We know that x^{2}+y^{2}=1 is the equation of a circle, but what happens when the degree of the equation is greater than two? In this project we investigate some of the geometry of algebraic curves – that is, solutions of polynomial equations P(x,y)=0. In particular, when P is a cubic polynomial, the resulting curve is called an elliptic curve and has some interesting properties (in the complex domain it looks like a doughnut), but higher degree curves are even more interesting.

Supervisor: Dr Colin Christopher