The Differential Geometry of Partial Differential Equations

偏微分方程的微分几何

• 批准号: 9870164
• 负责人: Robert Bryant
• 金额: $ 18.54万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9870164&HistoricalAwards=false
• 关键词: Differential; Geometry; Partial; Equations

Abstract Proposal: DMS-9870164 Principal Investigator: Robert Bryant The principal investigator plans to apply the theory of differential systems and the method of equivalence to problems in differential geometry and mathematical physics that have resisted more traditional approaches, emphasizing two main problems. A Finsler structure on a manifold M assigns a notion of length to each tangent vector in M, leading to a notion of length for paths in M. Riemannian geometry is a special case where the length is derived from an inner product on tangent spaces. Finsler structures are essentially geometrized calculus of variation problems and the fundamental problems involve studying paths that are extremals of length (i.e., the geodesics), their stability properties, their computability, and so forth. As in the familiar Riemannian case, the geometric object that controls stability of geodesics is a sort of curvature tensor, called the flag curvature. Bryant plans to develop classification and global existence theorems for Finsler structures with constant flag curvature, using exterior differential systems techniques. The second main problem arises in certain models of super-symmetric string theory that require the construction on a smooth manifold of a connection with reduced holonomy, perhaps with torsion, out of a Riemannian metric and a three-form. The problem is to classify which pairs of metric and three-form will allow the physical theory to be super-symmetric. Bryant has already done the classification in various low dimensions and is ready to study the intermediate dimensions (six through twenty-six) that are of physical interest, using the techniques of exterior differential systems that contributed to the solution of the holonomy problem in the classical case (in which the three-form was identically zero). Bryant also plans to continue his collaboration with Griffiths and Hsu on the geometry of PDE and their conservation laws and to generalize his recent structure theorems for harmonic morphisms. Optimization is a central problem in mathematics, in which one tries to select the 'best' configuration in the space of possible configurations in a model for a physical system. An example is the problem of navigating on a body of water in which one must take water currents into account in planning the 'best' path from origin to destination, where 'best' is taken to mean 'shortest time of traverse'. A path that is optimal for a short period (a 'geodesic') might not remain optimal if pursued long enough. This is known as instability. (For example, in a river where the current is faster in midstream it turns out that downstream geodesics are stable, but that upstream geodesics are not.) The geometric quantity that measures this notion of stability is known as 'curvature', since it was first identified in studies of the curvature of the Earth. Bryant's work studies curvature and 'over-determined' systems of differential equations, and is relevant to optimization problems in motion planning, control theory, robotics, and string theory models in high energy physics.

摘要 提案：DMS-9870164主要研究者：Robert Bryant 首席研究员计划应用微分理论 系统和方法的等价问题，在微分几何和数学物理，抵制更传统的方法，强调两个主要问题。 流形M上的芬斯勒结构赋予M中的每个切向量一个长度的概念，从而导致M中的路的长度的概念。黎曼几何是一个特殊的情况，其中长度是从正切空间上的内积导出的。 Finsler结构本质上是变分问题的几何化微积分，并且基本问题涉及研究长度为极值的路径（即，测地线），它们的稳定性，它们的可计算性，等等。 在熟悉的黎曼情形中，控制测地线稳定性的几何对象是一种曲率张量，称为旗曲率。 Bryant计划利用外部微分系统技术，为具有常旗曲率的Finsler结构建立分类和全局存在定理。 第二个主要问题出现在超对称弦理论的某些模型中，这些模型需要在光滑流形上构造一个具有约化完整性的联络，也许是具有挠率的联络，它来自黎曼度量和三形式。 问题是分类哪些度量和三形式对将允许物理理论是超对称的。 布莱恩特已经在各种低维中完成了分类，并准备研究物理上感兴趣的中间维（6到26），使用外部微分系统的技术，这些技术有助于解决经典情况下的完整性问题（其中三种形式都是零）。 布莱恩特还计划继续他的合作与格里菲斯和许的几何偏微分方程和他们的守恒定律，并推广他最近的结构定理调和态射。 最优化是数学中的一个中心问题，在这个问题中，人们试图在物理系统模型中的可能配置空间中选择“最佳”配置。 一个例子是在水体上航行的问题，其中必须考虑水流，以规划从起点到目的地的“最佳”路径，其中“最佳”是指“最短的航行时间”。 一条在短时间内是最优的路径（“测地线”），如果追求的时间足够长，可能就不会保持最优。 这被称为不稳定性。 (For例如，在一条河中，水流在中游更快，结果是下游测地线是稳定的，但上游测地线不是。） 测量这种稳定性概念的几何量被称为“曲率”，因为它首先在地球曲率的研究中被确定。布莱恩特的工作研究曲率和微分方程的“超定”系统，并与运动规划，控制理论，机器人和高能物理中的弦理论模型中的优化问题有关。