Reference on Persistent Homology
Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice book if the students had sufficient background, and the foundational paper of Zomorodian-Carlsson if they did not.
I haven't read Jose's recent article mentioned in Joe's answer, but here is what I remember of the good old days (with apologies to all the important stuff that got missed).
1992: Frosini introduces "size functions", which we would today consider equivalent to 0-dimensional persistent homology.
1995: Mischaikow + Mrozek publish a computer-assisted proof of chaos in the Lorenz equations; a key step involves computing Conley indices, which are relative homology classes. This produces considerable interest in machine computation of homology groups of spaces from finite approximations (eg large cell complexes).
1999: Robbins publishes this paper emphasizing that functoriality helps approximate the homology of an underlying space from Cech complexes of finite samples; meanwhile Kaczynski, Mischaikow and Mrozek publish their book on efficient homology computation via simple homotopy type reductions of cell complexes.
2002: Edelsbrunner, Letscher and Zomorodian introduce persistence from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.
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.
2008: Niyogi, Smale and Weinberger publish a paper solving the homology inference problem for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts Ayasdi, putting his money where his math is.
2009: Carlsson and Zomorodian use quiver representation theory to point out that getting finite barcodes for multiparameter persistence modules is impossible, highlighting dimension 2 as the new frontier for theoretical work in the field.
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.
2011: Nicolau, Levine and Carlsson discover a new type of breast cancer using 0-dimensional persistence on an old, and purportedly well-mined, tumor dataset.
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.
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.
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.
Good luck with your course!
Edelsbrunner and Harer's book seems good.
Edelsbrunner, Herbert; Harer, John L., Computational topology. An introduction, Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241 p. (2010). ZBL1193.55001.
This paper was released on the arXiv just this (12Sep2018) morning:
"A Brief History of Persistence." Jose A. Perea. 2018. arXiv abstract.
"Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas, starting from the subject's computational inception more than 20 years ago, to the more modern categorical and representation-theoretic point of view."
