The Differential Geometry of Partial Differential Equations

建议：DMS-9870164首席研究员：罗伯特·布莱恩特首席研究员计划将微分系统理论和等价方法应用于微分几何和数学物理中抵制更传统方法的问题，强调两个主要问题。流形M上的Finsler结构给M中的每个切向量分配了一个长度概念，从而引出了M中路径的长度概念。黎曼几何是一个特例，其中长度是从切线空间上的内积得到的。Finsler结构本质上是几何化的变分问题，基本问题涉及研究作为长度极值的路径(即测地线)、它们的稳定性、它们的可计算性等。与常见的黎曼情形一样，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。布莱恩特计划利用外微分系统技术，建立具有常曲率的Finsler结构的分类和整体存在定理。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出黎曼度规和三形式的约化完整联系，可能是扭转联系。问题是要对哪对度规和三种形式进行分类，以使物理理论具有超对称性。科比已经在不同的低维进行了分类，并准备研究具有物理意义的中间维度(从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问题(在经典情况下，三种形式相同为零)。科比还计划继续他与Griffiths和Hsu在偏微分方程组的几何及其守恒定律方面的合作，并推广他最近关于调和态射的结构定理。最优化是数学中的一个中心问题，在这个问题中，人们试图在一个物理系统的模型中的可能构型空间中选择“最佳”构型。一个例子是在一片水域上航行的问题，在规划从起点到目的地的最佳路径时必须考虑水流，其中最佳的意思是“穿越的最短时间”。如果追求足够长的时间，一条在短时间内最优的路径(测地线)可能不会保持最优。这就是所谓的不稳定。(例如，在一条中游水流较快的河流中，下游测地线是稳定的，但上游测地线不稳定。)衡量这种稳定性概念的几何量被称为“曲率”，因为它最早是在研究地球曲率时发现的。布莱恩特的工作研究了曲率和超定的微分方程组，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。

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.