Analysis, Spectra and Number Theory, a Conference in Honor of Peter Sarnak’s 61st Birthday

The Average Rank Conjecture states that, when elliptic curves over Q are ordered by naive height/conductor/discriminant/Faltings height, the average rank is 1/2. This conjecture was first formulated by Goldfeld for quadratic twist families, and for the Universityersal family by Katz and Sarnak. In this talk, we describe some of the data that do/don't support the conjecture, as well as some of the recent newer conjectures and theorems that have a bearing on the Average Rank Conjecture.