Topology and geometry of Lagrangian submanifolds and its applications

摘要奖：DMS-9971446主要研究人员：Yong-Geun OhLagrangian子流形是辛几何中最重要的几何对象。拉格朗日交理论是辛几何的核心，余切丛上的拉格朗日交理论是基的Morse理论的辛化。Oh，Chekanov，Seidel和Polterovich的最新结果表明，定义Floer同调是研究Lagrange子流形的拓扑学和几何的有力工具。在余切丛上，拉格朗日交理论也提供了Morse理论的一种解释，当与Floer同调理论结合时，它以规范的方式编码辛刚性。在余切丛之外，定义来自拉格朗日子流形的量子几何的Floer理论存在障碍。Oh建议进一步发展Floer理论中的函数结构，并将其应用于与辛拓扑和镜像对称有关的问题。另一项研究涉及Kaehler流形上的拉格朗日子流形的最小化体积的变分问题，该问题与Harvey和Lawson的特殊拉格朗日子流形理论密切相关。特殊的拉格朗日子流形是全纯保体积映射几何的关键，正如Gromov的伪全纯曲线理论在辛映射几何研究中所起的作用一样。吴建议研究复欧氏空间中全纯保体积映射的几何与辛映射的几何之间的各种类比(例如，非压缩定理的一种形式)。从经典近似的知识构造系统的量子形式的过程在物理学中被称为‘量子化’。从历史中得到的教训是，力学或几何学很难量子化，当一个技术系统可量化时，它揭示了一个非常特殊的物理和几何性质。然而，三维和四维拓扑的量子公式是最近低维拓扑突破的重要组成部分，而量子化经典(微分)拓扑是推导最精细的非平凡微分不变量的关键。拉格朗日子流形是辛几何中最重要的几何对象，在几何量子化以及最近物理中镜像对称和弦理论的发展中发挥着关键作用。本研究的目的一方面是为了实现微分拓扑学的量子化程序，另一方面是为了了解辛几何、黎曼几何和复几何之间的相互关系。

AbstractAward: DMS-9971446Principal Investigator: Yong-Geun OhLagrangian submanifolds are the most important geometric objectsin symplectic geometry. Lagrangian intersection theory is thecore of symplectic geometry, and on the cotangent bundle it isthe symplectification of Morse theory of the base. Recentresults by Oh, Chekanov, Seidel and Polterovich demonstrate thatFloer homology, when defined, is a powerful tool to investigatetopology and geometry of Lagrangian submanifolds. On cotangentbundles, Lagrangian intersection theory also provides aninterpretation of Morse theory, and when combined with the Floerhomology theory, it encodes symplectic rigidity in a canonicalway. Outside cotangent bundles there are obstructions todefining a Floer theory that come from the quantum geometry ofLagrangian submanifolds. Oh proposes to further developfunctorial constructions in the Floer theory and to apply them toproblems related to symplectic topology and mirror symmetry.Other research concerns the variational problem of minimizingvolume of Lagrangian submanifolds on Kaehler manifolds, which isclosely related to Harvey and Lawson's theory of specialLagrangian submanifolds. Special Lagrangian submanifolds are akey to the geometry of holomorphic volume preserving maps, in theway that Gromov's theory of pseudo-holomorphic curves played sucha role in the study of geometry of symplectic maps. Oh proposesto study various analogies between the geometry of holomorphicvolume preserving maps to that of symplectic maps in complexEuclidean spaces (e.g. a version of non-squeezing theorem).The process of constructing a quantum formulation of a systemfrom a knowledge of a classical approximation is called``quantization'' in physics. The lesson learned from history isthat it is hard to quantize mechanics or geometry, and when amechnical system is quantizable, it reveals a very particularphysical and geometrical nature. However, quantum formulationsof 3 and 4 dimensional topology have been essential ingredientsin recent breakthroughs in low dimensional topology and it wasessential to quantize classical (differential) topology to derivethe most delicate non-trivial differential invariants.Lagrangian submanifolds are the most important geometric objectsin symplectic geometry and play a key role in the geometricquantization and in the recent development of mirror symmetry andstring theory in physics. This research will aim at, on the onehand, carrying out this quantization program of differentialtopology and, on the other hand, understanding the inter-relationbetween symplectic, Riemannian and complex geometries.