Afra Zomorodian, Dartmouth College & MSRI 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 filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no complete discrete invariant exists for multidimensional persistence. We also propose a discrete invariant, the \emph{rank invariant}, for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.