Symplectic Geometry and Complex Geometry

辛几何和复几何

• 批准号: 9803192
• 负责人: Richard Schoen
• 金额: $ 27.38万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803192&HistoricalAwards=false
• 关键词: Symplectic; Geometry; Complex; Symplectic; Geometry; Complex

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.

摘要建议：DMS-9803192主要研究者：Simon Donaldson该项目的主要主题涉及从复杂几何形状到符号拓扑的方法的应用。研究人员将制定一项一般程序，将该领域的问题转化为涉及Codimension 2 Submanifold家族的单际疾病的组合问题。 预计这原则上将适用于象征性歧管的分类问题，也适用于拉格朗日的副型和同构。一旦考虑了一般的基础，将考虑到申请：这里的问题是查看组合问题是否可以施加到可拖动的形式中。 特别关注从单片中获得的浮子同源组的作用。该项目中的一个辅助主题涉及对卡勒指标的几何形状进行研究：这里的具体目标是证明在卡勒指标空间中存在某些地理位置，并将其应用于卡拉比的极端指标计划。这个存在的问题是均质蒙加 - 安培方程的迪里奇问题的一种版本，这是一个独立关注的话题。另一个子公司主题涉及对具有特殊全能组的流形的研究，尤其是搜索使用复杂的3倍作为基础的复杂示例的搜索。 由真实和虚构组成组成的复杂数字在整个数学中都是基本的。在几何形状中，自上世纪中叶以来已经意识到，复数系统的性质与二维空间的几何形状和拓扑密切相关。该主题的阐述及其扩展到更高的维度，一直是20世纪数学的主要成就之一。这些想法与数学物理学的众多分支有许多接触，包括潜力和领域的理论，量子理论和相对论。从肿瘤上讲，许多问题都归结为对非线性偏微分方程的详细分析。 拟议的研究将有助于这种育雏的发展，重点是许多具体和局部的问题，在所有这些问题中，复杂的变量都起着关键作用。