9 problems

Persistent homology equivalence conjecture. The infinity-Vietoris–Rips metric thickenings should have the same persistent homology as the classical Vietoris–Rips simplicial complex…

Let $(S^1)^n$ be the $n$-torus, let $\mathcal F_r$ be the set of all closed geodesic balls of radius $r$ in $(S^1)^n$, and let $\mathcal P((S^1)^n,\mathcal F_r)$ denote the associa…

Let $S^n$ be the $n$-sphere, and let $\mathrm{AVR}^m(S^n;\frac{2\pi}{3})$ denote the anti-Vietoris–Rips metric thickening at scale $\frac{2\pi}{3}$. Homotopy-dimension conjecture.…

Let $4\mathcal{U} be a uniformly bounded open cover of a separable metric space$X$. The Vietoris simplicial complex$4mathcal{V}(mathcal{U}) has vertex set $X$, with a finite subs…

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 $X$ 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…

Let $M=(X,K,\phi)$ be a simplicial metric thickening, with metric thickening $\mathcal{M}\subseteq\mathcal{P}(X)$. Let $\hat{M}$ be the closure of $\mathcal{M}$ in $\mathcal{P}(X)$…

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…