Topology and geometry of Lagrangian submanifolds and its applications

AbstractAward：DMS-9971446首席研究员：Yong-Geun OhLagrange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，它是莫尔斯理论在余切丛上的辛化。 Oh，Chekanov，Seidel和Polterovich最近的结果表明，当定义Floer同调时，Floer同调是研究Lagrange子流形拓扑和几何的有力工具。在余切丛上，拉格朗日交理论也提供了对莫尔斯理论的一种解释，当与弗洛尔同调理论相结合时，它将辛刚性编码为一个正则轨道。 在余切丛之外，从拉格朗日子流形的量子几何出发，定义Floer理论存在着障碍。 Oh提出进一步发展Floer理论中的函子结构，并将其应用于辛拓扑和镜像对称的相关问题，其他研究涉及Kaehler流形上的Lagrange子流形体积极小化的变分问题，这与Harvey和Lawson的特殊Lagrange子流形理论密切相关。特殊的Lagrange子流形是全纯保体积映射几何的一个关键，就像Gromov的伪全纯曲线理论在辛映射几何的研究中所起的苏查作用一样。 Oh建议研究复欧氏空间中保体积全纯映射几何与辛映射几何之间的各种类比（例如非压缩定理的一个版本）。从经典近似的知识构造系统的量子公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，很难解释力学或几何学，当一个机械系统可以量化时，它揭示了一个非常特殊的物理和几何性质。 然而，在这方面，3维和4维拓扑的量子公式是近年来低维拓扑研究取得突破的重要组成部分，（差分）拓扑结构，以推导出最微妙的非-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的镜像对称和弦理论的发展中起着关键作用。物理学本研究的目的一方面是实现微分拓扑的量子化，另一方面是理解辛几何、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.