29 problems

Root-of-unity conjecture. The eigenvalues of $M_i$ must be roots of unity.

Total-persistence conjecture. The extremal filtrations described in Section 4 achieve the maximal total persistence over any edgewise filtration of flag complexes in homology degre…

Linear-size conjecture. The filtration $\mathcal{D}_{\infty}^{1}(X)$ consists of $O(kn)$ simplices.

Polynomial relationship conjecture. For any triply-periodic minimal surface $M$, there exists a polynomial $P \in \mathbb{R}[t]$ such that

Let $X$) be a compact Riemannian manifold of dimension $d$, and let $f\in C^{r+\alpha}(X)$, with the persistence quantity $\Pers_p(H_k(X,f))$ defined as the $p$-persistence of the…

Limiting-distribution conjecture. The random variable

Let $X(n,p)$ be the random clique complex, let $M_k(n)$ denote the maximal persistence over all $k$-dimensional cycles in $X(n,p)$, let $f_k(n)=n^{1/k(k+1)}(\log n)^{1/(k+1)}$, and…

Left-skewed Gumbel conjecture. The universal limiting distribution is

Let $\mathbb{S}$ be a sampling model in the class $\mathcal{U}$. For a complex type $\mathcal{T}$ and homological degree $k\geq 1$, let $\mathcal{L}_n$ be the empirical measure of…

Let $S=\{\sigma_1,\dots,\sigma_N\}$ be a filtered simplicial complex, and let $\tau\in\mathfrak S_N$ be a permutation of its simplices. Write … For any sequence…

Let $C$ be the phase space for the universal quasiperiodic factorization, with $u=U\circ\Phi$ and $\Phi:S^1\to C$. An appropriate wavelet basis is used to decompose the signal $u$…

Fiber–base Betti-number conjecture. The fiber $\mathrm{PH}^{-1}(D_K)$ and $K$ have the same Betti numbers.

Let $n\geq 1$ be fixed, and let $d^p_{\mathcal{M}}$ and $d^p_{\mathcal I}$ denote the corresponding distances on finitely presented $n$-parameter persistence modules, with…

Let $G$ be the filtered simplicial complex under consideration, let $\mathrm{PD}_k(G)$ denote the degree-$k$ persistence pairs, and let $\Omega$ be the associated homology class wi…

Let $G$ be the filtered simplicial complex under consideration, let $\mathrm{PD}_k(G)$ denote the set of persistence pairs in degree $k$, and let $\Omega$ be a $k$-dimensional homo…

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

Let $P$ be a finite poset, and let $\field P$ denote its incidence algebra. For the critical and regular levels $a_0,t_1,a_1,t_2,\dots,t_n,a_n$ associated with the Morse function…

Hausdorff-measure conjecture. If $\mathit{PH}_0(\operatorname{supp}\mu)$ is trivial and $0<\alpha<d$, then $$

Let $X\subseteq\mathbb{R}^m$ be compact with positive Lebesgue measure, and let $\mu$ be Lebesgue measure restricted to $X$ and rescaled to have total mass one. Let…

Let $X_n$ be a random sample of $n$ points from a compact set $X\subseteq\mathbb{R}^m$, distributed according to a nonsingular probability measure $\mu$, where $m\geq 2$. Let…

Higher-dimensional barcode-functional conjecture. There exists a constant $\kappa_g>0$ such that, for any $\lambda>0$,

Let $X$ and $Y$ be metric spaces, let $i\geq 0$, and let $d_b$ denote the bottleneck distance between persistent homology modules. Write $\mathrm{PH}_i(\mathrm{VR}^m(X;-))$ and…

Cycle-network characterization conjecture. On cycle networks having any number of nodes, 1-dimensional PPH is isomorphic to 1-dimensional Dowker persistent homology. More specifica…

Three-node equivalence conjecture. On three-node networks, 1-dimensional PPH is isomorphic to 1-dimensional Dowker persistent homology.