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。 他打算研究的具体问题和领域包括： 在仿射几何中，他将研究他在研究仿射 Bonnet 问题时发现的新可积系统。 在黎曼几何中，他将继续对黎曼淹没从空间形式和其他对称空间进行分类，对特殊度量（例如孤子和曲率张量的代数类型限制的度量）进行分类，以及理解与更高同质性的特殊完整度量相关的可积系统。在芬斯勒几何中，布莱恩特将继续发展具有恒定旗曲率的芬斯勒流形理论，无论是在构造新的例子还是在理解其测地流的性质方面；将特别研究与扭量理论和奇异完整结构的关系。 在复杂几何中，布莱恩特将继续研究表征在全纯接触 $3$ 折叠中有理接触曲线的模空间上导出的有理法向结构，目的是了解它们如何与可积系统及其曲率特性联系起来。 所有这些（以及其他相关项目）都会产生本身有用的结果，但它们也将促进外微分系统和等价方法中新技术的发展，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步都依赖于发展对这些问题的几何理解，即只关注那些与特定（通常是任意的）问题无关的方面的理解。选择）坐标系。 例如，爱因斯坦在能够有效地运用黎曼几何的无坐标概念来分离和描述重力作为时空曲率的本质之后，才提出了广义相对论。 然而，问题的无坐标表述通常会揭示需要通过非标准工具（例如微分系统理论）来处理的简并性。 布莱恩特的研究计划应用了微分系统的无坐标理论和等价方法，将导致对空间表面研究中出现的许多问题、由于额外或隐藏对称性而出现的特殊结构以及许多几何环境中曲率本身的性质产生更基本的几何理解。

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.