11 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…

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…

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…

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