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:
On the practical side, what packages are you guys using? I'm familiar with JavaPlex for Matlab:
For my persistent homology calculations I always use Dionysus ( http://www.mrzv.org/software/dionysus/ ). Rumor has it a much improved parallelized version will be released soon.
Happy to answer questions here regarding TDA but am no longer with Ayasdi so won't comment on company specific stuff.
The material in the video is dated but still relevant - I now have a better understanding of how to use "Shape" to improve Machine Learning models in an operational setting.