It seems that Fourier analysis/harmonic analysis plays an important role in math, at least in number theory and statistics. It also seems to me that talking about it is talking about locally compact groups. So to have a big picture, can you explain to me how the local and non-local compact, compact and non-compact groups play a role in many branches of math?
I'm looking for an answer similarly to the one in What does $S^1$ do in many branches of math? My background is Dym & McKean, Fourier Series and Integrals. I still don't understand much what that group mean though.