Project

Objective

You’ll form small groups and explore a topic of choice that has some connection to real analysis or differential geometry. 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.

Description

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

1. A document (in LaTeX, ideally) containing an original exposition of the topic of your choice.
   • It should contain precise definitions of any concepts involved, at least one substantial theorem and its proof, and a few examples that illustrate the concepts involved.
   • By “original exposition,” I mean that you don’t need to prove a whole new theorem, but everything should be written up in your own words.
   • Pictures are always a great idea.
   • You should also cite references that you used.
   • If you want to do something involving applications and “at least one theorem and its proof” doesn’t really make sense for what you have in mind, that’s perfectly fine! Just talk to me first and we’ll figure what does make sense for what you have in mind.
   • 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 whatever is 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, MA375 and 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. 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. If you’d like my help finding references for you to look at for any of these topics, just ask! I’ll keep adding to this list as I think of more.

• Single variable calculus
  • Power rule for arbitrary exponents (including a rigorous construction of the real numbers)
  • Volterra’s construction (of a differentiable function \(f\) such that \(f'\) is continuous on an arbitrary dense \(G_\delta\) set)
  • Pompeiu’s construction (of a non-constant differentiable function \(f\) such that \(f' = 0\) on a dense subset of the domain)
  • Newton’s method (or other algorithms in numerical analysis)
  • L’Hôpital’s rule
  • Fundamental theorem of calculus (including a rigorous construction of the Riemann integral)
• Multivariable calculus
  • Morse’s lemma
  • Implicit function theorem
  • Constant rank theorem
• Manifolds
  • Exotic manifold structures (eg, exotic \(\mathbb{R}^4\)s, or exotic spheres)
  • Hairy ball theorem