Mathematical Sciences: Applications of Lie Theory to Control, Computer Science and Physics

1)线性微分系统的可积性理论和非线性控制理论中反馈线性化方程的摄动理论。这项工作将支持由George Meyer博士指导的NASA-Ames研究中心的航天器非线性控制工程研究计划。这项研究在最近的NSF控制理论报告中被引用为几何方法论应用的实际成功的主要例子。2)计算机科学中知识表示和计算理论方面的几何和层论方法论。这将为离散事件系统和人工智能的研究提供数学支持。3)数值分析中的李论方法为线性代数和微分方程组的数值算法发展了新的几何和李论方法。4)基本粒子物理中的无限伪群，利用无限李伪群结构的Cartan-Matshushima-Guillmin-Sternberg-Conn理论，寻找弦理论和标准模型规范理论的自然数学合成。

This research will focus on four areas: 1) integrability theory for linear differential systems and perturbation theory for the feedback linearization equations of nolinear control theory. This work will support the program of engineering research of nonlinear control of air-space-craft at NASA-Ames Research Center directed by Dr. George Meyer. This research was cited in the recent NSF Report on Control Theory as a prime example of the practical success of the application of geometric methodology. 2) Geometric and sheaf-theoretic methodology in the knowledge-representation and computation theory side of computer science. This will provide mathematical support for the Discrete Event System and Artificial Intelligence research. 3) Lie- theoretic methods in numerical analysis developing new geometric and Lie-theoretic methodology for the numerical algorithms of linear algebra and differential equations. 4) Infinite pseudogroups in elementary particle physics looking for the ntural mathematical synthesis of String and Standard Model gauge theories, utilizing the Cartan-Matshushima-Guillemin-Sternberg- Conn theory of structure of infinite Lie pseudogroups.