I understand the concept of the Fourier transform or (at least) what it does. You split a complex signal into simple parts.
But what does the microlocal analysis do, a generalisation of Fourier transforms? Does it let you decompose signals in a curved space?
https://en.wikipedia.org/wiki/Microlocal_analysis
The term microlocal implies localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point. This gains in importance on manifolds of dimension greater than one.
Can you provide an example of a "practical" use? At least in some fictional world?
I am little experienced in math, hence "ELI5" (if possible)