I am currently in my 4th year of a 5-year BS-MS course. I am majoring in Mathematics and I am quite interested in differential topology and would really appreciate if someone could tell me about possible areas of research in the field (With resources like books, survey papers etc. if possible).
As far as my background is concerned, I have done courses on Point-set topology, Calculus on Manifolds and Algebraic Topology and have dabbled a bit with 3-manifolds (Schultens' book titled "Introduction to 3-Manifolds") as well. I have studied "Differential Topology" by Guillemin and Pollock and was thinking of going towards Morse Theory after this.
Here are some:
Symplectic topology
Contact topology
Low-dimensional topology (dimensions 3 and 4)
Gauge theory (primarily in low dimensions) and gauge-theory inspired invariants.
But one needs an advisor familiar with these fields in order to succeed in any of these areas. (Yes, there are exceptions to this rule, but very few.)