Symplectic Geometry and Complex Geometry

摘要 提案：DMS-9803192主要研究者：Simon唐纳森 这个项目的主要课题涉及从复杂几何辛拓扑的方法的应用。调查员将开发一个通用程序，用于将这一领域的问题转化为涉及余维2子流形家族的单值性的组合问题。 预计这将适用于，原则上，既辛流形的分类问题，也拉格朗日子流形和辛同胚。一旦一般的基础是在适当的位置应用将被考虑：这里的问题将是看看是否组合问题可以转换成一个易于处理的形式。 将特别注意从单值性中得到的Floer同调群的作用。该项目的一个附属主题涉及研究卡勒度量的几何：这里的具体目标是证明卡勒度量空间中某些测地线的存在，并将其应用于卡拉比的极值度量程序。这个存在性问题是齐次Monge-Ampere方程的Dirichlet问题的一个版本，这是一个独立感兴趣的话题。另一个附属主题涉及研究流形与特殊的holonomy群，特别是寻找新的例子，获得使用复杂的3倍作为积木。 复数由真实的和虚的部分组成，是整个数学的基础。在几何学中，人们从上个世纪中期就认识到，复数系的性质与二维空间的几何和拓扑密切相关。这一主题的阐述及其向更高维度的扩展是二十世纪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、量子理论和相对论。数学上，许多问题归结为非线性偏微分方程的详细分析。 拟议的研究将有助于这一育雏的发展，重点放在一些具体的和专题的问题，在所有这些复杂的变量发挥了关键作用。

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.