Georgia Institute of Technology College of Sciences

The problem of finding rational solutions to cubic equations is central in number theory, and goes back to Fermat. I will discuss why these equations are particularly interesting, and the modern theory of elliptic curves that has developed over the past century, including the Mordell-Weil theorem and the conjecture of Birch and Swinnerton-Dyer. I will end with a description of some recent results of Manjul Bhargava on the average rank.