I have just graduated and I have to start thinking about topics for my PhD thesis and areas I am going to specialize in. The thing is that one thing that looks fun to me is classifying smooth manifolds up to diffeomorphism, searching for "differential invariants", which is the way I call invariants that allow to distinguish homeomorphic but not diffeomorphic manifolds. If this has already a standard name, please tell me what is it.
Is this area interesting for doing research?
I've read Milnor's paper on the existance of exotic 7-spheres (which I don't fully understand yet), and it seems like he hasn't any good tool for solving his problem, so he has to biuld his own invariant. Also, all of the homology theories I know so far are invariant under homotopy equivalence, so despite they are sophisticated, they are useless for this purpose, right? These are the reasons why I think this might be interesting.
If so, what should I learn from now on in order to get to the "differential invariants" already known?