Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers in a certain area of mathematics thought that their work could be of value to some field other than mathematics—maybe it was some kind of engineering, so I'll just call it "engineering"—but then it was found that interactions between engineers and mathematicians made substantial contributions to mathematical research but not to engineering. I don't remember what it was about, beyond that.
So my question is: What are the most edifying examples in recent centuries, of applications to fields other than mathematics greatly benefitting mathematical research when mathematicians had expected to be only the benefactors of those other fields?
Best Answer
If I may substitute "physics" for "engineering", one could argue that string theory is an example of a topic where mathematicians have interacted with a different field of research and the dominant benefit of that interaction was in mathematics (witness the award of a Fields medal to a physicist).