Topology and geometry of Lagrangian submanifolds and its applications

拉格朗日子流形的拓扑几何及其应用

• 批准号: 9971446
• 负责人: Yong-Geun Oh
• 金额: $ 7.88万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971446&HistoricalAwards=false
• 关键词: Topology; geometry; Lagrangian; submanifolds; its

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.

Abstractaward：DMS-9971446原理研究者：Yong-Geun Ohlagrangian submanifolds是最重要的几何对象，在象征性的几何形状上。 拉格朗日相交理论是象征性几何形状的概念，在cotangent束上，它是摩尔斯基础理论的符合性。 Oh，Chekanov，Seidel和Polterovich的最新回报表明，定义Floer同源性是Lagrangian Submanifolds的研究和几何学的强大工具。在cotangentBundles上，拉格朗日交集理论还提供了摩尔斯理论的解释，当与浮动机构理论结合使用时，它在规范通道中编码了象征性刚性。 外部束束外有障碍物，这些障碍物从llagrangian submanifolds的量子几何形状产生。 哦，建议在浮子理论中进一步开发调查结构，并将其应用于与象征性拓扑和镜像对称性有关的棘手问题。特殊的Lagrangian submanifolds是保留图形的几何形状，在Gromov的伪形曲线理论中，在符号图的几何学研究中起着这种作用。 OH PropoSeSto研究了整体形状的几何形状在复杂的欧克利德空间中保存图形图的几何形状（例如，非质量定理的一种版本）。构建系统近似近似值的系统的量子公式的过程。从历史上汲取的教训很难量化力学或几何形状，并且当障碍系统是可量化的，它揭示了一种非常特殊的物理和几何性质。 但是，3和4尺寸拓扑结构的量子配方是在低维拓扑中的最新突破中的必要成分，并且必须量化经典（差异）拓扑来衍生出最精致的非差异差异的不变性。物理学中的和弦理论。这项研究将旨在单人执行此差异化学量化程序，另一方面，了解符号，riemannian和复杂几何形状之间的相互关系。