Symplectic Geometry and its Interactions with Low-Dimensional Topology

Symplectic geometry shares deep connections with many other fields of mathematics such as complex geometry, smooth topology, algebraic geometry, and dynamics. However, we still do not have a good understanding of what a symplectic manifold should look like. Early work on understanding the topology of symplectic manifolds dates back to the groundbreaking work of Gromov in the 1980s, with the invention of pseudoholomorphic curves. This project aims to construct new tools based on pseudoholomorphic curves while further exploiting the fruitful connection to low-dimensional topology to improve understanding of symplectic manifolds. In addition, the PI endeavors to make mathematics accessible to a broader audience through mentoring graduate and undergraduate students, as well as organizing conferences and workshops. The project is organized into two primary themes. The first theme is to use the moduli spaces of pseudoholomorphic curves and the algebraic and geometric invariants extracted from them to classify symplectic manifolds and study their Lagrangian submanifolds. The second theme is to build more tools in Floer theory, including embedded contact homology, which has many interactions with other areas of low-dimensional topology. Many aspects of the project can form research subunits for graduate and undergraduate students, aligning with the PI's commitment to provide mentorship and serve as a resource to spread knowledge within the broader mathematical community. 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.