Speaker: Aleksander Doan (University of Cambridge and University College London)

Title: A differential-geometric approach to local enumerative invariants

Abstract: It is a long-standing open problem to find a differential-geometric interpretation of enumerative invariants of Calabi-Yau three-folds. One approach involves counting J-holomorphic curves C in such manifolds, for a generic almost complex structure J, with weights derived from moduli spaces of certain geometric data on C. These weights can be seen as the local contributions of C to the invariants. Notably, Gromov-Witten, Gopakumar-Vafa, and recently defined Bai-Swaminathan invariants can be expressed as such weighted counts.

In this talk, based on joint work with Thomas Walpuski, I will present a new construction of such weights associated with curves. We conjecture these to be related to known algebraic invariants (PT and higher rank DT). Importantly, these weights depend on the germ of J around C; understanding this dependence leads to an interesting analysis problem about the space of Cauchy-Riemann operators on C.