NASA has fueled the development of lots of spin-off technologies: smoke detectors, memory foam, infrared ear thermometers, etc. NASA didn’t pursue these things directly, but they were a useful side effect.

Something analogous happens in mathematics. While pursuing one goal, mathematicians spin off tools that are useful for other purposes. Algebraic topology might be the NASA of pure mathematics. It’s worth taking a course in algebraic topology even if you don’t care about topology, just to see a lot of widely used ideas in their original habitat.

Number theory has developed an enormous amount of technical machinery. If a graduate student in number theory described to you what she’s working on, you probably wouldn’t recognize it as number theory. But as far as I know, not much of the machinery of number theory has been applied to much besides number theory. Maybe there’s a arbitrage opportunity, not to apply number theory per se, but to apply theĀ *machinery* of number theory.

Very interesting proposal. I strongly agree that breakthroughs can occur when we cross pollinate different domains. Could you give a few examples of number theory applied outside its natural habitat.

An example of ideas spreading from topology to other areas is homotopy. This topological concept has lead to homotopy type theory (HoTT) which touches on logic, formal theorem proving, type theory, computer science, foundations of math, etc.

Another example is category theory. Algebraic topology associates algebraic structures with topological structures, and so provided the natural environment for developing functors. (“Natural” here is sort of a pun. The first goal of category theory was to rigorously define what it means for a construction, such as many of the constructions that appear in algebraic topology, to be “natural.”)

@David: I believe big-O notation (and related notations for asymptotics) originated in number theory. That’s the only example that comes to my mind right now, although I expect there are others.