In Oksendal's Stochastic Differential Equations he writes in the first few pages:
The Kalman-Bucy filter is an example of a recent mathematical discovery that has proved to be useful - it is not just "potentially" useful.
It is also a counterexample to to the assertion that "applied mathematics is bad mathematics" and to the assertion that "the only really useful mathematics is the elementary mathematics." For the Kalman-Busy filter - as the whole subject of stochastic differential equations - involves advanced, interesting, and first-class mathematics.
My question is this:
What are other examples of advanced, interesting, and first-class mathematics that is genuinely and presently useful (not just "potentially")?
I would ask that an answer to this question provide evidence that the mathematics is genuinely useful. The standard will be "useful to the point that a non-mathematician who cares about the application described would either: a. devote time to learning at least the basics of the requisite mathematics or b. put a specialist mathematician in the field on payroll" That the mathematics is advanced, interesting, and first-class will likely be readily apparent.
If there is something unclear about the question, or if it can be improved, please let me know!
One example is that the Fourier transform is essential for magnetic resonance imaging. The MRI machine is somehow measuring Fourier coefficients. We can apply the inverse Fourier transform (or inverse discrete Fourier transform) to obtain an MR image.
And then compressed sensing is another piece of first-class math that is useful for reducing the number of measurements that an MRI machine must make in order to obtain a clear image.