Lie Theory and Continuous Symmetry in the Undergraduate Curriculum

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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.