Applications of Moving Frames

移动框架的应用

• 批准号: 0807317
• 负责人: Peter Olver
• 金额: $ 29.01万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807317&HistoricalAwards=false
• 关键词: Applications; Moving; Frames

OlverDMS-0807317 The aim of this project is to further develop and extend therange of applications of the investigator's equivariant method ofmoving frames, for both finite-dimensional andinfinite-dimensional group actions. The specific areas ofresearch in the proposal are as follows: Development ofcomputational algorithms for differential invariant algebras,with particular emphasis on methods for determining minimalgenerating sets; exploitation of the invariant variationalcalculus based on moving frames to analyze minimizers ofinvariant variational problems and the behavior of invariantsubmanifold flows, focusing on the evolution of differentialinvariants and their signatures; algorithms for analyzing thestructure of symmetry pseudo-groups, their use in classifying andfinding explicit solutions to nonlinear partial differentialequations, with applications in fluid mechanics, gauge theories,integrable systems, and meteorological models; furtherdevelopment of practical tools for object recognition andsymmetry detection in digital images based on differentialinvariant and joint invariant signature methods; and explorationand application of new invariant numerical algorithms forintegrating nonlinear differential equations and approximatingdifferential invariants such as curvatures, torsion, etc. Exploitation of symmetry, equivalence and invarianceproperties remains one of the most important mathematical methodsfor tackling the complex nonlinear systems that arise in a widevariety of applications. The investigator's reformulation of thegeometrically-based method of moving frames has led to a widerange of new and, as yet, underexploited computational tools, aswell as new and unexpected areas of application, includingcomputer vision, recognition and surveillance, materials scienceand physical gauge theories, fluid mechanics, interface flows,DNA mechanics, relativity, and elsewhere. The project continuesto develop this powerful new approach to analyzing very generalgroup actions and their invariants, directly driven by currentneeds in these areas of application.

OlverDMS-0807317 该项目的目的是进一步开发和扩展研究者的移动框架等变方法的应用范围，适用于有限维和无限维群动作。 该提案的具体研究领域如下：微分不变代数计算算法的开发，特别强调最小生成集的确定方法；利用基于移动框架的不变变分微积分来分析不变变分问题的最小化和不变子流形流的行为，重点关注微分不变量及其特征的演化；分析对称伪群结构的算法，它们在分类和寻找非线性偏微分方程的显式解中的应用，以及在流体力学、规范理论、可积系统和气象模型中的应用；进一步开发基于微分不变和联合不变签名方法的数字图像中的对象识别和对称性检测实用工具；探索和应用新的不变数值算法来积分非线性微分方程并近似微分不变量，如曲率、扭转等。利用对称性、等价性和不变性仍然是解决各种应用中出现的复杂非线性系统的最重要的数学方法之一。 研究人员对基于几何的移动框架方法的重新表述催生了一系列新的、尚未充分利用的计算工具，以及新的、意想不到的应用领域，包括计算机视觉、识别和监视、材料科学和物理规范理论、流体力学、界面流、DNA 力学、相对论等。 该项目继续开发这种强大的新方法来分析非常普遍的群体行为及其不变量，直接由这些应用领域的当前需求驱动。