Background: I'm at the end of my BsC and in the next semester I'll start my MsC. I'm already familiar with analysis on manifolds (at the level of Tu's "An Introduction to Manifolds" and Spivak's "A Comprehensive Introduction to Differential Geometry, Volume 1") and classic differential geometry (at the level of Do Carmo's "Differential geometry of curves and surfaces").
My main goal is to deeply understand how smooth manifolds work in terms of their topology and geometry. unfortunately I can't spend the whole 2 years of the MsC in this endeavor, so I need a more pinpoint focus in the future, so I'd like some help in deciding a sort of pre-project given the following (I'll try to be as specific as possible):
- Understanding the algebraic and differential topology of manifolds is a big deal for me. Right now I'm studying Milnor's "Topology from the differentiable viewpoint" and I'd really like to know what's out there beyond the topics he already covers (like, where does one go after Hopf's theorem?)
- The interplay of geometry and topology. Like the Bonnet-Myers theorem, (I think) Morse's theory, the sphere theorem and Hamilton's theorem, that kind of stuff. I'm also particularly interested in deeply understanding some of the topics Wellington de Melo covers in his "Topology of manifolds", like bundle geometry and the morphism of Chern-Weil.
I know most of you will say "ask your advisor, he's the best one to guide you in this" but to be honest he's more or less as lost as I am. If it were possible I'd just spend the next 2 years of the MsC just trying to understand the topics I mentioned, but this is not doable and eventually I'm gonna have to focus on something a lot more specific. So I'm looking for advice: is this too much to expect for 2 years? Should I focus on something in particular on the list above? What have people been up to with diffential topology of manifolds now?