Nonlinear Problems in Symplectic Geometry and Complex Geometry

辛几何和复几何中的非线性问题

• 批准号: 9802479
• 负责人: Gang Tian
• 金额: $ 64.44万
• 依托单位: Massachusetts Institute of Technology
• 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 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802479&HistoricalAwards=false
• 关键词: Nonlinear; Problems; Symplectic; Geometry; Complex

AbstractProposal: DMS-9802479Principal Investigator: Gang TianThe project addresses some fundamental problems from symplectic geometry and Kaehler geometry. The first part concerns the existence of Kaehler-Einstein metrics with positive scalar curvature. These metrics provide solutions of the Einstein equation on Riemannian manifolds. In the second part, the principal investigator (PI) intends to further study new invariants, often refered as GW-invariants, of general symplectic manifolds and find more applications to Hamiltonian systems, symplectic topology and geometry. This part also contains very basic problem of understanding structure of GW-invariants and its possible relation to particular integrable systems. The PI also suggests to study some related nonlinear PDE problems which arise from physics and geometry.Problems in this project were motivated by our desire of probing mathematics of basic physical laws under suitable conditions. For example, GW-invariants were inspired by a sigma model theory coupled with gravity in mathematical physics, and they extend classical enumerative geometry which involves counting curves through a number of points or intersections of curves. The resolutions of these problems will provide new mathematical insights of Einstein equation in general relativity, quantum field theory, mirror symmetry phenomenon in string theory. Our study will also deepen our understanding properties of symplectic manifolds, algebraic manifolds and Calabi-Yau spaces.

摘要项目主要研究辛几何和Kaehler几何中的一些基本问题。第一部分讨论了具有正标量曲率的卡勒-爱因斯坦度量的存在性。这些度量提供了黎曼流形上爱因斯坦方程的解。在第二部分中，首席研究员（PI）打算进一步研究一般辛流形的新不变量，通常称为gw不变量，并在哈密顿系统，辛拓扑和几何中寻找更多的应用。这一部分还包含了理解gw -不变量结构及其与特定可积系统的可能关系的非常基本的问题。PI还建议研究一些相关的非线性偏微分方程问题，这些问题来自物理和几何。这个项目的问题是由我们想在合适的条件下探索基本物理定律的数学的愿望所激发的。例如，gw不变量的灵感来自于数学物理中的sigma模型理论和引力，它们扩展了经典的枚举几何，包括通过许多点或曲线的交叉点来计数曲线。这些问题的解决将为广义相对论中的爱因斯坦方程、量子场论、弦理论中的镜像对称现象提供新的数学见解。我们的研究也将加深我们对辛流形、代数流形和Calabi-Yau空间性质的理解。