This is just a quick list of resourses on TDA that I put together for @rickasaurus after he was asking for links to papers, books, etc on Twitter and is by no means an exhaustive list.
Both Carlsson's and Ghrist's survey papers offer a very good introduction to the subject
- Topology and Data by Gunnar Carlsson
- Barcodes: The Persistent Topology of Data by Robert Ghrist
- Extracting insights from the shape of complex data using topology A good introductory paper in Nature on the
Mapperalgorithm.