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)提供资金的。这个项目将应用外微分系统的几何理论和Cartan的等价法来解决微分方程中出现的各种问题。Clelland将继续研究亚Finsler几何，最初由Clelland和Christopher Moseley引入，作为亚Riemannian几何的自然推广，并应用于控制理论。此外，Clelland将研究更一般类别的控制-仿射系统(具有“漂移”的系统)的几何，特别关注非常数类型的系统的几何，其中漂移向量场沿着状态空间的一个独特的子流形消失。“t-协分布”的概念最初是由埃尔金提出的，预计将促进卡尔坦的等价法在这一重要情况下的应用。这项研究最终将包括引入和分析这类系统的类似次黎曼几何和/或次Finsler几何的度量结构，并将其应用于控制-仿射系统的最优控制研究。Clelland将继续她在Backlund变换几何学方面的工作，重点是在许多(但不是所有)Backlund变换中出现的任意参数的几何意义，以及双曲Monge-Ampere偏微分方程组的“自动Backlund变换”的分类，它将一个双曲Monge-Ampere偏微分方程组的解与相同PDE的其他解联系起来。微分方程组有巨大的应用范围，从工程和物理到生物和金融，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式的结构特征，这些特征可能被在特定坐标选择下的方程式表达式所掩盖。这种方法是由Elie Cartan在20世纪初提出的，在加深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Finsler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，其应用包括在信号质量方面具有最小失真的长距离有效信号传播。

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.