The Gopakumar-Vafa finiteness conjecture
Abstract: Gromov-Witten invariants, which encode information about pseudo-holomorphic curves contained in a given symplectic manifold, provide powerful tools for solving problems in symplectic geometry. However, their geometric meaning is not always clear. These invariants are rational numbers and therefore cannot be simply interpreted as a count of pseudo-holomorphic curves. Moreover, a single curve can contribute to infinitely many Gromov-Witten invariants. The Gopakumar-Vafa conjecture predicts that for symplectic manifolds of dimension six, the information encoded in the Gromov-Witten invariants can be repackaged into a collection of numbers, called the BPS invariants, which don't suffer from these drawbacks: they are integers and only finitely many of them are nonzero in every homology class. The first part of the conjecture was proved in 2018 by Ionel and Parker. I will discuss a proof of the second part, which relies on combining Ionel and Parker's cluster formalism with results from geometric measure theory. The talk is based on joint work with Eleny Ionel and Thomas Walpuski.