Manjul Bhargava, Ph.D.

Manjul Bhargava pursues algebraic number theory and the geometry of numbers in the tradition of Gauss and Minkowski. Bhargava has inspired an extraordinary resurgence of this field, with wonderful applications. His overarching goal in this work is to count the basic objects of number theory and to make computational conclusions about their asymptotics. For example, it is conjectured that, in a certain natural sense, the average rank of the group of rational points of an elliptic curve defined over the rationals is 1/2. Bhargava and his student Shankar recently showed that it is less than 1. Previously, it was not even known whether the average rank is finite. In joint work with Dick Gross, Bhargava has also shown that the number of rational points on the majority of hyperelliptic curves is bounded by a certain small number independent of the genus of the curve. This work opens up remarkable new vistas in arithmetic and suggests exciting conjectures.