(This is a restatement of a question asked on the Mathematics.SE, where the solutions were a bit disappointing. I'm hoping that professional mathematicians here might have a better solution.)
What are some problems in pure mathematics that require(d) solution techniques from the broadest and most disparate range of sub-disciplines of mathematics? The difficulty or importance or real-world application of the problem is not my concern, but instead the breadth of the range of sub-disciplines needed for its solution. The ideal answer would be a problem that required, for instance, number theory, group theory, set theory, formal logic, homotopy theory, graph theory, combinatorics, geometry, and so forth.
Of course, most sub-branches of mathematics overlap with other sub-branches, so just to be clear, in this case you should consider two sub-branches as separate if they have separate listings (numbers) in the Mathematics Subject Classification at the time of the result. (Later, and possibly in response to such a result, the Subject Classifications might be modified slightly.)
One of the reasons I'm interested in this problem is by analogy to technology. More and more problems in technology require a range of disciplines, e.g., electrical engineering, materials science, perceptual psychology, optics, thermal physics, and so forth. Is this also the case in research mathematics?
I'm not asking for an opinion—this question is fact-based, or at minimum a summary of the quantification of the expert views of research mathematicians, mathematics journal editors, mathematics textbook authors, and so forth. The issue can minimize the reliance on opinion by casting it as an objectively verifiable question (at least in principle):
What research mathematics paper, theorem or result has been classified at the time of the result with the largest number of Mathematics Subject Classification numbers?
Moreover, as pointed out in a comment, the divisions (and hence Subject Classification numbers) are set by experts analyzing the current state of mathematics, especially its foundations.
The ideal answer would point to a particular paper, a result, a theorem, where one can identify objectively the range of sub-branches that were brought to bear on the proof or result (as, for instance, might be documented in the Mathematics Subject Classification or appearance in textbooks from disparate fields). Perhaps one can point to particular mathematicians from disparate sub-fields who collaborated on the result.
Best Answer
The proof of the Ramanujan conjecture by Deligne. It uses:
number theory
algebraic geometry
topology
representation theory
commutative algebra
complex analysis