Abstract: In this lecture we will introduce elliptic curves
and some of the fundamental questions about them. The rank
of an elliptic curve is a measure of the number of solutions
of the equation which 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 talk we
will survey what is known, as well as what is conjectured but
not known, about ranks of elliptic curves.