Average ranks of curves

Rational points and ranks of elliptic curves are subjects of many important conjectures, such as the Birch-Swinnerton-Dyer conjecture and conjectures on `typical’ and `maximal’ ranks. In a recent series of papers, Manjul Bhargava and his collaborators made several fundamental breakthroughs on average ranks and Selmer ranks of elliptic curves over the rationals. In particular, they prove that the average rank of all elliptic curves over Q is less than 1 (this average was not even known to be bounded), and deduce that a positive proportion of elliptic curves satisfy the Birch-Swinnerton-Dyer conjecture. This beautiful work combines techniques from invariant theory, Selmer groups, geometry and analytic number theory. In this lecture I will give a brief and elementary overview of their approach and explain some related results.