Counting embedded curves in symplectic six-manifolds

The number of embedded pseudo-holomorphic curves in a symplectic
manifold typically depends on the choice of an almost complex
structure on the manifold and so does not lead to a symplectic
invariant. However, I will discuss two instances in which such naive
counting does define a symplectic invariant, which turns out to be
related to the Gopakumar-Vafa conjecture inspired by string theory.
The talk is based on joint work with Thomas Walpuski.