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:
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.
- 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].
- An extension of class work, which might be one of the following:
- 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]
- computer program you write and report back on how that went
- demo you create
- historical timeline you create
- classroom worksheet that you create
- the beginnings of a more extensive research project...
- lots of possibilities here - I encourage you to be creative
- 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
Here is a
sample project for a different class (2240)
Russell Chamberlain and Dalton Cook.
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 final paper,
which I would recommend is on the same topic as your final project.
For 5530 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
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
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
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
Oddly shaped wheels for nonflat surfaces, like
A Bicycle with Flower-Shaped Wheels
Spirograph parametrizations like
Economics and curvature
The Gauss map
Best Way to Hold a Pizza Slice
Visualization in differential geometry
Differential geometry humor
Physics in differential geometry
Poincare conjecture (no longer a conjecture)