What are these useful for?

homology
homotopy
diffeomorphism
differential forms
unitary spaces, quadratic form, bilinear form
prime number theorem
contour integration, holomorphic functions

Please explain why I should know these things.

They are useful for TESTING YOUR FUCKING ABILITY TO USE GOOGLE, AND FUCKING GAWWWWD D-A-M-N DID YOU JUST FAIL IT!