Lie Theory and Continuous Symmetry in the Undergraduate Curriculum

Roger E. Howe DUE 9555134耶鲁大学1996财政年度$ 77,373康涅狄格州纽黑文1997财政年度$ 59,742财政年度1998财政年度$ 37,673课程与课程数学题目：本科课程中的李论与连续对称虽然对称的概念在数学的群概念中得到了正式体现，在过去的一个世纪里，它一直是数学和物理学的主要推动力，但到目前为止，它在本科课程中只得到很少的体现。群论通常作为抽象代数课程的一部分出现，主要考虑的群是有限群。另一方面，影响我们生活和宇宙结构的主要群体是连续群体（李群）。这个项目开始通过创建一个展示连续对称的统一性质的课程来补救这种情况。本课程将从讨论与物理空间概念直接相关的对称性开始：膨胀、刚性运动、相似性。然后将介绍线性代数的基本概念，并讨论2、3和4维的几何问题，最后将介绍狭义相对论。示例和应用将贯穿始终。本课程将强调互动式教学，并将包括有小组活动可能性的扩展项目。一个重要的项目组成部分将涉及与PIMMS合作，PIMMS是康涅狄格州领先的数学和科学教育工作。通过这一努力，该项目将影响职前教师群体。一篇课文连同计算机实验室材料一起计划。这项工作的总体目标是让学生接触到过去一个世纪以来数学知识进步的主要方向，并将数学教育改革的思想进一步融入本科课程。特别地，建议的课程将是一个有吸引力的替代通常的二年级水平线性代数课程。

Roger E. Howe DUE 9555134 Yale University FY1996 $ 77,373 New Haven , CT 06520 FY1997 $ 59,742 FY1998 $ 37,673 Course and Curriculum Mathematics Title: Lie Theory and Continuous Symmetry in the Undergraduate Curriculum Project Summary: Although the idea of symmetry, which is embodied formally in the mathematical concept of group, has been a major driving force in mathematics and physics during the past century, it is so far only meagerly represented in the undergraduate curriculum. Group theory usually appears as part of a course in abstract algebra, and the main groups considered are finite groups. On the other hand, the main groups which affect our lives and the structure of the universe are continuous groups (Lie groups). This project begins to remedy this situation by creating a course which exhibits the unifying nature of continuous symmetry. The course will begin by discussing symmetries which are directly related to concepts of physical space: dilations, rigid motions, similarities. It will then introduce the basic concepts of linear algebra, and discuss geometric issues in 2, 3 and 4 dimensions, culminating in the special theory of relativity. Examples and applications will be emphasized throughout. The course will emphasize interactive teaching, and will include extended projects with possibilities for group activities. An important project component will involve working with PIMMS, the leading mathematics and science education effort in the State of Connecticut. Through this effort the project will impact the pre-service teacher population. A text is planned along with computer laboratory materials. The overall goal of this work is to put students in touch with a major direction of intellectual advance in mathematics over the past century, and to take the ideas of mathematics education reform f urther into the undergraduate curriculum. In particular, the proposed course will be an attractive alternative to the usual sophomore-level linear algebra course.