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    $\begingroup$ I wa going to mention Topological Data Analysis which is very pure in its background and intuitions but is very useful in new areas of data analysis. Brendan beat me to it in mentioning Gunnar Carlsson and his team. This is exciting modern maths and deserves being talked about to general audiences. There is also a paper on the discovery of new classes of breast cancer using TDA. Check out Topological Data Analysis with a web search. $\endgroup$ Commented Jul 19, 2013 at 13:40