Hey HN folks - I am the co-founder and CEO of Ayasdi. If you have questions about the math/CS aspects of this, happy to answer.
I'd love to read a couple of journal articles that you recommend to learn about TDA. I do large scale data analysis on health care data at my university and am always on the look-out for interesting techniques.
Pair it with Afra's book (http://www.amazon.com/Computing-Cambridge-Monographs-Computa...)