Holomorphic Curves in Symplectic and Complex Geometry

项目编号：dms -0505778项目负责人：Richard hindd该项目将进行辛和复杂几何的研究。辛几何的主题旨在应用和扩展由Eliashberg， Givental和Hofer提出的辛场论的思想和定理。目前的主要应用是辛4流形中拉格朗日球的哈密顿微分同态的分类，目前正在进行的工作旨在建立更高属表面的类似结果。在辛拓扑中，这种分类结果只存在于维4中。然而，这个提议也将努力增加我们对更高维度的理解。donaldson的工作为这种理解提供了一条途径，该工作表明，在吹胀操作之后，每个积分辛6流形都可以实现为Lefschetzfibration，其中纤维是辛4流形和消失循环拉格朗日球。在基础理论方面，辛场论允许我们沿着超曲面分裂辛流形，并应用全纯曲线方法分别研究每个部分。这是一个非常强大的想法，上面的结果在很大程度上依赖于它，但它在根本上是有限的，因为它可能不可能沿着这样的超曲面将辛流形分成足够小的部分，以完全理解。因此，该提议也将推广辛场论，以允许更一般的分裂，例如沿着带角的超曲面。在复杂几何中，本提案计划继续Burns和Hind的工作，研究与实际解析黎曼流形相关的复流形，特别关注黎曼流形是非紧型非对称空间的情况。然后我们有一类复流形它给出了经典有界对称域的一个很好的推广。辛几何起源于古典力学和量子力学的现代数学语言。该提案将解决该领域的基本数学问题，特别是它将应用和扩展被称为辛场论的强大新技术。辛场论是一个令人兴奋的新发展，它有望帮助我们快速实现描述辛流形和哈密顿系统的全局性质的目标。由于这些问题本质上是全球性的，它们最初似乎是难以解决的，但新方法提供了将我们的分析分解为更易于管理的局部部分的前景。最终应该有可能开发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和动力系统。

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.