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.
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.