# Researching for differential invariants

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?

Thank you.

• I think the term is "smooth invariants." It sounds to me like you'll be interested in Floer homology. You might also like Scorpan's "The Wild World of 4-Manifolds." But you say you've just graduated. Why do you "have to start thinking about topics for [your] PhD thesis"? It seems to me like you only need to know your general area of interest. Have you been accepted to grad school? – Jesse Madnick Jun 28 '13 at 23:43
• My (very limited) understanding is that yes, this is an interesting and active area of research, but it is also very difficult. By "difficult" I mean creating new invariants and distinguishing smooth manifolds is probably a best-case scenario. More likely, I think, one would spend lots of time exploring technical properties of currently known, recently defined invariants. – Jesse Madnick Jun 28 '13 at 23:50
• I live in Spain, so things here are a bit different. Once you're graduated, you get into a one year master degree and then you choose a topic for your thesis and start writing it. This should take about three or four years. – rf1x Jun 28 '13 at 23:51

In dimension $1-3$ it is well known the $Diff$ category coincide with $Top$ for manifolds. So the classification we have for compact surfaces and the geometrizion is "suffice". In dimension $4$, it seems there are uncountablely many inequivalent differential structure available. This is a result by Taubes using Selberg-Witten theory. There is an high level "expository" article available written by Witten, but may not be readable to a beginning graduate student. There are also various treastises on this subject like the one by John Morgan.