The shape and global behavior of physical systems arising in classical mechanics, a purview of the field of symplectic geometry, is known to be largely determined by the appearance and quantity of certain area-minimizing surfaces known as pseudoholomorphic curves. This research project aims to further develop systematic rules for understanding and counting such pseudoholomorphic curves in a large class of physical systems, by establishing formulae reducing the study of area-minimizing surfaces of different shapes (spheres, surfaces with many holes, etc.) to the study of a simpler type of area-minimizing surfaces (disks), and subsequently by understanding the degree to which such latter counts (of disks) can be assembled from a decomposition of the physical system into elementary pieces. Applications will be studied to mirror symmetry, a far-reaching geometric duality first discovered in string theory involving (on one side) counts of such curves. The educational component of the project aims to create a series of online virtual research activities in symplectic geometry and related areas, including organization of a virtual seminar (ongoing), workshops, and mini-courses. The PI will also train and encourage mathematics students through traditional workshops, new course and seminar content, undergraduate and graduate advising, and outreach to K-12 students through science fairs.
The long term research project funded by this award is to establish and further develop systematic frameworks for computing invariants in symplectic geometry and mirror symmetry coming from pseudoholomorphic curve theory. In one direction, it aims to develop and apply new structural results such as local-to-global principles to simplify the study of Fukaya categories of closed symplectic manifolds, with applications to homological mirror symmetry. In another, the project aims to further elucidate the relationship between Gromov-Witten invariants and the Fukaya category, with applications to enumerative mirror symmetry. The third and final direction is to study and further explain the appearance and coincidences of certain integral lattices in mirror symmetry. Many of the latter invariants can be packed into the notion of a non-commutative Hodge structure, giving a useful framework for the proposed work.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.