Mathematics is an extraordinary human achievement. This course will survey important ideas from the major areas of mathematics. Topics will be selected on the basis of their beauty and lasting impact. The goal will be to develop a sense of what mathematics is - its vitality and variety. Students will be required to write commentaries, prepare lab reports, and complete a final paper.
Please note topic and course description in catalog are incorrect.
Topic for fall 1998: Differential Geometry
In Differential Geometry, we continue the study of curves and surfaces we began in Calculus and perhaps Advanced Calculus. We use a blend of approaches -- intuitive, visual, computational, analytic. There are two kinds of properties of the objects we study: local properties, which depend only behavior in a neighborhood of a point (e.g. derivatives, curvature, tangent lines and planes), and global properties, which depend on the entire surface (e.g. orientability, compactness).
We will use a new textbook, Differential Geometry: A Geometric Introduction (Prentice Hall, 1998). The approach here is an attempt to blend the intuition and the analysis more successfully than is sometimes done. It begins with a long chapter of examples intended to form the basis of the development of both the intuition and the formal tools required for understanding curves, surfaces, and other objects - both in Euclidean space and abstractly defined. The approach is "problem-based" and there is plenty of room for exploration. I would hope to study the first six chapters thoroughly and to have students investigate an additional topic independently or in a group. There are some programs and problems (using Maple) in the text, but it seems they are relatively independent of the text, and that students with a fair amount of computer experience could develop even more interesting ways to use the computer to understand and describe the subject. The course may be of interest to majors in Physics and Computer Science as well as to Mathematics Majors.