A cubic equation in the plane has solutions which satisfy a group law. These are called elliptic curves. At a very basic level one could look at the situation over finite fields, where everything is very hands-on and computable, and where the subject plays a key role in cryptography. Fancier versions would involve looking at it over the integers, where the problems are much harder; they touch on very modern mathematics, ranging from the proof of Fermat's Last Theorem to the conjecture of Birch and Swinnerton-Dyer.
Suitable for student who have completed MATH2322.