41 problems

Let $T_{3n-2,3n-2}$ be the $(3n-2)\times(3n-2)$ torus grid with its induced $l^1$ metric, and let $\mathrm{VR}(T_{3n-2,3n-2};n)$ be the Vietoris–Rips complex at scale $n$. 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…

Conjecture 3. As $N,M\to\infty$,

Conjecture 2. As $N,M\to\infty$,

Conjecture 1. As $N,M\to\infty$,

Let $M$ be a smooth, simple closed curve in $\mathbb{R}^N$. For a finite set $S\subseteq\mathbb{R}^N$, write $d_H(S,M)$ for the Hausdorff distance between $S$ and $M$, and let…

The side-length extrema conjecture. There are real values $1<a_1^-<a_1<a_1^+<a_2^-<a_2<a_2^+<\sqrt{2}$, with $a_1\approx1.3299$ and $a_2\approx1.4123$, such that…

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…

Let $\mathbb{Z}^n$ carry the Manhattan metric, also called the standard word metric, and let $\mathcal{VR}(\mathbb{Z}^n;r)$ denote its Vietoris–Rips complex at scale $r$. Zaremsky'…

3-sphere conjecture. For a countable family of $(n,k)$ pairs, potentially including the pairs listed above, $\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to $S^3$. The pair…

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

Let $n$ be a positive integer, equip $\mathbb{Z}^n$ with its standard word metric, and for $t\in\mathbb{R}\cup\{\infty\}$ let $\operatorname{\mathcal{VR}}_t(\mathbb{Z}^n)$ denote t…

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 $Q_n=\\{\pm1\\}^n$ be the Hamming cube, and let $\operatorname{\mathcal{VR}}(Q_n;r)$ denote its Vietoris–Rips complex at scale $r$. Homology-dimension conjecture. For some scal…

Let $Q_n=\\{\pm1\\}^n$ be the Hamming cube, with Hamming distance, and let $\operatorname{\mathcal{VR}}(Q_n;r)$ be its Vietoris–Rips complex at scale $r$. Write…

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 $S^n$ be the $n$-sphere, and let $\mathrm{VR}(S^n;r)$ denote its Vietoris–Rips complex at scale $r\ge 0$. Countability conjecture. The number of different homotopy types of…

Shukla's conjecture. For $n\geq r+2$,

Consider the original New-VR algorithm, its simplified version, the original Incremental-VR algorithm, and its simplified version, with complexity measured for the constructions de…

Let $X$ be a separable metric space and let $r\ge 0$. The Vietoris--Rips complex $4\mathrm{VR}(X;r)\ and metric thickening$4mathrm{VR}^mathrm{m}(X;r)…

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…