# Reference on Persistent Homology

I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.

I am quite new to the ideas of persistent homology and have not done a single computations in this field. But, I like to learn on the subject. More is that I like to lead the course that I will be teaching so that towards the end, I can give some taste of persistent homology to students. But, I am not sure if there is any well written set of lecture notes on the material, or should we dive into the literature and start with some papers!?! The course involves of $3/2\times 30$ hours of lectures.

Do you think this is possible or should I use some simplicial approaches instead? or you think it is more suitable to give this as a task to students to start as a project and discover the ideas for themsevles?!?!

I would be very grateful for any advise in terms of addressing to main references on the subject. I also would be grateful if you can give me some advise on the history of the subject; for instance when people decided to use homology to study biological problems and whether or not the main stream researchers in biology or data analysis really consider these kind of tools?!

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.

• this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!! – user51223 Sep 13 at 4:20
• This background is great - thank you for writing all that up. I'm a data science practitioner by trade but a topologist at heart, so in a sense I'm "rooting" for persistent homology to succeed. The problem is that I have yet to see a convincing statistical inference / classification / regression problem that is solved by persistent $H_i$ for $i > 0$ - are you aware of any such examples? – Paul Siegel Sep 13 at 20:18
• Hi Paul, what exactly are you looking for? This paper uses both H_1 and H_2 (but no H_0) to predict protein compressibility directly from XRay crystallography data: link.springer.com/article/10.1007/s13160-014-0153-5 – Vidit Nanda Sep 13 at 23:02

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."

Maybe the following papers will be useful:

https://www.cambridge.org/core/journals/acta-numerica/article/topological-pattern-recognition-for-point-cloud-data/BB0DA0F0EBD79809C563AF80B555A23C (Topological pattern recognition for point cloud data, by Gunnar Carlsson).

https://escholarship.org/uc/item/2h33d90r (Persistent Homology: Theory and Practice, by Herbert Edelsbrunner and Dimitry Morozov).

The book by Steve Oudot is an alternative: Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs).

There is also a relatively new tutorial by Paweł Dłotko: Computational and applied topology, tutorial.

This introduction by Fugacci and others may also help you: Persistent homology: a step-by-step introduction for newcomers.