Invariants of Morse complexes and persistent homology
Résumé
There is canonical partition of set of critical values of smooth function into pairs "birth-death" and a separate set representing basis in Betti homology, as was established in S.Barannikov "The Framed Morse complex and its invariants" Adv. in Sov. Math., vol 21, AMS transl, (1994). This partition arises from bringing the Morse complex to so called "canonical form" by a linear transform respecting the filtration defined by the order of the critical values. These "canonical forms" are combinatorial invariants of R-filtered complexes. Starting from the beginning of 2000s these invariants became widely used in applied mathematics under the name of "Persistence diagrams" and "Persistence Bar-codes". Currently there are over 5000 articles of more than 500 researchers applying these invariants in different disciplines ranging from symplectic geometry, biology, cosmology to artificial neural nets.
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