Symbolic Group-Invariant Computation

自然界中观察到的现象通常具有对称性，描述它们的微分、积分和代数方程系统也是如此。然而，已经观察到，对称方程通常是最具挑战性的处理标准符号计算算法。群体行为提供了对称的数学描述。不变量，即不受群体行为影响的量，在设计这种算法时起着至关重要的作用。符号群不变量计算包括计算不变量，用不变量重写问题，并用不变量处理问题。长期目标是开发和实现利用对称信息的符号算法，并且足够通用，可以适用于数学，科学和工程中的广泛问题，涉及群体行为，对称性和不变量。研究者通过让研究生和大四本科生参与该项目，将她的研究和教育目标结合起来。本研究的最初重点是以下三个开放性问题。(1)研究者致力于设计和实现经过各种几何变换的几何形状的鲁棒分类算法。计算机图像识别问题为本课题提供了动力。该方法是基于积分不变量，而不是更传统的微分不变量。发展积分不变量的基础理论是这个项目的一部分。(2)研究变量线性变化多项式的分类算法的开发和实现。这个长期开放的问题是解决使用新颖的代数公式的移动框架方法。除了理论兴趣外，这些算法在多项式求解和信号处理方面也有应用。(3)研究微分方程和变分问题的自动对称约简算法的开发和实现。目标是利用对称性来获得显式解，或者至少是关于解集的重要信息。该方法结合了运动帧法、微分代数算法和同调代数技术。

Phenomena observed in nature often have symmetry properties, and so do the systems of differential, integral, and algebraic equations describing them. It has been observed, however, that symmetric equations are often the most challenging for processing by standard symbolic computation algorithms. Group actions provide a mathematical description of symmetries. Invariants, that is the quantities that are unaffected by the group action, play a crucial role in designing such algorithms. Symbolic Group-Invariant Computation involves computing invariants, rewriting a problem in terms of invariants, and processing the problem in terms of invariants. The long term objective is to develop and implement symbolic algorithms that take advantage of the symmetry information, and are general enough to be applicable to a wide range of problems in mathematics, science and engineering, that involve group actions, symmetries and invariants. The investigator integrates her research and educational objectives by involving graduate and senior undergraduate students in the project.The initial focus of this research is on the following three open problems. (1) The investigator works on designing and implementing robust classification algorithms for geometric shapes undergoing various geometric transformations. The problem of computer image recognition provides motivation for this project. The method is based on integral invariants, rather than on the more traditional differential invariants. Development of the underlying theory of integral invariants is a part of this project. (2) The investigator works on the development and implementation of the classification algorithms for polynomials undergoing linear changes of variables. This long-standing open problem is addressed using a novel algebraic formulation of the moving frame method. Besides theoretical interest, such algorithms have application to polynomial solving and to signal processing. (3) The investigator works on the development and implementation of the automated symmetry reduction algorithms for differential equations and variational problems. The goal is to exploit the symmetries in order to obtain either explicit solutions or, at least, vital information about the solution set. The approach is based on the combination of the moving frame method, differential algebra algorithms and homological algebra techniques.