Topics in Differential Topology: Symplectic Methods in Low-dimensional Topology

Heegaard Floer homology is an invariant for low-dimensional manifolds constructed using methods in symplectic geometry (Lagrangian Floer homology). A related invariant for knots can also be constructed, whose Euler characteristic, in a suitable sense, is the Alexander polynomial of that knot. This course gives the construction of Heegaard Floer homology and the knot invariant, and with a view towards topological applications, and a special emphasis on modern computational tools.