The Curvature of 4-Manifolds

4-流形的曲率

• 批准号: 1205953
• 负责人: Claude LeBrun
• 金额: $ 31.87万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-15 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205953&HistoricalAwards=false
• 关键词: Curvature; Manifolds

AbstractAward: DMS-1205953 Principal Investigator: Claude R. LeBrunThe principal investigator plans to study a family of related global problems in 4-dimensional Riemannian geometry. The primary goal of the proposed research is to discover and develop fundamental links between the curvature of a Riemannian 4-manifold and the differential topology of the underlying space. The main problems to be investigated focus on the existence, uniqueness, and detailed structure of several classes of canonical metrics. Topics of study will include Einstein metrics, extremal Kaehler metrics, Bach-flat metrics, and self-dual metrics. Some related problems in higher dimensions will also be considered, in anticipation of the discovery of further fundamental differences between the 4-dimensional and higher-dimensional settings.This research program aims both to explore fundamental issues in the mathematical field of differential geometry, and to discover new links between mathematics and theoretical physics. Much of the planned research activity takes its inspiration from current attempts to bridge the gulf separating Einstein's theory of gravitation from the quantum field theories that describe the forces of nature on a microscopic scale. A major focus of the research is the influence of the large-scale (topological) structure of a 4-dimensional universe on whether or not it admits Riemannian-signature solutions of Einstein's gravitational field equations.

摘要奖：DMS-1205953主要研究者：Claude R. LeBrunThe首席研究员计划研究四维黎曼几何中的一系列相关全局问题。该研究的主要目标是发现和发展黎曼4流形的曲率与基础空间的微分拓扑之间的基本联系。主要研究的问题集中在几类典型度量的存在性、唯一性和详细结构上。课程将包括爱因斯坦度量，极端Kaehler度量，巴赫平坦度量和自对偶度量。在更高维度的一些相关问题也将被考虑，在预期的发现之间的进一步的基本差异4维和更高维的设置。这个研究计划的目的是探索微分几何的数学领域的基本问题，并发现数学和理论物理之间的新的联系。许多计划中的研究活动的灵感来自目前试图弥合爱因斯坦引力理论与量子场论之间的鸿沟，量子场论描述了微观尺度上的自然力。研究的一个主要焦点是四维宇宙的大尺度（拓扑）结构对爱因斯坦引力场方程是否允许黎曼签名解的影响。