90 problems

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

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

Higher-dimensional recovery conjecture. A similar procedure can be used to recover cycles in higher dimensions.

Barcode cardinality conjecture. The number of intervals in the barcode satisfies

Left-skewed Gumbel conjecture. The universal limit $\mathcal{L}^*$ might be the left-skewed Gumbel distribution.

Let a measuring function be given, and let an $i$-essential value mean a critical value at which homology in degree $i$ changes. Assume that torsion is not involved, and allow othe…

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

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

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

The generalized persistence diagram and the spread-Euler characteristic are invariants associated with finitely presented multiparameter persistence representations; on a fixed spr…

Topological critical-exponent conjecture. The exponent $\alpha_{H_1}$ is a genuinely new exponent characterizing the topological content of alpha-complex persistent homology at cri…

Interval-factor conjecture. Such a module $M$ does not contain an indecomposable uniquely generated submodule of dimension larger than $1$. In particular, every semistable factor o…

Let $M$ be a multiparameter persistence module and let an $varepsilon$-pruning be its decomposable approximation obtained by discarding features below scale $varepsilon$. Bjerkevik…

The Lorenz '63 system produces a trajectory sampled into bins, and its Dowker persistence contains a very long-lived one-cycle, provided there are enough bins. Lorenz '63 trajector…

Let $\mathbb{X}_{n, 0}^*$ and $\mathbb{X}_{n, \lambda}^*$ be samples produced by the subsampling procedure with threshold values $0$ and $\lambda$, respectively, and let…

Polynomial barcode-growth conjecture. The upper bound from the theorem for strictly convex or concave toric domains is sharp: ${\mathfrak b}_\epsilon(s)$ grows as a polynomial of d…

Subexponential barcode-growth conjecture. ${\mathfrak b}_\epsilon$ grows at most subexponentially, or perhaps even slower, when the flow and the first integrals are real analytic o…

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.

Let $S^1$ be the circle with its Vietoris–Rips metric thickening $\mathrm{VR}^\mathrm{m}(S^1;r)$ at scale $r$, equipped with the natural $S^1$-action, and let $u$ be a class of deg…

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

The experiments compare graphcodes with one-parameter methods on orbit datasets. The graphcode-GNN pipeline is evaluated against several one-parameter persistence methods as the da…

Homology–persistence correspondence conjecture. The (co)homology functors have associated THDs that are intimately related to the barcodes or diagrams of persistence modules and th…

Finiteness conjecture. The set

A persistence diagram records homology classes by their birth and death times; for a class, its persistence ratio is the death-to-birth ratio. The noise consists of the features in…