Applications of Symmetry & Invariants

对称性的应用

• 批准号: 9500931
• 负责人: Peter Olver
• 金额: $ 11万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-15 至 1998-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500931&HistoricalAwards=false
• 关键词: Applications; Symmetry; & Invariants

DMS-9500931 Olver, Peter The research project will be devoted to the theory and applications of Lie groups and the Cartan equivalence method to a variety of problems arising in physics and engineering. Basic theoretical research goals include the classification of the Lie transformation groups in three-dimensional space, their differential invariants, and the associated Lie algebra cohomology. The applications of these results fall into three broad categories. In computer vision, a new paradigm of image processing based on differential geometric diffusion equations relies on classifying the differential invariant signatures of visual symmetry groups. In quantum mechanics, the analysis of quasi-exactly solvable Schrodinger operators continues to produce significant new developments in the study of Lie algebras of differential operators, including the remarkable phenomenon known as "quantization of cohomology"; in the present project, the concentration will shift to real planar quantum operators, as well as extensions to a fully three-dimensional theory. In continuum mechanics, the application of equivalence methods and exterior differential systems to the classification of variational problems will be further analyzed with a view to providing canonical forms, geometric invariants, and conservation laws for nonlinear variational problems arising in elasticity. The goals of this research project are to further our understanding of the mathematical theory of symmetry and invariants in three-dimensional space, motivated by several key physical applications, of importance in physics, engineering and medicine. In computer vision, new methods of image processing promise to have an immediate practical impact in medical imaging, such as the ultrasound detection of breast tumors. In elasticity, the classification of simple forms and symmetries of general materials should have significant consequences in the study of crack propagation and waves. In quantum mechanics, new types of problems some of whose fundamental states are classified by algebraic tools have known applications to molecular spectroscopy and scattering theory.

DMS-9500931 Olver，Peter该研究项目将致力于李群的理论和应用以及Cartan等价方法在物理和工程中出现的各种问题。基本的理论研究目标包括三维空间中李变换群的分类，它们的微分不变量，以及相关的李代数上同调。这些结果的应用分为三大类。在计算机视觉中，基于微分几何扩散方程的图像处理的新范例依赖于对视觉对称群的微分不变签名进行分类。在量子力学中，准精确可解薛定谔算子的分析继续在微分算子的李代数研究中产生重要的新发展，包括被称为“上同调量子化”的显着现象;在本项目中，重点将转移到真实的平面量子算子，以及扩展到全三维理论。在连续介质力学中，将进一步分析等价方法和外微分系统在变分问题分类中的应用，以期为弹性力学中出现的非线性变分问题提供规范形式、几何不变量和守恒定律。 该研究项目的目标是进一步了解三维空间中对称性和不变量的数学理论，其动机是几个关键的物理应用，在物理学，工程学和医学中具有重要意义。在计算机视觉中，新的图像处理方法有望对医学成像产生直接的实际影响，例如乳腺肿瘤的超声检测。在弹性力学中，一般材料的简单形式和对称性的分类在裂纹扩展和波的研究中具有重要意义。在量子力学中，一些基本状态被代数工具分类的新类型问题在分子光谱学和散射理论中有已知的应用。