Symplectic instanton homology
Floer's instanton homology was originally defined as an invariant of integral homology 3-spheres. The Atiyah-Floer Conjecture claims that there should be a symplectic counterpart to instanton theory, based on Lagrangian Floer homology. Starting from a Heegaard decomposition of a 3-manifold, I will explain one way to make sense of the symplectic side of the Atiyah-Floer conjecture, for arbitrary 3-manifolds. This is joint work with Chris Woodward.