16 problems

Let $P$ be a poset and let $M$ and $N$ be persistence modules over $P$. Let $d_I(M,N)$ denote their interleaving distance, and let $d_{DG}(M,N)$ denote the derived convolution metr…

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…

Bjerkevik's conjecture. On persistence modules of bounded pointwise dimension, the pruning distance is stable and Lipschitz equivalent to the interleaving distance. More precisely,…

Let $\mathscr{P}$ be a finite poset, and let $B_1$ and $B_2$ be $\mathscr{P}$-persistence sets. Write $L_{B_1}$ and $L_{B_2}$ for their linearizations, and let…

Let $M$ and $N$ be upset decomposable persistence modules, where an upset decomposable module is a direct sum of upset modules. The upset matching conjecture. There is a constant…

Let $M$ and $N$ be persistence modules that are $\epsilon$-interleaved, with $\epsilon\geq 0$. Let … Here $B(M)$ is the relevant neighborhood of $M$ defined in the paper, and an…

Let $M$ and $N$ be pfd persistence modules, and let $d_P$ denote the pruning distance and $d_I$ the interleaving distance. Write $r=\operatorname{supdim}M<\infty$. The pruning-dist…

Cubic Betti-number bound. We have

Finite minimal resolution conjecture. Every tame module $M$ over a discrete polyhedral group $Q$ has finite minimal downset and upset resolutions of the corresponding type.

Finite dense resolution conjecture. Every tame, semialgebraic, or PL module $M$ over a real polyhedral group $Q$ has finite dense downset and upset resolutions of the corresponding…

Let $S^2$ be the two-dimensional sphere, let $\bar{\mathcal C}_{S^2}$ be the relevant completion or quotient space of persistence-module objects associated with $S^2$, and let…

Indicator-dimension conjecture. The indicator-dimension of $Q$ equals the rank of $Q$ as a free module over the field $\mathbb{R}$ in the real case or the group $\mathbb{Z}$ in the…

Let $M$ and $N$ be interval decomposable $\mathbb{R}^n$-modules such that $B(M)$ and $B(N)$ contain only convex intervals. For $\epsilon>0$, they are $\epsilon$-matched when there…

Let $M$ and $N$ be $n$-dimensional p.f.d. interval decomposable modules whose barcodes contain only convex intervals. The isometry conjecture claimed that … Here $d_I$ denotes the…

Composition conjecture. Any composition of suitable left and right interleavings yields an equivalence with interleavings.

Nonadditivity conjecture. The failure of the stated composition property is not merely an artifact of the proof technique: the corresponding statement does not hold for this type o…