The theory of multidimensional persistence

Persistent homology captures the topology of a filtration--a one-parameter family of increasing spaces--in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the ...

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers $k\ge 0$ and $n\ge 1$ we consider the dimension k Vietoris–Rips persistence ...${\mathbf{}}_{}^{}$${\mathbf{}}_{\mathrm{}\mathrm{}}^{}$$\mathrm{}\mathrm{}$

We introduce and study $$A_\infty $$A -persistence on the homology, with coefficients in a field, of a filtration of topological spaces. This is a family, one for each $$n\ge 1$$n 1, of homological invariants which provide information not readily ...