I will introduce relative Seiberg-Witten (or SW) moduli spaces for an arbitrary pair (X,S) of a closed oriented 4-manifold X and a closed oriented 2-dimensional submanifold S in X with positive genus. Using these moduli spaces, we obtain an SW sum formula (aka a product formula) that relates the SW invariants of a sum X of two closed oriented 4-manifolds X1 and X2 along a common oriented surface S with dual self-intersections to the relative SW invariants of (X1,S) and (X2,S). This formula generalizes Morgan-Szabo-Taubes' product formula. This is joint work with Pedram Safari.

The Gopakumar-Vafa conjecture concerns the Gromov-Witten invariants of symplectic manifolds of dimension six. The first part of the conjecture, the integrality conjecture, which was proved recently by Ionel and Parker, asserts that the Gromov-Witten invariants can be expressed in terms of simpler, integer invariants called the BPS numbers. The second part of the conjecture, the finiteness conjecture, predicts that only finitely many of the BPS numbers are nonzero for every homology class. In this talk, based on joint work with E. Ionel and T. Walpuski, I will discuss a proof of the second part of conjecture. The proof combines ideas from the theory of pseudo-holomorphic curves, including Ionel and Parker's proof of the integrality conjecture, and methods of geometric measure theory, especially Allard's regularity theorem for currents with bounded mean curvature.