Sure. Gunnar's "Topology and Data" paper (already linked to by another poster) is the first place to go for an overview of TDA (although it's getting dated). A more updated version (but behind paywall) is "Topological pattern recognition for point cloud data" ( http://dx.doi.org/10.1017/S0962492914000051 )

Robert Ghrist has another take on TDA ( https://www.math.upenn.edu/~ghrist/preprints.html ) that is more "engineering" focused and also uses a different mathematical tool set than the above (eg. Sheaves/Co-Sheaves). I particularly like the sensor coverage results.

Rob Ghrist and John Harer ( http://fds.duke.edu/db/aas/math/faculty/john.harer/publicati... )both have textbooks available if you want to get into the fundamentals in the field.

Jose Perea has some nice results using ideas from TDA in a variety of contexts. eg texture classification ( https://www.math.msu.edu/user_content/docs/KleinBottleTextur... ) and signal processing ( https://www.math.msu.edu/user_content/docs/Sw1Pes_Theory2015... )

Here's a talk I gave at ICERM last summer using persistent homology as a feature generating method for drug discovery:

https://icerm.brown.edu/video_archive/#/play/726

(slides available for download if you poke around on the icerm web site).

This is my "part two" of the video linked to in the parent article:

https://www.youtube.com/watch?v=3Z73Wd2T1xE