Techniques in Symplectic Geometry and Applications

The central goal in geometry and topology is to understand mathematical spaces. A mathematical space can be characterized by both local and global information, such as how curved it is (local) and how connected it is (global). As a subfield, symplectic geometry studies a special kind of space called a symplectic manifold. These spaces are all the same locally but can have a variety of global shapes. The main tools of study fall into two categories: algebraic and analytic. The algebraic tools are the frameworks of wrapping up global information and making calculations, while the analytic tools are techniques for solving differential equations and for constructing the algebraic frameworks. This research project aims to improve existing techniques and develop new tools to solve longstanding difficult questions, with emphasis on analytic methods. At the same time, the project will enrich the K-12 outreach (Math Circle) program at Texas A&M University, build a community of researchers in Texas and Central South America, and enhance connections among mathematics faculty and students. On the technical level, the project involves three topics. First, using the technique of virtual cycle and adiabatic limit, the research aims to establish the relationship between Witten's gauged linear sigma model and the nonlinear sigma model. In contrast to many algebraic approaches, the method is analytic, allowing one to extend to the open-string case where algebraic methods are not yet available. Second, using a new method, the PI plans to prove compactness results in situations where traditional approaches do not apply. This includes a compactness result in Atiyah-Floer conjecture, compactness for pseudoholomorphic curves regarding reversed surgery for cleanly-intersecting Lagrangian submanifolds, and compactness for pseudoholomorphic curves with certain singular Lagrangian boundary conditions. Third, the project aims to develop a new counting method to define Gromov-Witten invariants and Floer homology over integers. The PI will also train PhD students and advise undergraduate students. 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.