallest facet in $\operatorname{\mathcal{VR}}(Q_n;r)$. The source proves $s(n;r)\leq2r$ when a Hadamard matrix of order $2r$ exists and…

Let $Q_n=\\{\pm1\\}^n$ be the Hamming cube with Hamming distance. A subset is maximally diameter-$r$ if its diameter is $r$ and adjoining any point outside it produces diameter gre…

Let $S^n$ be the $n$-sphere, let $r\ge 0$, and let $\mathrm{VR}(S^n;r)$ denote the Vietoris–Rips complex formed using the strict-distance convention of the paper. Finite-CW conject…

Let $S^n$ be the $n$-sphere, let $r_n$ be the scale defined in the paper, let $\mathrm{SO}(n+1)$ be the special orthogonal group, let $A_{n+2}$ be the alternating group, and let…

Let $S^n$ be the $n$-sphere, and write $\mathrm{VR}(S^n;r)$ for its Vietoris–Rips complex at scale $r\ge 0$. Critical-scale conjecture. There are countably many critical scales … s…

Let $\mathbb{I}_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{VR}(\mathbb{I}_n;r)$ be its Vietoris–Rips complex at scale $r$. Let the collapsibility number of a…

Let $\mathbb{I}_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{VR}(\mathbb{I}_n;r)$ be its Vietoris–Rips complex at scale $r$. For $r\in\{2,3\}$, write…

Let $K\subseteq {\mathbb R}^d$ be a convex body with smooth boundary, and let the distribution be uniform on $K$. Write $r_n$ for the radius parameter and ${{\mathcal R}}(n,r_n)$ f…

Let $X$ be a metric space, potentially with additional assumptions, let $r\geq 0$, and let … Here $\operatorname{VR}^{\mathrm{m}}(X;r)$ denotes the metric Vietoris–Rips thickening…

ANR Vietoris-Rips conjecture. There exists $r(X)>0$ such that

Longest-interval conjecture. For every $I\in\mathrm{barc}^{\mathrm{VR}}_k(M;\mathbb{F})$ and every $k\geq 1$,

AAF17's sphere conjecture. There exists an $\varepsilon>0$ such that

Lower semicontinuity conjecture. The complex $\mathrm{VR}_r(X)$ is homotopy equivalent to $\mathrm{VR}_{r-\varepsilon}(X)$ whenever $\varepsilon>0$ is sufficiently small.

Projective-space stability conjecture. There exists a sufficiently small $\varepsilon>0$ such that, for every $0<\delta<\varepsilon$, both…

Let $k\geq 1$, let $r$ satisfy … let $\mu\in\mathrm{VR}^m(S^1;r)$, and let $\iota$ map the boundary of the Barvinok–Novik orbitope $\mathcal{B}_{2k}$ into the Vietoris–Rips metric…

Let $S^1$ be the unit circle, let $r>0$, and let $\mathrm{VR}^m(S^1;r)$ denote its Vietoris–Rips metric thickening. For each integer $k\geq 1$, let $\mathcal{B}_{2k}$ be the Barvin…

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…

Let $S^n$ be the $n$-sphere, let $r_n$ be the scale under consideration, and let $\mathrm{VR}^m_\leq$, $\mathrm{VR}^m_<$, $\mathrm{VR}_<$, and $\mathrm{VR}_\leq$ denote the corresp…