Curve counting and spaces of Cauchy-Riemann operators

It is a long-standing open problem to generalize sheaf-counting invariants of complex projective three-folds to symplectic manifolds of real dimension six. One approach to this problem involves counting  J-holomorphic curves  C, for a generic almost complex structure J, with weights depending on J. Various existing symplectic invariants (Gromov-Witten, Gopakumar-Vafa, Bai-Swaminathan) can be expressed as such weighted counts. In this talk, based on joint work with Thomas Walpuski, I will discuss a new construction of weights associated with curves and a closely related problem about the structure of the space of Cauchy-Riemann operators on  C.