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Persistent homology – a survey. (English) Zbl 1145.55007
Goodman, Jacob E. (ed.) et al., Surveys on discrete and computational geometry. Twenty years later. AMS-IMS-SIAM summer research conference, Snowbird, UT, USA, June 18–22, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4239-3/pbk). Contemporary Mathematics 453, 257-282 (2008).
Summary: Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.
For the entire collection see [Zbl 1134.52001].

55N99 Homology and cohomology theories in algebraic topology
68W01 General topics in the theory of algorithms
57M99 General low-dimensional topology
55T05 General theory of spectral sequences in algebraic topology