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Can anyone describe to me in Layman's terms what kind of use does Topology have through Persistent Homology in Data Analysis...that's can you give me some real life examples in which which this has been used with success.

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Identifying a subgroup of breast cancer is the most significant one that I know of: http://www.pnas.org/content/108/17/7265.abstract

But it has also been used to classify players in basketball (finding that there are 13 types of players rather than the usual five positions they are grouped into, http://offthedribble.blogs.nytimes.com/2012/03/06/trading-small-forward-for-scoring-rebounder/?_r=0), to study swarming behavior in insects (https://arxiv.org/abs/1412.6430), and I'm sure it has found countless other applications as well (https://www.ayasdi.com/).

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To add to the list, it was used to study images and to help develop new compression algorithms: http://redwood.berkeley.edu/vs265/carlsson-ijcv08.pdf .

This is a nice survey I found: https://www.math.upenn.edu/~ghrist/preprints/nieuwarchief.pdf

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