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Author Barannikov, S.
Source Hyper Articles en Ligne (HAL)
Content type Text
File Format PDF
Language English
Subject Keyword Morse complex ♦ Spectral sequences ♦ Morse theory ♦ critical values ♦ persistence diagrams ♦ singularity theory ♦ Persistence modules ♦ Persistence diagram ♦ persistent homology ♦ info ♦ math ♦ Computer Science [cs]/Computational Geometry [cs.CG] ♦ Mathematics [math]/Algebraic Topology [math.AT] ♦ Mathematics [math]/Symplectic Geometry [math.SG] ♦ Mathematics [math]/Algebraic Geometry [math.AG] ♦ Mathematics [math]/K-Theory and Homology [math.KT]
Abstract 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 S.Barannikov "Framed Morse complex and its invariants"(1994). This partition arises from bringing filtered 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. This is an introduction to these invariants and their applications in mathematics and data analysis.
Educational Use Research
Learning Resource Type Proceeding
Publisher Date 2019-03-07