The PDF file TDA Background Reading gives a brief two page introduction to the topic, with links to several downloadable papers and an extensive list of others. Two good starting points are
- Topology and Data by Gunnar Carlsson, Bull. Am. Math. Soc. 46(2) (2009) 255-308
- Topological Estimation Using Witness Complexes by Vin DeSilva and Gunnar Carlsson, in Eurographics Symposium on Point-Based Graphics (2004), M. Alexa, S. Rusinkiewicz, (Editors)
Some more specific references to topics in lectures (full citations to be added):
- Papers by Bubenik on Persistence Landscapes, for example The Persistence Landscape and Some of its Properties at https://arxiv.org/abs/1810.04963, also available here
- The Ring of Algebraic Functions on Persistence Bar Codes by Aaron Adcock, Eric Carlsson and Gunnar Carlsson in Topology, Homotopy and Applications 18(1) (2013), DOI: 10.4310/HHA.2016.v18.n1.a21