Symplectic Topology, Mirror Symmetry and Analysis of Pseudoholomorphic Curves

自Gromov伪全纯曲线出现以来，在辛拓扑学的近二十年发展中，伪全纯曲线方法结合几何分析和哈密顿动力学的工具，表现了许多真正的辛现象。花理论和它的表亲辛场论，通过同调代数的语言作为分析、动力学和几何的统一力量而出现。本项目以花理论和全纯曲线的相关分析为研究的常用工具，围绕辛拓扑和哈密顿动力学的更深层次的研究，直至拓扑范畴的水平。本项目还旨在通过绝热退化和爆破分析，研究开和闭伪全纯曲线的模空间，超越众所周知的稳定映射型紧化及其在辛拓扑和镜像对称中的应用。本文的研究将使伪全纯曲线的方法在辛拓扑中得到更好的理解，并进一步促进几何分析、辛拓扑和动力系统之间存在的相互作用。哈密顿形式主义不仅对解决经典力学中的问题，而且对将经典力学转化为量子力学起着根本性的作用。当考虑约束系统中的力学时，即在弯曲空间上的力学，如“球摆”或“台球”，相应相空间和微分方程的描述需要在辛结构和辛流形的背景下使用哈密顿形式。辛几何有两个方面的研究，一个是动力学方面的研究，另一个是几何方面的研究。理解这两个方面之间的相互作用是辛拓扑的核心。Oh提出的研究旨在更深入地理解辛拓扑及其与物理学中的弦理论的关系。它还提议编写一本研究生水平的关于弗洛尔理论及其应用的教科书，以方便来自相关领域的新研究生和数学家获得吴的研究成果。

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.