New aspects of curve counting and applications

The central goal of geometry is to understand the structure of mathematical spaces, which may or may not be directly related to physical space. Such spaces can be described by both local properties, like curvature, and global properties, such as connectivity. Symplectic geometry, a specialized branch of this field, focuses on symplectic manifolds: spaces that are locally identical but can exhibit a wide range of global structures. These objects have originated in the study of motion and classical mechanics and play an important role in mathematical physics and applications. The primary tools used in their study fall into two broad categories: algebraic and analytic. Algebraic methods provide frameworks for encoding global information and facilitating computations, while analytic techniques involve solving differential equations and constructing these algebraic frameworks. This research project seeks to refine existing methods and develop new analytic tools to tackle longstanding challenges in the field. It will also contribute to outreach activities aimed at K–12 students through afterschool programs and support the training and professional development of graduate students via summer schools and seminars. At the technical level, this project focuses on several ambitious goals within symplectic geometry and mathematical physics. These goals all require refining existing techniques and developing new methods. First, it builds upon the method developed jointly by the PI and Bai to establish novel applications of Floer theory, including the longstanding Arnold–Givental conjecture. Second, the same framework will be used to investigate the relationship between integer-valued Gromov–Witten invariants and the Gopakumar–Vafa invariants of Calabi–Yau threefolds, a problem of interest to both geometers and physicists. Third, the project will continue the study of the framework known as the gauged linear sigma model, with particular emphasis on its role in classical and homological mirror symmetry. The PI will also set up afterschool outreach activities aimed at advancing the mathematical learning of K–12 students, as well as continue organizing summer schools and seminars to foster the training of Ph.D. 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.