Applications of Symmetry & Invariants

该研究项目将致力于李群和Cartan等效方法在物理和工程中出现的各种问题的理论和应用。基础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。这些结果的应用可分为三大类。在计算机视觉中，基于微分几何扩散方程的图像处理新范式依赖于对视觉对称群的微分不变特征进行分类。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进展，包括被称为“上同调的量子化”的显著现象；在本项目中，重点将转移到真正的平面量子算子，以及扩展到一个完全三维的理论。在连续介质力学中，将进一步分析等效方法和外部微分系统在变分问题分类中的应用，以期为弹性中的非线性变分问题提供标准形式、几何不变量和守恒定律。该研究项目的目标是进一步理解三维空间中对称和不变量的数学理论，受到几个关键物理应用的启发，在物理、工程和医学中具有重要意义。在计算机视觉领域，图像处理的新方法有望对医学成像产生直接的实际影响，例如乳房肿瘤的超声检测。在弹性学中，一般材料的简单形式和对称性的分类对裂纹扩展和波的研究具有重要意义。在量子力学中，一些新类型的问题，其中一些基本状态是由代数工具分类的，已知应用于分子光谱学和散射理论。

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.