Final Project

You may work alone or with one other person. Choose a topic related to differential geometry as evidenced by its inclusion in a differential geometry text.
You project will be graded based on the depth of your differential geometry connections, and the clarity and creativity of:
  1. Maximum 3-Page Typed (Single-spaced) Review of Material We Covered in Class that is Related to your topic
    Include the relevant definitions, mathematical symbols and notation, pictures, theorems, demonstrations, and examples from class, homework and tests that relate to your topic. Part 1 is purely class review. Any new material belongs in part 2. For instance, say your extension incorporates Gauss curvature in some way. Then the new extension is in part 2, while a review of what we already did related to Gauss curvature belongs in part 1 (so part 1 will not cover all class topics--just the ones that relate). Some past students reported that they have found it helpful to think of part 1 as a review of class notes and hw as if they were studying for a final exam [without the exam component - instead the product is finding the connections we covered that relate to your topic].
  2. An extension of class work, which might be one of the following:
    1. summary of what you have learned (in your own words) after researching a topic [1 or 2 pages should be sufficient in many cases and it could be in bullet point or paragraph format]
    2. computer program you write and report back on how that went
    3. demo you create
    4. historical timeline you create
    5. classroom worksheet that you create
    6. the beginnings of a more extensive research project...
    7. lots of possibilities here - I encourage you to be creative
  3. An annotated reference list (to turn in). The annotations are brief comments about how you used each reference in your project. Most topics should utilize journal articles or books from the library and/or my office.
All components must be typed products that you create yourself in your own words, and that look professional and flow well. Mathematics symbols and notation should be typed in a program like LaTeX or Maple. Here is a grading rubric I'll use.

Here is a sample project for a different class (2240) by Russell Chamberlain and Dalton Cook. Here is Russell and Dalton's LaTeX file. If you want to use LaTeX, I can help, and you can copy their file into a site like Overleaf and modify it from there.

Research Session Presentations on the final exam day Bring a printed version of your all of your work. We will divide up the class into two sessions (half the class will stand next to their work as the other half examines the projects, and then we will switch roles). During your session, you must stand by your work to discuss your topic and answer questions. If you work with another person, they will be in the other session so you should be prepared to present the entire project. The presentation sessions are similar to research day at Appalachian, poster presentations at research conferences, or science fairs. In addition, when you are viewing other projects, you will conduct peer review and a self evaluation.

For 4141 Students In addition, 4141 students turn in a Capstone Project, which has different requirements. It can be on the same topic as your final project.

For 5500 Students In addition, 5530 students will research the literature (mathematics and/or physics and/or cs journals) and discuss some recent work, and if possible an open problem, that relates to your topic. Summarize what you found in your own words, and be sure to list the journal articles.

Here are some final project ideas, just to give you a sense of some possibilities:

  • See p. 453-454 of our textbook, which lists some final project ideas
  • 5.7 in our textbook: an industrial application of wrapping and unwrapping
  • Rudy Rucker's Software related to How Flies Fly
  • Designing a Baseball Cover - the article by Richard B. Thompson - The College Mathematics Journal, Vol. 29, No. 1 (Jan., 1998), pp. 48-61. Published by: Mathematical Association of America
  • Write a related computer program
  • Explore a curve: the Brachistrone or any other
  • Explore a theorem or topic from class or the textbook or a related idea.
  • Explore a related journal article, like The Klein Bottle as an Eggbeater by Richard L.W. Brown.
  • Subdivision Surfaces (Geometry and Computing) by by Jorg Peters and Ulrich Reif explores the connections between differential geometry and the popular technique for representing surfaces. For example in 2005, Tony DeRose won a Technical Achievement Academy Award (Oscar) for his work on subdivision surfaces at Pixar. DeRose said he was on "a mission to show kids how cool math and science can be...[and] this award will help get that message across."
  • Explore a Maple file related to differential geometry, such as Robert Jantzen's demo on geodesics on the torus
  • Compute the Christoffel symbols and Ricci curvature for your metric from the presentation by using commands similar to those found in Wormhole metric
  • Oddly shaped wheels for nonflat surfaces, like A Bicycle with Flower-Shaped Wheels
  • Spirograph parametrizations like Spirotechnics!
  • Economics and curvature
  • The Gauss map
  • Minimal surfaces
  • Schwarzschild solutions
  • Tensors
  • Developable surfaces
  • Best Way to Hold a Pizza Slice
  • Visualization in differential geometry
  • Differential geometry humor
  • Physics in differential geometry
  • Poincare conjecture (no longer a conjecture)