In the last two decades of development in symplectic topology since the advent of Gromov's pseudoholomorphic curves, the method of pseudoholomorphic curves manifested many genuinely symplectic phenomena combining tools from geometric analysis and Hamiltonian dynamics. Floer theory and its cousin, symplectic field theory, have emerged as a unifying force of analysis, dynamics and geometry via the language of homological algebra. With Floer theory and the relevant analysis of holomorphic curves as the common tools of investigation, this project centers around the investigation of deeper aspects of symplectic topology and Hamiltonian dynamics up to the level of the topological category. The project also aims at the study of the moduli space of pseudoholomorphic curves, both open and closed, beyond the well-known stable map type compactifications and its applications to symplectic topology and to mirror symmetry through adiabatic degeneration and the blow-up analysis. The proposed research will put the method of pseudoholomorphic curves in symplectic topology in better perspective and further promote the existing interactions between geometric analysis, symplectic topology and dynamical systems. The Hamiltonian formalism played a fundamental role not only for solving problems in classical mechanics but also for transforming the classical mechanics into quantum mechanics. When one considers mechanics in a constrained system, i.e., mechanics on a curved space like `spherical pendulum' or `billiards', description of the corresponding phase space and the differential equation requires the Hamiltonian formalism in the context of the symplectic structure and symplectic manifolds.
In symplectic geometry, there are two aspects of study, one the dynamical aspect and the other the geometric aspect. Understanding the interplay between the two aspects is the core of symplectic topology. Oh's proposed research aims at deeper understanding of symplectic topology, and of its relation to the string theory in physics. It also aims at easing an access of new coming graduate students and mathematicians from related fields to the results derived from Oh's research by proposing to write a graduate level textbook on Floer theory and its applications.
