[adrianm]

I don't agree with your suggestion about olympiad math since it often has little relationship to applying advanced mathematics, but there definitely is merit to the idea that you might need to put theory into practice in order to gain insight about these things.

I recommend (surprise surprise) programming. Implement fast fourier transform in C and then Common Lisp. Write a finite difference PDE solver. Try solving actual problems to motivate you. Signals analysis can be a fun way to exercise your knowledge. Try analyzing your favorite songs and figuring out what makes them sound the way they do. Maybe implement some audio filters. If that's not your cup of tea, write physics or chemistry simulations instead. Then use OpenGL to visualize them. Then make them interactive.

I can go on and on, but I'll just leave two book recommendations for those who might enjoy programming advanced mathematics.

Structure and Interpretation of Classical Mechanics : https://mitpress.mit.edu/classical_mech

Functional Differential Geometry : https://mitpress.mit.edu/books/functional-differential-geome...

[FabHK]

Several Project Euler problems require some mathematical nuggets and provide nice motivation to dive deeper (and provide an application right there, too).