The role of geometry in the education of our math majors has sadly fallen by the wayside. Below I quote from the famous mathematician H. Busemann as he discusses the decline of geometry in the curriculum:
. . . do we have the right to completely disrupt historical continuity whenever a subject moves out of the focus of contemporary interest? Do we really expect or agree that our present mathematical efforts will be altogether junked (at least from courses) as soon as the interests change? This disregard for historical continuity obviously leads to dangerous absurdities if taken seriously and must be fought."
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 differential geometry.
This web site on geometric probability is part of a large project to make publicly available a comprehensive series of web sites for discovery-based, interactive courses on geometry. The web courses will be written by students for students, and are to be modular in design to facilitate ease of use by students 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.
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 an interactive exercise written in HTML and Java. The students can then submit their own contributions to the web course as appropriate (a beautiful new solution, an interesting find in an old journal, a surprising application, etc.)..
In addition there is further educational value to any student who helps 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.
Curriculum
1. Finite Geometry
2. 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