Beginner's Guide to Persistent Homology (focusing on Theoretical Mathematical Aspects)

Question

I am interested in seeking out a reference for learning Persistence Homology (more of the theoretical mathematical aspects rather than the applications/ computer science aspects).

Currently, the two books I have attempted to read:

• Elementary Applied Topology by R. Ghrist (Not very elementary at all)
• Computational Topology - An Introduction by Herbert Edelsbrunner (also not an easy read)

One issue I faced reading the above two books is that the format of the books is like a story or "novel", rather than the theorem/proof style which is more organized. Also, the level of the books are quite difficult (despite the words "elementary" and "introduction").

Does anyone know a more gentle introduction to the subject of persistent homology? Ideally a self-contained book would be good.

Answer 1

I'm having trouble resolving the tension between you wanting a reference for beginners but also a reference focusing on theoretical math. The beginner bit makes me want to recommend something like

• Persistent Homology: A Non-Mathy Introduction with Examples

or any of the various other blog posts or YouTube videos giving an intuitive description of persistent homology. But since you want theoretical math:

• Persistence Theory: From Quiver Representations to Data Analysis by Steve Oudot – An survey book that treats the algebraic aspect and topological aspect of persistent homology separately. I like what I've read from it so far.

• Topological Pattern Recognition for Point Cloud Data by Gunnar Carlsson – This is kinda like a fundamental paper in the field, but Carlsson does provide a nice exposition.

And I should really also link to this analogous question over on MathOverflow.