Washington and Lee University

The role of geometry in the education of our math majors has sadly fallen by the wayside. This is unfortunate as geometry is beautiful, practical, and its development over the centuries up to the present day has required major efforts by some of the greatest mathematicians. Below I quote from the famous mathematician H. Busemann as he discusses the decline of geometry in the curriculum:

The focus of most undergraduate math curricula is on algebra and analysis, which is to be expected as these two subjects introduce powerful tools that are used throughout all of mathematics. Few departments have the faculty resources to offer a wide spectrum of courses in geometry, and typically can only offer a course in classical geometry (primarily aimed at majors with an interest in secondary education) and/or differential geometry.

The purpose of this project, a portion of which is near completion, is to make publicly available a comprehensive series of web sites for discovery-based, interactive courses on geometry. The courses would be modular in design to facilitate ease of use by faculty worldwide. They will be offered here at W&L as part of our independent study program in modern geometry, allowing the student to choose those branches of geometry of greatest interest to her. Among its aims are: to exercise the student in the visualization of mathematical concepts, to present a unified view of classical and modern geometry, to develop and exercise problem-solving abilities, and to give the student a sense of the unity of mathematics by relating geometry to other fields, both in and outside of pure mathematics

Ideas and definitions at each step in the learning process will be both motivated and illustrated, sometimes via interactive and dynamic geometric tools. The theory will then proceed to develop through problems, reading assignments, and projects. Some of these problems will take the form of missing steps in interactive exercises written in HTML and Java. The students will then add their own contributions to the web course as appropriate (a beautiful new solution, an interesting find in an old journal, a surprising application, etc.).

The first course (on finite geometry) is now nearly complete, and can be seen by clicking on the appropriate link on my homepage at www.wlu.edu/~mcraea. The next course or courses would be completed during the summer of 2002, and then could be used in my winter 2003 offering of geometry.

The following software is available for my use: Adobe Acrobat, Adobe AfterEffects, Adobe GoLive, Adobe Illustrator, Adobe LiveMotion, Painter, and Macromedia Web Design Studio. Since I am comfortable with all of these applications, I will not need to take advantage of the Fellow-In-Residence program.

In the summer of 2001 I received the following grants from Washington and Lee:

1. R. E. Lee Student Research Grant (provided support for one student researcher)
2. Glenn Grant (provided a small stipend and funds to purchase software)
3. Culpepper Grant (provided a small stipend and funds to purchase software)

I intend to apply for both a R. E. Lee Student Research Grand and a Glenn Grant for the summer of 2002.

Geometry allows the student to see the beauty and unity of mathematics, the 'big picture' if you will. Instead of a course with unmotivated general definitions, few examples, and little or no ties to other courses, my web courses in geometry are about those examples that are central in mathematics and have inspired generations of mathematicians. The student of geometry acquires a greater appreciation for mathematics as a creative act. And they would get to share their work with the world by contributing to the web page.

In addition there is further educational value to a student who helps me build and add to the web courses. She has the opportunity to learn enough about a field of mathematics not only to know what some of the unanswered questions are, but also to start thinking about them. In addition to her intellectual pursuits (reading, problem solving, questioning, generalizing), she may think about how best to relate what she has learned to others within the context of a web environment.

Web courses for the first four topics listed below are ultimately to be written. The first course is nearly complete, and attention needs to be given to the next three, as very few books suitable for undergraduate study are available for these subjects, while there are many excellent resources in projective, algebraic, classical, and differential geometry. I would ultimately like to build a web course in space-time geometry in support of a course on time machines that I teach every few years here at W&L. These ten independent courses cover the subject of geometry fairly well, and would easily fit into the curriculum of most math undergraduate programs. Faculty outside of W&L could use these web materials either within their own courses or as independent study courses.

1. Finite Geometry
2. Integral Geometry and Geometric Probability
3. Convex Geometry
4. Minkowski Geometry
5. Space-Time Geometry
6. Projective Geometry
7. Geometry of Singularities
8. Algebraic Geometry
9. Classical Geometry
10. Differential Geometry

I recently gave a demonstration of a typical web lecture (in projective geometry) at a joint meeting of the American Mathematical Society and the Mathematical Association of America. Afterwards I was asked by the editor of the MAA on-line journal JOMA to contribute the presented module for publication. I believe that the time is ripe, not only to re-introduce geometry into the math curriculum in a vigorous way, but to support this endeavor with Java and other web-based technologies, and to disseminate the project as widely as possible. Assessment by peers and students is vital, and will be accomplished as follows:

The average scientific article is read by only two people. My students will have the chance to be a part of a project that could be viewed and used by thousands. All web courses for this project are to be publicly available using W&L's servers. In addition, material of this project is to be used to further