>>8331383This is virtually impossible to answer. The majority of pure math published is not only useless in the real world, it's also useless to most other areas of math because it's not motivated by anything. It's a rare result that finds wide applicability. Usually mathematicians just pick some problem that looks interesting and then sort of file away the obstacles (i.e. things they can't prove) they reach along the way. Eventually, when a bunch of people run into the same obstacle over and over again, you sort of know that it would be important to solve it, and it becomes a big conjecture like the Riemann Hypothesis.
Less rarely do people start by trying to create a framework and definitions that will be adapted to the general principles behind a situation. Usually if you do so in the right way, it becomes very easy to prove things, and your results are more widely applicable. This is the "avocado approach" of Grothendieck.
