The rank of an elliptic curve is a measure of the number of solutions of the equation that defines the curve. In recent years there has been spectacular progress in the theory of elliptic curves, but the rank remains very mysterious. Even basic questions such as how to compute the rank, or whether the rank can be arbitrarily large, are not settled.
In this lecture we will introduce elliptic curves and discuss what is known, as well as what is conjectured but not known, about ranks.