Some new results on the matching distance in biparameter persistent homology

When studyng monoparameter persistent homology, the bottleneck distance is the most standard choice in literature. The advantage of this choice is that persistence diagrams in the monoparameter setting are a complete invariant as shown by the Isometry Theorem. However, they are not a complete invariant in multiparameter persistent homology. This makes the multiparameter case much more harder to study. In this work we study some properties of the biparametric matching distance, which is an analogue of the bottleneck distance for higher dimensions, between persistence diagrams, along with the set of special values (a,b) in which the matching realising this distance may change abruptly. We prove a result that drastically reduces the cost of the computation of the matching distance, under suitable regularity conditions on the filtered space (X, φ) and on the special set. Moreover, we give a proof for an extension of the Position Theorem, which is a central step in the proof of our main theorem.