A little while ago, I was reading Cathy O'Neil's post Why is math research important (subtext: why does Pure Math deserve funding), where she discusses 3 possible answers. One of these is the usual "because there may be applications some day." Of course, this is not too compelling, or an explanation of why people do Pure Math (nor does she claim it is).
I wondered if one can make a compelling argument that Pure Math is important because it binds all of Science together---it is, in some sense, the most interdisciplinary area of science because it is an exploration of ideas from all parts of Science and how they interact with each other. That is, Mathematics is not just the common language of Science, but a sort of playground where different Science ideas meet together, and sometimes (frequently?) copulate. If this is true, we should have a lot good examples where a question in Science A was turned into a Pure Math problem, and was solved based on idea from Science B. I'm not an expert on history, or applications, so I have no idea if there are many such examples, but this is my question.
What are examples of important problems in Science, which led to Pure Math research questions that were later solved by using an idea from a different area of science, which in turn resulted in a solution to or significant progress towards the original science problem?
Let me emphasize that I'm not looking for examples of solutions to pure math problems, which serendipitously found some important application later. Rather these should be pure math problems that originally studied because they were very clearly motivated from a "practical" question from field X. Preferably in situations where it is reasonable to posit that the other part(s) of science which led to the key insight(s) in solving this problem had little-to-no direct contact with field X, and Pure Math really was the essential conduit.