Gunnar wrote a review article a few years ago called Topology and Data (http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/ http://www.ams.org/jo

Gunnar wrote a review article a few years ago called Topology and Data (http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/ http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01...). It is an amazingly well written and accessible paper for a technical audience. Pair it with Afra's book (http://www.amazon.com/Computing-Cambridge-Monographs-Computational-Mathematics/dp/0521136091/ref=sr_1_1?ie=UTF8&qid=1444973590&sr=8-1&keywords=topology+for+computing http://www.amazon.com/Computing-Cambridge-Monographs-Computa...)