Final Project

Objective

You’ll form small groups and explore a topic of choice that has some connection to differential equations. It can be something purely mathematical, or it can involve an application to an area of your choice. Then you’ll share what you’ve learned with your peers

The final product you’ll be working towards will have two parts:

1. A typewritten document containing an original exposition of the topic of your choice.
   • It should give an introduction to the topic, contain precise definitions of any new concepts involved, and incorporate a detailed and substantial proof and/or example.
   • By “original exposition,” I mean that you don’t need to prove a whole new theorem or something like that, but everything should be written up in your own words.
   • Pictures are always a great idea!
   • You should cite references that you used.
   • If something I’ve mentioned above doesn’t really make sense for what you have in mind, that’s fine! Just talk to me first and we’ll figure what does make sense for what you have in mind.
   • It’s not required, but I strongly suggest using LaTeX. There’s a bit of a learning curve, but the end result looks far more polished and professional than anything you can accomplish using other word processing software. It’s probably a useful skill to have going forward.
   • Send the document to me as a PDF before the start of class on the last day of class (fourth Wednesday).
2. A presentation to share the highlights of the topic with the rest of the class.
   • It should take 10-15 minutes.
   • You don’t need to go through all the details that you discuss in your document, but you should work through some illustrative examples.
   • Pictures are always a great idea.
   • You could do slides (“Beamer” is the TeX way of doing slides), or you could give a chalk talk. Or do whatever is most appropriate for what you have in mind. Feel free to be creative!
   • You’ll present during class on the last day of class (fourth Wednesday).

For both of these, your intended audience is one of your classmates outside of your group: someone who’s familiar with topics we’ve discussed in our class and has taken all of the prerequisite classes (eg, MA220), but not necessarily anything more than that.

There will be two deadlines related to the project before the last day. They will show up on the course calendar. Here’s what you need to have done by each of the deadlines.

• Proposal deadline: Nothing fancy is required. Just form your group (of 2-3 people), pick out a topic, and then one person in your group should send me an email by the deadline saying who’s in the group and what you’ll be working on. I’ll want different groups to working on different topics, so I may suggest some re-organizing.

• Draft deadline: I’d like to see a a draft of the document. It doesn’t need to be complete, of course. It might just contain some of the basic definitions and theorem statements, perhaps an example or a picture. It could be a rough outline. I’m mostly looking for an indication that you’ve started thinking about your project topic in earnest.

Grading

Your project score will be based on the following criteria.

• Content (out of 4 points). There should be correct and substantial mathematical content involved (during the presentation, or on the handout).
• Clarity (out of 3 points). The topic should be clearly explained and logically organized, in a way that is appropriate for the intended audience.
• Participation (out of 3 points). You’ll get full credit on participation if it is clear that everyone in the group understands the topic at the same level (in particular, everyone should have a substantial role in the presentation).

Possible Topics

There are many possibilities! I encourage you to be creative. I only ask that it not be something that we’ve spent time discussing as a class.

If you’re struggling to find things, here’s a short list of things that I think might make interesting projects. I’ll keep adding to this list as I think of more.

• Pendulums.
• Lotka-Volterra model for predator-prey systems.
• Compartmental models in epidemiology (besides SIR).
• Runge-Kutta methods.
• Bifurcation.
• Laplace transform.
• Lyapunov stability.
• Ricatti equation.
• Sturm-Liouville theory.
• Chaos theory.
• Singularities of linear homogeneous ODEs.
• Vector fields on manifolds.
• Proof of Picard’s existence and uniqueness theorem for first order ODEs.
• Proof of the fact that a homogeneous first order linear system with \(n\) equations and variable coefficients has a solution space thats \(n\) dimensional (this involves a multidimensional generalization of Picard’s existence and uniqueness theorem for first order ODEs).