60 problems

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

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

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

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

Let $T$ be a merge tree, and let a path be a 1-dimensional simplicial complex with no cycles. A discrete Morse function on a path induces a merge tree by filtering the path through…

Persistence diagrams can be augmented to depth posets, which equip their points with a partial ordering and can be applied to the radial transform of a loop around a singularity. C…

Let $K$ be a hyperbolic knot with signature $4\sigma(K)4$, natural slope $\text{slope}(K)$, volume $\text{vol}(K)$, and injectivity radius $\text{inj}(K)$. Final conjecture. There…

Let $M$ be a smooth, simple closed curve in $\mathbb{R}^N$. For a finite set $S\subseteq\mathbb{R}^N$, write $d_H(S,M)$ for the Hausdorff distance between $S$ and $M$, and let…

Finiteness conjecture. Under generic conditions on the input pair $(\bo\p,\bo\s)$, this special subset of $[0,1]$ is finite.

Geodesic-space conjecture. Under mild assumptions, the space of persistence diagrams with the PPD induces a geodesic space structure that is computationally tractable and topologic…

Lifted toroidal-cycle conjecture. All cycles in $K$ which are trivial in $G$ can be written in terms of lifts of toroidal cycles.

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…

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…

Topological complexity conjecture. Since LSUN Kitchen is a much larger dataset containing diverse images, its underlying lower-dimensional submanifold has more complex topological…

Torus extension conjecture. It should be possible to extend the Poisson approximation results of Theorems 1–3 to $mathbb T^d$.

Let $(X,d,cmu)$ be a metric-measure space, and let the sample size increase for the data-driven procedures estimating the real cohomology groups of $(X,d,cmu)$. Consider the Hodge-…

The attracting sets of the dynamical systems under consideration are expected to have relatively low dimensions. Topological-inference sufficiency conjecture. Available topological…

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…

The input manifold is partitioned by the rank decomposition of a neural network, and in the contractible case the topologically destructive regions are finite collections of indivi…

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…

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…

Manifold reconstruction conjecture. Given the Manifold Hypothesis that time series data lie on a manifold, this method reconstructs this topological space from the input time serie…

Let neural networks with sigmoidal activation functions have a fixed architecture, and let their topological expressivity measure the Betti numbers of decision regions realizable b…

Persistent homology encodings are represented by persistence images (PIs), and the downstream model may be a complex model such as a convolutional neural network (CNN). Let the ave…