- Ghrist’s Elementary Applied Topology
- Section 4.1 discusses homology
- Section 5.3 discusses persistence/persistent homology
- Edelsbrunner and Harer’s Computational topology
- Section III.1 and III.4 discuss simplicial complexes, and then alpha complexes
- Section IV.1 discusses homology
- Section VII.1 discusses persistence
- Section VIII.2 discusses stability
- Section 2 discusses cubical complexes and homology computations
- Ghrist’s Barcodes: the persistent topology of data
- A very readable introduction to what TDA is all about, with (relatively) minimal math pre-requisites. Builds up to an example of TDA in natural image processing.
- Giusti, Ghrist, and Bassett’s Two’s company, three (or more) is a simplex
- Another very readable introduction to TDA, with a focus on neuroscience applications. Less formal math shows up in the exposition, but they include formal definitions/statements in side boxes.
- Ghrist’s Homological algebra and Data
- Notes from Ghrist’s lectures on the topic at the PCMI “The Mathematics of Data” summer program. Written with more mathematical maturity in mind.
- Carlsson’s Topology and Data
- An introduction to TDA with non-specialist mathematicians in mind; maybe a “step-up” from Ghrist’s Barcodes survey.
- Lum et.al.’s Extracting insights from the shape of complex data using topology
- An interesting overview of the various ways topological methods can be used to visualize (and get insights from) various kinds of complex data.
An algebraic topology/neuroscience bibliography by Chad Giusti
Optimal cycles by Wu et. al.