Lagrangian Floer Theory and Quantum Invariants of Symplectic Manifolds

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Symplectic geometry is the mathematical study of the foundations of classical mechanics. Despite its classical origins, this research area has recently experienced an explosion of progress centered on the study of quantum invariants defined using geometric objects known as holomorphic curves. These quantum invariants have appeared not only in geometric analysis but also in low-dimensional topology, which studies three-dimensional space as well as four-dimensional space-time, and in certain models in high-energy physics. The investigator will study fundamental questions about these quantum invariants, including their behavior under mathematical operations known as surgery and symmetry reduction. Applications will be of interest in topology and physics. The investigator will also continue his outreach activities for middle-school geometry teachers.Specifically, the project studies the behavior of quantum invariants such as the Fukaya category and quantum K-theory under operations such as flips, blow-ups, and symplectic reduction. In the first part of the project, the investigator will construct generators of the Fukaya category associated to surgeries on the symplectic manifold that arise naturally as the symplectic structure is varied. Each of these gives rise to a collection of objects in the Fukaya category, and conjecturally a factor in the quantum cohomology. The main technique here involves the Abouzaid-Ganatra generation criterion and a restricted version of symplectic field theory for the Fukaya category developed using stabilizing divisors. In the second part of the project, the investigator and collaborators will construct a homotopy version of Kirwan's map from the quasimap Fukaya category of a Hamiltonian group action to the Fukaya category of the symplectic quotient. This study will have applications to the disk potentials of Lagrangians in symplectic quotients. In a third collaborative project, the investigator will study the behavior of K-theoretic Gromov-Witten invariants under wall-crossing and aims to prove that the potentials are unchanged, generically, in the case of crepant birational transformations associated to variation of symplectic quotient.