Ranks of elliptic curves
Abstract: Elliptic curves have surprising and important applications both inside and outside of number theory. Elliptic curves over finite fields have cryptographic applications, while elliptic curves over the rational numbers were used to prove Fermat's Last Theorem. The rank of an elliptic curve is a measure of the number of rational points on the curve. We will explore questions about the distribution of ranks, especially in families of quadratic twists.