The geometry of partial differential equations and applications

偏微分方程的几何及其应用

• 批准号: 1359583
• 负责人: Robert Bryant
• 金额: $ 13.29万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1359583&HistoricalAwards=false
• 关键词: geometry; partial; differential; equations; applications

Robert Bryant (the PI) plans to apply the theory of exterior differential systems, the method of equivalence, and methods from the calculus of variations to study a variety of problems in differential geometry, mathematical physics, and econometrics. Among the specific problems and areas that he intends to investigate are these: In affine geometry, he will investigate the new integrable systems that he has discovered while working on the affine Bonnet problem. In Riemannian geometry, he will continue his work on classifying Riemannian submersions from space forms and other symmetric spaces, work on classifying special metrics (such as solitons and metrics with restrictions on the algebraic type of the curvature tensor), and on understanding the integrable systems associated with special holonomy metrics of higher cohomogeneity. In Finsler geometry, Bryant will continue to develop the theory of Finsler manifolds with constant flag curvature, both in constructing new examples and in understanding the properties of their geodesic flows; the relations with twistor theory and exotic holonomy structures will be especially examined. In complex geometry, Bryant will continue his work on characterizing the rational normal structures induced on the moduli space of rational contact curves in holomorphic contact $3$-folds, with the goal of understanding how these are connected to integrable systems and their curvature properties. All these (and other related projects) will produce results that are useful in themselves, but they will also motivate the development of new techniques in exterior differential systems and the method of equivalence and will facilitate the training of graduate students and postdoctoral fellows in these techniques.Many of the important advances in our ability to use mathematics in physics and other sciences have depended on developing a geometric understanding of those problems, i.e., an understanding that focuses only on the aspects of those problems that are not tied to specific (often, arbitrarily chosen) coordinate systems. For example, Einstein's formulation of General Relativity came only after he was able to employ the coordinate-free concepts of Riemannian geometry effectively to isolate and describe the essential nature of gravity as the curvature of space-time. However, coordinate-free formulations of a problem often reveal degeneracies that need to be approached by nonstandard tools, such as the theory of differential systems. Bryant's plan of research, which applies the coordinate-free theories of differential systems and the method of equivalence, will lead to a more fundamental, geometric understanding of a number of problems that arise in the study of surfaces in space, the special structures that arise because of extra or hidden symmetries, and the nature of curvature itself in a number of geometric contexts.

罗伯特·布莱恩（Robert Bryant）（PI）计划运用外部差异系统的理论，等效性的方法以及从变化的计算中的方法来研究差异几何，数学物理学和计量经济学的各种问题。 在他打算调查的具体问题和领域中，有这些问题：在仿射几何形状中，他将研究他在研究仿射引擎盖问题时发现的新的可集成系统。 在Riemannian的几何形状中，他将继续他的工作，从而从空间形式和其他对称空间中分类riemannian稀释，致力于对特殊指标（例如，孤子和指标和对代数类型的曲率张量类型）进行分类，并理解与更高的cohomogene的特殊整体系统相关的可相关的系统。在Finsler的几何形状中，科比将继续发展具有恒定标志曲率的Finsler歧管理论，无论是在构建新的示例和理解其大地测量流的特性时；将特别研究与扭曲理论和异国情调的全能结构的关系。 在复杂的几何形状中，科比将继续他的工作，以表征全体形态接触中有理接触曲线的模量空间所引起的合理正常结构$ 3 $ - 折，目的是了解它们如何连接到可集成的系统及其曲率属性。 所有这些（和其他相关项目）都将产生其本身有用的结果，但它们还将激励外部差异系统和等效方法的新技术的发展，并促进对这些技术的研究生和博士后研究员的培训。在我们在物理学和其他方面使用数学方面的重要能力的重要进步，这些理解是依靠这些问题和其他理解的，即依靠这些问题。这些问题的各个方面与特定（通常是任意选择的）坐标系无关。 例如，爱因斯坦（Einstein）的一般相对性表述仅在他能够有效地利用黎曼几何形状的无坐标概念来隔离并描述重力的基本本质为时空的曲率。 但是，问题的无坐标表述通常揭示了非标准工具（例如差分系统理论）需要处理的脱模构。 科比的研究计划应用了差异系统的无坐标理论和等效性的方法，将导致对空间表面研究中出现的许多问题的基本，几何理解，这些问题，由于额外的或隐藏的对称性而出现的特殊结构，以及在数量上的曲率本身中出现的曲率本身。