My area of research is symplectic and contact topology and I am currently focused on classifying exact Lagrangian fillings of Legendrian links. The main methods I like to use are pseudoholomorphic curves and Symplectic Field Theory. Here are two great introductory articles on each of those topics: What is... a Pseudoholmorphic Curve? and Sympletic Field Theory: an Overview.
Publications and Preprints:
The Legendrian Hopf Link has exactly two Lagrangian fillings, submitted. [arxiv]