LEAPS: MPS Four dimensions from two in symplectic topology and dynamics

Our universe (space-time) has an underlying mathematical structure called a four-dimensional manifold, and yet surprisingly, we understand very little about the geometry of four-dimensional spaces. Topology studies the fundamental properties of space that remain unchanged even when the structure is continuously bent and deformed, while dynamics studies the self-mappings of the space that preserve a given structure. Initially formulated from the equations of Hamiltonian mechanics, symplectic structures have grown into an important abstract mathematical topic that is particularly powerful for studying four-dimensional spaces. This project focuses on understanding the topology and dynamics of four-dimensional manifolds with a symplectic structure, using cutting-edge tools to make novel advances. In addition, the PI will organize a variety of outreach activities and engage in mentoring programs for undergraduate and graduate students, especially among underrepresented minority students. The investigator also plans to hold career preparation workshops, increasing the participation of underrepresented groups in STEM and enhancing the research environment at his institution. The investigator will study how the properties of four-manifolds and their symmetries are encoded by two-dimensional information. The primary goal is to explore whether results about braids and Dehn twists along Lagrangian spheres can be extended to more general four-manifolds while comparing the Diff and Symp groups in dimension four. Further goals include comparing the symplectic and Kähler cones, solving the isotopy problem of certain Lagrangian submanifolds and symplectic surfaces, and exploring the dynamical properties of symplectic maps. The project will employ various mathematical tools to study symplectic 4-manifolds and surfaces: surgery on Lefschetz fibrations, Gromov-Witten theory, almost complex inflation, and family Seiberg-Witten invariants. 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.