Topics in the Geometry of Differential Equations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本专题将应用外微分系统的几何理论和嘉当的等价方法来解决微分方程中出现的各种问题。 Clelland将继续研究分芬斯勒几何，最初介绍了Clelland和克里斯托弗莫斯利作为一个自然推广的分黎曼几何与应用控制理论。 此外，Clelland将研究更一般类别的控制仿射系统（系统与“漂移”）的几何形状，特别注意系统的几何形状的非常数型，其中漂移向量场消失沿着一个杰出的子流形的状态空间。 最初由埃尔金引入的“t-协分布”概念，预计将有助于将Cartan的等效方法应用于这一重要情况。 这项研究最终将包括介绍和分析度量结构类似于子黎曼和/或子芬斯勒几何等系统，与控制仿射系统的最优控制的研究应用。 Clelland将继续她的工作几何的Backlund变换，重点是几何意义的任意参数，出现在许多（但不是全部）Backlund变换，以及双曲Monge-Ampere偏微分方程的“自Backlund变换”的分类，其中涉及一个双曲蒙日的解决方案，微分方程有着广泛的应用，从工程和物理到生物学和金融，仅举几例。 所有拟议项目的一个共同主题是使用几何技术来研究微分方程的结构特征，这些特征可能会因方程在特定坐标系中的表达而变得模糊。 这种方法，在世纪早期由埃利·嘉当开创，在促进对许多类型的微分方程及其解的理解方面取得了巨大的成功。 随着对亚芬斯勒几何和控制仿射系统的拟议研究，Clelland将把这些技术应用于与控制理论相关的问题的研究，其应用包括机器人和量子计算等领域。 随着对Backlund变换的拟议研究，Clelland将研究与可积系统相关的问题，其应用包括长距离有效的信号传播，信号质量失真最小。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project will apply the geometric theory of exterior differential systems and Cartan's method of equivalence to a variety of problems arising in differential equations. Clelland will continue the study of sub-Finsler geometry, originally introduced by Clelland and Christopher Moseley as a natural generalization of sub-Riemannian geometry with applications to control theory. In addition, Clelland will study the geometry of the more general category of control-affine systems (system with "drift"), with particular attention to the geometry of systems of non-constant type, where the drift vector field vanishes along a distinguished submanifold of the state space. The notion of a "t-codistribution," originally introduced by Elkin, is expected to facilitate the application of Cartan's method of equivalence to this important case. This study will eventually include the introduction and analysis of metric structures akin to sub- Riemannian and/or sub-Finsler geometry for such systems, with applications to the study of optimal control for control-affine systems. Clelland will continue her work on the geometry of Backlund transformations, with emphasis on the geometric significance of the arbitrary parameter that appears in many (but not all) Backlund transformations, as well as the classification of "auto-Backlund transformations" for hyperbolic Monge-Ampere PDEs, which relate solutions of one hyperbolic Monge-Ampere PDE to additional solutions of the same PDE.Differential equations have an enormous range of applications, from engineering and physics to biology and finance, just to name a few. A common theme in all the proposed projects is the use of geometric techniques to study structural features of differential equations which may be obscured by the expression of an equation in a particular choice of coordinates. This approach, pioneered in the early 20th century by Elie Cartan, has enjoyed great success in furthering the understanding of many types of differential equations and their solutions. With the proposed research on sub-Finsler geometry and control-affine systems, Clelland will apply these techniques to the study of problems related to control theory, whose applications include areas such as robotics and quantum computing. With the proposed research on Backlund transformations, Clelland will study problems related to integrable systems, whose applications include efficient signal propagation over long distances with minimal distortion in signal quality.