Heegaard Floer homology is an invariant for knots and three-manifolds
defined using methods from Lagrangian Floer homology and symplectic
geometry. I will describe a concrete model for computing this
invariant for knots in S3. Time permitting, I will turn to some
further developments in the theory. The material I will describe is
joint with collaborators, including Robert Lipshitz, Ciprian
Manolescu, Dylan Thurston, and Zoltán Szabó.