The geometry of partial differential equations and applications

罗伯特·布莱恩特（PI）计划应用外微分系统理论，等价方法和变分法来研究微分几何，数学物理和计量经济学中的各种问题。 在具体的问题和领域，他打算调查的是这些：在仿射几何，他将调查新的可积系统，他发现，而工作的仿射Bonnet问题。 在黎曼几何，他将继续他的工作分类黎曼淹没从空间形式和其他对称空间，工作分类特殊度量（如孤子和度量的限制代数类型的曲率张量），并了解可积系统与特殊holonomy度量较高的cohomogeneity。在芬斯勒几何，布莱恩特将继续发展芬斯勒流形的理论与常旗曲率，无论是在建设新的例子，并在了解其测地流的性质;与扭量理论和异国情调的holonomy结构的关系将特别审查。 在复杂的几何中，布莱恩特将继续他的工作，描述全纯接触$3$-folds中有理接触曲线的模空间上诱导的有理正规结构，目的是了解这些如何与可积系统及其曲率性质相联系。 所有这些（以及其他相关项目）将产生本身有用的结果，但它们也将促进外微分系统和等效方法的新技术的发展，并将促进研究生和博士后研究员在这些技术方面的培训。我们在物理学和其他科学中使用数学的能力的许多重要进展都依赖于发展几何理解这些问题，即，一种理解，只关注那些问题的各个方面，而不依赖于特定的（通常是任意选择的）坐标系。 例如，爱因斯坦的广义相对论公式是在他能够有效地使用黎曼几何的无坐标概念来分离和描述引力的本质为时空曲率之后才出现的。 然而，一个问题的无坐标公式往往揭示退化，需要处理的非标准工具，如微分系统理论。 布莱恩特的研究计划，适用于坐标自由理论的微分系统和方法的等效性，将导致一个更根本的，几何理解的一些问题，出现在研究中的表面在空间中，特殊的结构，出现因为额外的或隐藏的对称性，和性质的曲率本身在一些几何背景。

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.