Abstract:Symplectic 4-manifolds come to us in two types: those which posess symplectic spheres with semi-positive self intersection, and those which do not. The first sort we, roughly speaking, understand -- we can prove strong theorems about their diffeomorphism type, how many symplectic structures they carry, the homotopy type of their symmetry groups etc... Of the second sort, those which lack such spheres, we understand very little and can answer none of these questions.
The reason for this dichotomy is that a foliation by j-holomorphic spheres is generic, and one of higher genus is not- If one perturbs the almost complex structure these higher genus curves dissappear. In this talk we shall investigate this dissappearance; we shall prove results which suggest that while the curves themselves dissappear under perturbation, they unwind to form a j-holomorphic foliation with an invariant area form.