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
- Extracting insights from the shape of complex data using topology A good introductory paper in Nature on the Mapperalgorithm.
- Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition A more technical presentation of Mapper.
- Topological Data Analysis and Machine Learning Theory Applications of TDA to machine learning.
- Ayasdi, the company founded by Gurjeet Singh and Gunnar Carlsson, have several good videos and whitepapers on how they use Mapperand TDA in machine learning pipelines.
- Computational Topology: An Introduction A good introducgtory book on persistent homology
- Elementary Applied Topology A book by Robert Ghrist that goes beyond applications of algebraic toplogy to data analysis, but a very good read. There is a very inexpensive print version and the PDF is available for free.
Giotto is a collection of open-source Python libraries for TDA: giotto.ai, https://github.com/giotto-ai