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][1] if the students had sufficient background, and the foundational [paper][2] 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). 

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**1992**: Frosini introduces "[size functions][3]", which we would today consider equivalent to 0-dimensional persistent homology.

**1995**: Mischaikow + Mrozek publish [a computer-assisted proof][4] 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][5] 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][6] on efficient homology computation via simple homotopy type reductions of cell complexes. 

**2002**: Edelsbrunner, Letscher and Zomorodian [introduce persistence][7] from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.

**2005**: This was a big year.

1. Zomorodian and Carlsson [reinterpret persistence][2] 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][8] for sensor networks.

**2008**: Niyogi, Smale and Weinberger publish a paper [solving the homology inference problem][9] for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts [Ayasdi][10], 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][11], 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][12]. 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][13] 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][14]. 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][15]. 

**2013**: Mischaikow and I [retool the simple homotopy-based reductions][16] 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][17] of the interleaving distance in the context of multiparameter persistence modules.

**2018** MacPherson and Patel concoct [bisheaves][18] to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.

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Good luck with your course!


  [1]: https://bookstore.ams.org/surv-209/
  [2]: https://link.springer.com/article/10.1007/s00454-004-1146-y
  [3]: http://spie.org/Publications/Proceedings/Paper/10.1117/12.57059
  [4]: https://arxiv.org/abs/math/9501230
  [5]: http://www.topology.auburn.edu/tp/reprints/v24/tp24222.pdf
  [6]: https://www.springer.com/gp/book/9780387408538
  [7]: https://link.springer.com/article/10.1007/s00454-002-2885-2
  [8]: https://projecteuclid.org/euclid.agt/1513796672
  [9]: https://link.springer.com/article/10.1007/s00454-008-9053-2
  [10]: https://www.ayasdi.com
  [11]: https://link.springer.com/article/10.1007/s00454-009-9176-0
  [12]: https://link.springer.com/article/10.1007/s10208-010-9066-0
  [13]: http://www.pnas.org/content/108/17/7265
  [14]: https://arxiv.org/abs/1207.3674
  [15]: https://link.springer.com/article/10.1007/s00454-014-9573-x
  [16]: https://link.springer.com/article/10.1007%2Fs00454-013-9529-6
  [17]: https://link.springer.com/article/10.1007/s10208-015-9255-y
  [18]: https://arxiv.org/abs/1805.02539