Symplectic Geometry and Complex Geometry

摘要提案：DMS-9803192首席研究员：Simon Donaldson本项目的主要课题涉及从复杂几何到辛拓扑方法的应用。研究者将开发一个将这一领域的问题转化为涉及余维2子流形族的单态的组合问题的一般程序。我们期望这在原则上既适用于辛流形的分类问题，也适用于拉格朗日子流形和辛形态。一旦有了基本的基础，就可以考虑应用程序了：这里的问题是看看组合问题是否可以转换成可处理的形式。将特别注意从单系中获得的花同源群的作用。该项目的一个附属主题涉及对Kahler度量几何的研究：这里的具体目标是证明Kahler度量空间中某些测地线的存在，并将其应用于Calabi的极值度量程序。这个存在性问题是齐次蒙日-安培方程的狄利克雷问题的一个版本，是一个独立感兴趣的话题。另一个附属主题涉及对具有特殊完整群的流形的研究，特别是寻找使用复杂3折叠作为构建块获得的新示例。复数由实部和虚部组成，是整个数学的基础。在几何领域，自上世纪中叶以来，人们已经认识到，复数系统的性质与二维空间的几何和拓扑结构密切相关。对这一主题的阐述，以及将其扩展到更高维度，是20世纪数学的主要成就之一。这些思想与数学物理的许多分支有许多联系，包括势场理论、量子论和相对论。从数学上讲，许多问题归结为对非线性偏微分方程的详细分析。所提出的研究将有助于这一育雏发展，重点关注一些具体和局部问题，在所有这些复杂变量发挥关键作用。

Abstract Proposal: DMS-9803192 Principal Investigator: Simon Donaldson The main topic of this project involves the application of methods from complex geometry to symplectic topology. The investigator will develop a general procedure for translating problems in this area into combinatorial questions involving the monodromy of a family of codimension 2 submanifolds. It is expected that this will apply, in principle, both to the classification problem for symplectic manifolds and also to Lagrangian submanifolds and symplectomorphisms. Once the general foundations are in place applications will be considered: the question here will be to see if the combinatorial problems can be cast into a tractable form. Particular attention will be paid to the role of the Floer homology groups obtained from the monodromy. One subsidiary topic in the project involves research into the geometry of Kahler metrics: the specific goals here are to prove the existence of certain geodesics in the space of Kahler metrics, and apply these to Calabi's extremal metric program. This existence question is a version of the Dirichlet problem for the homogeneous Monge-Ampere equation, a topic of independent interest. The other subsidiary topic involves research into manifolds with exceptional holonomy groups, and particularly the search for new examples, obtained using complex 3-folds as building blocks. Complex numbers, made up of real and imaginary components, are fundamental throughout mathematics. In geometry, it has been realised since the middle of the last century that properties of the complex number system are intimately bound up with the geometry and topology of 2-dimensional spaces. The elaboration of this theme, and its extension to higher dimensions, has been one of the main achievements of twentieth century mathematics. The ideas have many contacts with numerous branches of Mathematical Physics, including the theory of potentials and fields, quantum theory and relativity. Mat hematically, many of the questions come down to the detailed analysis of nonlinear partial differential equations. The proposed research will contribute to this brood development, focusing on a number of specific and topical questions, in all of which complex variables play a key role.