Counting embedded curves in symplectic 6-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.