Holomorphic Curves in Symplectic and Complex Geometry

摘要奖：DMS-0505778首席研究员：理查德·辛德这项提议将进行辛几何和复几何的研究。辛几何的主题旨在应用和推广由Eliashberg，Givental和Hofer发展的辛场论的思想和定理。到目前为止，主要应用于4维辛流形上拉格朗日球面的哈密顿微分同态的分类，也有旨在建立更高类曲面的类似结果的工作正在进行中。在辛拓扑中，目前这样的分类结果只存在于4维。然而，该建议也将努力增加我们对高维的理解。Donaldson的工作提供了一条通向这种理解的途径，证明了在爆破操作之后，每个积分辛6-流形都可以被实现为Lefschetz纤维，其中的纤维是辛4-流形和零圈拉格朗日球面。关于底层理论，辛场理论允许我们沿着超曲面分割辛流形，并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割成足够小的碎片，以便完全理解。因此，该提议还将致力于推广辛场理论，以允许更一般的分裂，例如沿着带角的超曲面。在复几何中，Burns和Hind计划继续研究与实解析黎曼流形规范相关的复流形，特别是当黎曼流形是非紧致的非对称空间时。然后我们得到了一类复流形，它很好地推广了经典的有界对称域。辛几何起源于经典和量子力学的现代数学语言。这项建议将解决该领域的基本数学问题，特别是它将应用和推广被称为辛场理论的强大的新技术。辛场论是一个令人振奋的新发展，它有望帮助我们迅速朝着描述辛流形和哈密顿系统的整体性质的目标前进。由于这些问题是全球性的，它们最初往往看起来难以处理，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，开发算法来解决辛几何中的典型问题应该是可能的。这将立即在理论物理和动力系统中得到应用。

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.