What are good references to self study persistent homology?

For a quick introduction, you can read this AMS survey.

• What is … Persistent Homology? by Shmuel Weinberger

A basic notion in persistent homology is a barcode. The following article gives an introduction to the subject with an emphasis on shape recognition, and tells you what a barcode its.

• Barcodes: The Persistent Homology of Data by Robert Ghrist

Here is another introductory survey article giving you more background material about the theory and implementation of persistent homology. This one also talks about some heavier stuff from algebraic topology, like spectral sequences.

• Persistent Homology – a Survey by Herbert Edelsbrunner and John Harer

Here are a few references, some old, some new:

• Topological Pattern Recognition for Point Cloud Data by Gunnar Carlsson – A foundational paper on the subject of topological data analysis, with a good exposition.

• A Brief History of Persistence by Jose A. Perea – A recent (2018) eleven page introduction that covers persistence modules and quiver representations.

• Tamal K Dey's 2017 course on Computational Topology and Data Analysis

• This question on MathOverflow is quite insightful.