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 an