Holomorphic Curves in Symplectic and Complex Geometry

辛复几何中的全纯曲线

• 批准号: 0505778
• 负责人: Richard Hind
• 金额: $ 9.95万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505778&HistoricalAwards=false
• 关键词: Holomorphic; Curves; Symplectic; Complex; Geometry

AbstractAward: DMS-0505778Principal Investigator: Richard HindThis proposal will conduct research in symplectic and complexgeometry. Topics in symplectic geometry aim to both apply andextend the ideas and theorems of the Symplectic Field Theory asdeveloped by Eliashberg, Givental and Hofer. The main applicationso far is to the classification up to Hamiltonian diffeomorphismof Lagrangian spheres in symplectic 4-manifolds, there is ongoingwork aimed at establishing similar results for higher genussurfaces. In symplectic topology such classification results atpresent only exist in dimension 4. Nevertheless the proposal willalso strive to increase our understanding of higher dimensions.One route to such an understanding is provided by work ofDonaldson showing that, after a blowing-up operation, everyintegral symplectic 6-manifold can be realized as a Lefschetzfibration in which the fibers are symplectic 4-manifolds and thevanishing cycles Lagrangian spheres. Regarding the underlyingtheory, Symplectic Field Theory allows us to split a symplecticmanifold along a hypersurface and apply holomorphic curve methodsto study each part separately. This is a very powerful idea, theresults above rely heavily upon it, but it is fundamentallylimited in that it may not be possible to split a symplecticmanifold along such hypersurfaces into pieces which aresufficiently small to be completely understood. Therefore theproposal will also work to generalize the Symplectic Field Theoryto allow more general splittings, for example along hypersurfaceswith corners. In complex geometry the proposal plans to continuework of Burns and Hind studying complex manifolds canonicallyassociated to real analytic Riemannian manifolds, focussingespecially on the case when the Riemannian manifold is asymmetric space of the noncompact type. Then we have a class ofcomplex manifolds which gives a nice generalization of theclassical bounded symmetric domains.Symplectic geometry originated as the modern mathematicallanguage of classical and quantum mechanics. This proposal willaddress basic mathematical problems in the area, in particular itwill apply and extend the powerful new techniques known asSymplectic Field Theory. The Symplectic Field Theory is anexciting new development which promises to help us move rapidlytowards our goal of describing the global nature of symplecticmanifolds and Hamiltonian systems. As the problems are global innature often they initially appear intractible, but the newmethods offer the prospect of breaking down our analysis intomore manageable local pieces. Eventually it should be possible todevelop algorithms to solve typical problems in symplecticgeometry. This will have immediate applications to theoreticalphysics and dynamical systems.

摘要奖：DMS-0505778首席研究员：理查德·欣德这项提案将在辛和复杂几何中进行研究。辛几何的主题旨在应用和扩展Eliashberg，Givental和霍费尔发展的辛场论的思想和定理。到目前为止，主要的应用是对辛4-流形中拉格朗日球的Hamilton同构的分类，目前正在进行的工作旨在建立更高亏格曲面的类似结果。在辛拓扑中，这种分类结果目前只存在于4维空间.然而，该建议也将努力增加我们的理解更高的dimensions.One路线，这样的理解是由工作提供的唐纳森显示，经过爆破操作，每个积分辛6-流形可以实现为一个Lefschetzfibration其中的纤维是辛4-流形和消失的周期拉格朗日球。关于基础理论，辛场论允许我们沿着一个超曲面沿着分裂一个辛流形，并应用全纯曲线方法分别研究每一部分。这是一个非常强大的想法，上面的结果很大程度上依赖于它，但它是根本上的限制，因为它可能不可能分裂一个辛流形沿着这样的超曲面成小到足以完全理解。因此，该提案还将推广辛场论，以允许更一般的分裂，例如沿着带角的超曲面。在复杂的几何建议计划继续工作伯恩斯和欣德研究复杂的流形canonicallyassociated真实的解析黎曼流形，focusingespecially对案件时，黎曼流形是非对称空间的非紧型。然后我们得到了一类复流形，它给出了经典有界对称域的一个很好的推广。辛几何起源于经典和量子力学的现代语言。这一建议将解决该领域的基本数学问题，特别是它将应用和扩展强大的新技术称为辛场论。辛场论是一个令人兴奋的新发展，它有望帮助我们朝着描述辛流形和Hamilton系统的整体性质的目标快速前进。由于这些问题本质上是全球性的，它们最初看起来很棘手，但新方法提供了一个前景，即把我们的分析分解成更易于管理的局部部分。最后，应该有可能发展出算法来解决辛几何中的典型问题。这将直接应用于理论物理学和动力系统。