Applications of Moving Frames

本项目的目的是进一步发展和扩大研究人员的移动标架等变法在有限维群作用和无限维群作用中的应用范围。建议中的具体研究领域如下：发展微分不变代数的计算算法，特别强调确定最小生成集的方法；利用基于移动标架的不变变分演算来分析变分问题的极小项和不变子流形的行为，重点研究微分不变量及其特征的演变；分析对称伪群的结构，它们在分类和寻找非线性偏微分方程显式解的算法，以及在流体力学、规范理论、可积系统和气象模型中的应用；基于微分不变量和联合不变量签名方法，进一步开发用于数字图像中目标识别和对称性检测的实用工具；以及积分非线性微分方程和逼近曲率、挠率等微分不变量的新的不变数值算法的探索和应用。对称性、等价性和不变性的开发仍然是处理出现在广泛应用中的复杂非线性系统的最重要的数学方法之一。这位研究人员对基于几何的运动标架方法的重新表述导致了一系列新的、迄今尚未得到充分利用的计算工具，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、材料科学和物理量规理论、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通用的组动作及其不变量，直接受到这些应用领域的当前需求的驱动。

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.