Symplectic Topology, Mirror Symmetry and Analysis of Pseudoholomorphic Curves

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In the last two decades of development in symplectic topologysince the advent of Gromov's pseudoholomorphic curves, the methodof pseudoholomorphic curves manifested many genuinely symplecticphenomena combining tools from geometric analysis and Hamiltoniandynamics. Floer theory and its cousin, symplectic field theory,have emerged as a unifying force of analysis, dynamics andgeometry via the language of homological algebra. With Floertheory and the relevant analysis of holomorphic curves as thecommon tools of investigation, this project centers around theinvestigation of deeper aspects of symplectic topology andHamiltonian dynamics up to the level of the topological category.The project also aims at the study of the moduli space ofpseudoholomorphic curves, both open and closed, beyond thewell-known stable map type compactifications and its applicationsto symplectic topology and to mirror symmetry through adiabaticdegeneration and the blow-up analysis. The proposed research willput the method of pseudoholomorphic curves in symplectic topologyin better perspective and further promote the existinginteractions between geometric analysis, symplectic topology anddynamical systems. The Hamiltonian formalism played a fundamentalrole not only for solving problems in classical mechanics but alsofor 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 thedifferential equation requires the Hamiltonian formalism in thecontext of the symplectic structure and symplectic manifolds.In symplectic geometry, there are two aspects of study, one thedynamical aspect and the other the geometric aspect. Understandingthe interplay between the two aspects is the core of symplectictopology. Oh's proposed research aims at deeper understanding ofsymplectic topology, and of its relation to the string theory inphysics. It also aims at easing an access of new coming graduatestudents and mathematicians from related fields to the resultsderived from Oh's research by proposing to write a graduate leveltextbook on Floer theory and its applications.