Applications of Moving Frames

该项目的目的是进一步发展和扩展研究者的运动框架的等变方法的应用范围，用于有限维和无限维群体动作。具体的研究领域如下：微分不变代数的计算算法的发展，特别强调确定最小生成集的方法；利用基于运动框架的不变变分微积分来分析不变变分问题的极小解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，它们在非线性偏微分方程的分类和显式解中的应用，在流体力学、规范理论、可积系统和气象模型中的应用；基于微分不变和联合不变签名方法的数字图像目标识别和对称检测实用工具的进一步开发探索和应用新的不变数值算法来积分非线性微分方程和近似微分不变量，如曲率、扭转等。利用对称性、等价性和不变性仍然是处理复杂非线性系统的最重要的数学方法之一，在各种应用中出现。研究者对基于几何的移动框架方法的重新制定导致了广泛的新的和尚未开发的计算工具，以及新的和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，界面流动，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性，直接由这些应用领域的当前需求驱动。

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.