11 problems

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Write $\operatorname{conn}(K)$ for the connectivi…

Equally spaced configuration conjecture. The configuration attaining maximal lifetime consists of $m$ points equally spaced on the unit circle.

For each dimension $n\ge 1$ and scale $r>0$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r$. Finite CW-complex conjecture…

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Countable homotopy-type changes conjecture. The h…

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)…

Steinhaus–Čech equivalence conjecture. The Čech filtration constructed from $X$ is isomorphic to the cover filtration on $X$ constructed from $\check{C}_R(X)$.

Steinhaus–Čech equivalence conjecture. The Čech filtration and the Steinhaus Nerve constructed from its terminal cover are isomorphic.

Let $\mathcal{C}_r$ be the random Čech complex generated by a homogeneous Poisson process on the flat torus, let $H_k(\mathcal{C}_r)$ denote its $k$-th homology, and let $\Lambda$…

Let $G_k^{{\mathcal C},{\mathcal U}}$ denote the relevant $k$-dimensional random Čech complex, let $n$ be the number of sampled points, let $r_n$ be its scale parameter, and let…

Let $c_{m,k}^{\vartheta}$ denote the maximal $\vartheta$-persistent Betti number $\beta_k^{\vartheta}(P_m,r)$ over all point sets $P_m$ of $m=m(\vartheta,k)$ vertices and all…

Let $M$ be a compact Riemannian manifold, let $\mathcal{C}_r$ be the Čech complex of a sample of $n$ points at scale $r$, and let $H_k$ denote $k$-th homology. Write $\Lambda$ for…