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The best way to understand mathematics is to learn and discover it on one's own.
The best way to describe this project is to imagine that you are responsible for creating a "Mathematical Idea You Really Need to Know".
You will select a mathematical topic outside of (or beyond) those covered in our class, read and teach yourselves any necessary background to understand it, and then write up a four-page description. You are strongly encouraged to work with a partner (only one) on this project: by working together, you can learn from each other and share the burdens and joys of the experience.
Your peers will evaluate your work (as will I, and perhaps other members of the department). During our final exam slot, we'll present our work to each other.