2005: This was a big year.

1. Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.

2. Edelsbrunner, Cohen-Steiner and Harer show that the map $$\text{[functions X to R]} \to \text{[barcodes]}$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.

2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks.

2012 Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.

2015 Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.

2018 MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.

2005: Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.

2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks. Edelsbrunner, Cohen-Steiner and Harer show that the map $$\text{[functions X to R]} \to \text{[barcodes]}$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.

2012: Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.

2015: Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.

2018: MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.

1. Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.

2. Edelsbrunner, Cohen-Steiner and Harer show that the map $$\text{[functions X to R]} \to \text{[barcodes]}$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated {\em stability theorem}.

2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.

2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with fast (at the time!) software package Perseus.

1. Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.

2. Edelsbrunner, Cohen-Steiner and Harer show that the map $$\text{[functions X to R]} \to \text{[barcodes]}$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.

2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.

2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with a fast (at the time!) software package Perseus.