Isometric Embedding and Other Problems in Geometry and Differential Equations

项目编号：DMS 1206272，首席研究员：Jeanne Nielsen clelland首席研究员计划应用外微分系统理论和Cartan的等效方法，单独或与其他合作者研究微分几何和微分方程中的几个问题。她将与陈桂强、Marshall Slemrod、王德华和Deane Yang合作，研究高维黎曼流形在欧几里得空间等距嵌入的局部存在性问题，结合弹性静力学和双曲守恒定律的思想，寻找保证等距嵌入局部存在的条件。她将继续与Thomas Ivey共同研究某些类别偏微分方程的Backlund变换的分类结果，特别关注自动Backlund变换，它可以用来从同一偏微分方程的已知解中生成新的偏微分方程解。她将扩展她与Christopher Moseley和George Wilkens在sub-Finsler几何方面的合作，并研究与仿射分布相关的度量结构的几何性质，以及最优控制理论的潜在应用。在与她以前的博士生马修·斯塔克波尔的联合工作中，她将研究控制系统之间的动态等价现象，重点是寻找计算不变量的方法来区分动态不等价系统。在与她以前的硕士学生Nathaniel Bushek的合作中，她将研究5维中仿射空间中表面的几何性质，包括齐次例子和其他特殊族的分类。微分方程有广泛的应用，从工程和物理到生物和金融，仅举几例。在所有提议的项目中，一个共同的主题是使用几何技术来研究微分方程的结构特征，这些特征可能被在特定坐标选择中的方程表达所掩盖。这种方法由Elie Cartan在20世纪早期首创，在进一步理解许多类型的微分方程及其解方面取得了巨大成功。由于在蛋白质折叠科学中的潜在应用，该研究解决了DARPA最近提出的23个数学挑战之一。本文研究的是可积系统，其应用包括在信号质量失真最小的情况下进行长距离有效的信号传播。拟建的仿射分布和动态等价项目与控制理论有关，其应用包括机器人和量子计算等领域。

AbstractAward: DMS 1206272, Principal Investigator: Jeanne Nielsen ClellandThe principal investigator plans to apply the theory of exterior differential systems and Cartan's method of equivalence to study several problems in differential geometry and differential equations, both individually and with various collaborators. In joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang, she will investigate the local existence problem for isometric embedding of higher-dimensional Riemannian manifolds into Euclidean space, incorporating ideas from elastostatics and hyperbolic conservation laws to find conditions which guarantee local existence of isometric embeddings. She will continue her joint work with Thomas Ivey on classification results for Backlund transformations for certain categories of PDEs, paying particular attention to auto-Backlund transformations, which may be used to generate new solutions of a PDE from a known solution of the same PDE. She will expand upon her joint work with Christopher Moseley and George Wilkens on sub-Finsler geometry, and investigate geometric properties of metric structures associated to affine distributions, as well as potential applications to optimal control theory. In joint work with her former Ph.D. student Matthew Stackpole, she will investigate the phenomenon of dynamic equivalence between control systems, with a focus on finding ways to compute invariants to distinguish between dynamically inequivalent systems. In joint work with her former M.A. student Nathaniel Bushek, she will investigate geometric properties of surfaces in 5-dimensional centroaffine space, including the classification of homogeneous examples and other special families.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. The proposed research on isometric embedding addresses one of the 23 mathematical challenges recently posed by DARPA, due to potential applications to the science of protein folding. The proposed research on Backlund transformations is related to integrable systems, whose applications include efficient signal propagation over long distances with minimal distortion in signal quality. The proposed projects on affine distributions and dynamic equivalence are related to control theory, whose applications include areas such as robotics and quantum computing.