Symplectic Topology, Mirror Symmetry and Analysis of Pseudoholomorphic Curves

自Gromov的伪全纯曲线出现以来，在辛拓扑学发展的近二十年中，伪全纯曲线方法结合了几何分析和Hamilton代数的工具，表现了许多真正的辛现象。弗洛尔理论和它的表亲辛场论，通过同调代数的语言，已经成为分析、动力学和几何学的统一力量。本项目以Floer理论和全纯曲线的相关分析为研究工具，深入研究辛拓扑和Hamilton动力学的拓扑范畴，并研究伪全纯曲线的模空间，包括开曲线和闭曲线，除了著名的稳定映射型紧化及其应用辛拓扑和镜像对称通过绝热退化和blow-up分析。本文的研究将使伪全纯曲线方法在辛拓扑中得到更好的应用，并进一步促进几何分析、辛拓扑和动力系统之间的相互作用。哈密顿形式主义不仅对经典力学问题的解决起着基础性的作用，而且对经典力学向量子力学的转化也起着基础性的作用。对于“球摆”或“台球”等弯曲空间上的力学，相应的相空间和微分方程的描述需要辛结构和辛流形背景下的Hamilton形式，辛几何有两个方面的研究，一个是动力学方面，另一个是几何方面。理解这两个方面之间的相互作用是辛拓扑学的核心。Oh提出的研究旨在更深入地理解辛拓扑，以及它与物理学中的弦理论的关系。它还旨在通过建议编写一本关于Floer理论及其应用的研究生水平教科书，方便新来的研究生和相关领域的数学家获得Oh的研究成果。

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.