**Aleksander Doan,
Columbia University**

**Title:** 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.