# The Theory of Multidimensional Persistence

@article{Carlsson2007TheTO, title={The Theory of Multidimensional Persistence}, author={Gunnar E. Carlsson and Afra Zomorodian}, journal={Discrete \& Computational Geometry}, year={2007}, volume={42}, pages={71-93} }

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 similar complete discrete invariant exists for multidimensional…

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