Invariants of Morse complexes, persistent homology and data analysis

There is canonical partition of set of critical values of smooth function into pairs "birth-death" and a separate set representing basis in homology, as was shown in the speaker's paper in 1994. This partition arises from bringing Morse complex, defined by gradient trajectories of the function, to so called "canonical form" by a linear transform respecting the filtration given by order of the critical values. These "canonical forms" are combinatorial invariants of 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 400 scientists working on applications of these invariants in different domains ranging from biology and medicine to artificial neural nets. The talk is devoted to these invariants and their applications in mathematics and data analysis.