Critical symplectic geometry, Lagrangian cobordisms, and stable homotopy theory

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Symplectic geometry originated as a geometric approach to classical Newtonian mechanics, unifying previously disconnected ideas and providing a qualitative understanding of dynamical systems in cases where explicit solutions are not possible. More recent discoveries have revealed that symplectic geometry plays a key role in many other fields of mathematics like algebraic geometry, low-dimensional topology, and representation theory. This project will incorporate ideas from homotopy theory and category theory into symplectic geometry in order to prove structural results about symplectic manifolds and maps between them. The project also has a significant educational component. The PI will organize graduate school panels and math outreach events at UMass Boston, supervise undergrad research, mentor graduate students and postdocs are other universities, and serve as a judge for nationwide math competitions.The PI will investigate critical symplectic geometry: the study of certain symplectic manifolds called Weinstein domains up to stabilization and subcritical handles. Critical symplectic geometry was introduced in the PI's previous work in order to define a symplectic analog of topology localization in rational homotopy theory and generalize symplectic flexibilization. Furthermore, critical symplectic geometry is the natural setting to study J-holomorphic curve invariants like the Fukaya category, which is invariant under these two operations. In this project, the PI will relate critical symplectic geometry to Lagrangian cobordisms, show that Lagrangian cobordisms can detect symplectic flexibility, develop a geometric approach to Floer homotopy theory, and investigate non-Weinstein examples arising from Anosov dynamical systems. The project will use modern techniques from homotopy theory, higher algebra, and dynamical systems and import ideas from symplectic geometry into these areas of mathematics.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.