I suspect there’s a huge opportunity in moving mathematics from the pure column to the applied column. There may be a lot of useful math that never sees application because the experts are unconcerned with or unaware of applications.
In particular I wonder what applications there may be of number theory, especially analytic number theory. I’m not thinking of the results of number theory but rather the elegant machinery developed to attack problems in number theory. I expect more of this machinery could be useful to problems outside of number theory.
I also wonder about category theory. The theory certainly finds uses within pure mathematics, but I’m not sure how useful it is in direct application to problems outside of mathematics. Many of the reported applications don’t seem like applications at all, but window dressing applied after-the-fact. On the other hand, there are also instances where categorical thinking led the way to a solution, but did its work behind the scenes; once a solution was in hand, it could be presented more directly without reference to categories. So it’s hard to say whether applications of category theory are over-reported or under-reported.
The mathematical literature can be misleading. When researchers say their work has various applications, they may be blowing smoke. At the same time, there may be real applications that are never mentioned in journals, either because the work is proprietary or because it is not deemed original in the academic sense of the word.