The Differential Geometry of Partial Differential Equations

摘要：DMS-9870164首席研究员：Robert Bryant首席研究员计划将微分系统理论和等效方法应用于微分几何和数学物理中难以采用更传统方法的问题，重点关注两个主要问题。流形M上的Finsler结构赋予M中的每个切向量一个长度的概念，从而得到M中路径长度的概念。黎曼几何是由切空间上的内积导出长度的一种特殊情况。芬斯勒结构本质上是几何变分问题，其基本问题包括研究长度极值的路径（即测地线）、稳定性、可计算性等。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，称为标志曲率。Bryant计划使用外部微分系统技术，开发具有恒定旗曲率的Finsler结构的分类和全局存在性定理。第二个主要问题出现在超对称弦理论的某些模型中，这些模型要求在光滑流形上构造一个黎曼度规和三形式的简化完整连接，也许还有扭转。问题是要分类哪一对度规和三形式将允许物理理论是超对称的。Bryant已经完成了各种低维的分类，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，这种技术有助于解决经典情况下的完整问题（在这种情况下，三形式等于零）。Bryant还计划继续与Griffiths和Hsu在PDE几何及其守恒定律方面的合作，并推广他最近关于谐波态射的结构定理。优化是数学中的一个核心问题，在这个问题中，人们试图在物理系统模型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，在规划从起点到目的地的“最佳”路径时必须考虑水流，这里的“最佳”是指“最短的穿越时间”。在短时间内最优的路径（“测地线”），如果追求的时间足够长，可能就不会保持最优。这就是所谓的不稳定性。（例如，在一条中游流速较快的河流中，下游的测地线是稳定的，而上游的测地线则不是。）测量这种稳定性概念的几何量被称为“曲率”，因为它最初是在对地球曲率的研究中被确定的。Bryant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相关。

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.