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?
A fascinating and beautiful strain of math that only recently came to a (near) conclusion is Thurston's Geometrization Program in 3-manifold topology. The idea is that every topological 3-manifold can be cut up along $2$-spheres and $2$-tori so that the interiors of the pieces support one of eight geometric structures—that is, as a quotient of a homogeneous space by a lattice in the corresponding Lie group—and that one can recognize the geometric structure more or less from the manifold's fundamental group. (There are a few cases where two different geometric structures are possible).
The most recently solved parts of the program are the Virtually Haken and Virtually Fibered conjectures, now theorems due to Agol and Wise (building on work of several others including Kahn and Marcović). The Virtually Haken theorem states that every topological 3-manifold which is compact, irreducible, orientable and has infinite fundamental group has a finite cover which is Haken (that is, contains an embedded, incompressible, two-sided surface). The Virtually Fibered theorem states that every closed, orientable, atoroidal, irreducible topological 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.
Personally what I find beautiful about these theorems is that they use geometry (usually hyperbolic geometry) to prove entirely topological statements. Additionally, the proofs make decidedly large detours away from manifold topology and into the realm of Geometric Group Theory, particularly the geometry of objects called CAT(0) cube complexes.