Topology and geometry of Lagrangian submanifolds and its applications

摘要：dms -9971446项目负责人：Yong-Geun ohlagrange子流形是辛几何中最重要的几何对象。拉格朗日交点理论是辛几何的核心，在余切束上是基的莫尔斯理论的化简。Oh, Chekanov， Seidel和Polterovich最近的研究结果表明，当花同调被定义时，是研究拉格朗日子流形拓扑和几何的有力工具。在共切束上，拉格朗日交点理论也提供了莫尔斯理论的一种解释，当与弗洛尔同调理论结合时，它以一种标准的方式编码辛刚性。在余切束之外，有一些障碍来定义弗洛尔理论，这些障碍来自于拉格朗日子流形的量子几何。Oh建议在flower理论中进一步发展函数结构，并将其应用于辛拓扑和镜像对称的相关问题。另一个研究是关于Kaehler流形上拉格朗日子流形体积最小化的变分问题，这与Harvey和Lawson的特殊拉格朗日子流形理论密切相关。特殊拉格朗日子流形是全纯保体映射几何的关键，就像Gromov的伪全纯曲线理论在辛映射几何的研究中所起的作用一样。Oh建议研究复欧几里德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（例如非挤压定理的一个版本）。从一个经典近似的知识中构造一个系统的量子公式的过程在物理学中被称为“量子化”。历史的教训是，力学或几何是很难量化的，当一个机械系统是可量化的，它揭示了一个非常特殊的物理和几何性质。然而，3维和4维拓扑的量子公式已经成为低维拓扑最近突破的重要组成部分，而对经典（微分）拓扑进行量子化以推导出最精细的非平凡微分不变量是必不可少的。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、镜像对称和弦理论的最新发展中起着关键作用。本研究一方面旨在实现微分拓扑的量化规划，另一方面了解辛几何、黎曼几何和复杂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.