The Laureates

Copyright © Klaus Tschira Stiftung / Peter Badge

Manjul Bhargava

Born 8 August 1974, Hamilton, Ontario, Canada

Fields Medal (2014) “for developing new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.”

Manjul Bhargava was born in Canada but grew up in Long Island, New York. He was a child prodigy; his mother was a maths teacher, his father worked as a chemist. After high school, Bhargava studied at Harvard University (Bachelor 1996). In the same year he received the Morgan Prize of the American Mathematical Society for his undergraduate research. He started then his Ph.D. studies at Princeton, advised by number theorist Andrew Wiles who is famous for his proof of Fermat’s Last Theorem from the 1990s. In Bhargava’s thesis from 2001, the Gaussian law for the composition of binary quadratic forms is reformulated and generalized with surprising results. Bhargava worked afterwards as a visiting scholar at the Institute for Advanced Study in Princeton for one year, and at Harvard University for one year. Since 2003 he is a Professor of Mathematics at Princeton University.

Besides the Fields Medal, Bhargava received many other prizes and awards, for instance the Blumenthal Award for the Advancement of Research in Pure Mathematics (2005), the Packard Foundation Fellowship in Science and Engineering (2004), and the Merten M. Hasse Prize for Exposition of the Mathematical Association of America (2003).

He is also an accomplished tabla player who has performed in concert halls, and he is an admirer of Sanskrit poetry.

Manjul Bhargava works in the field of number theory. He became famous with a series of four papers in the “Annals of Mathematics” in which he generalized the composition of binary integral quadratic forms. Binary quadratic forms are polynomials in two variables x and y looking like that: f(x,y) = ax² + bxy + cy². If the coefficients a, b, and c are integers, one speaks of an integral form. Such objects are classics in number theory: as early as at the beginning of the 19th century, Carl Friedrich Gauss (1777-1855) developed in section V of his famous “Disquisitiones Arithmeticae” (1801) a method to compose such forms and to classify them. Gauss was mainly interested in the question for which numbers x and y the form f(x,y) has integral solutions (the form “represents” such solutions then).

Manjul Bhargava found methods to compose forms of higher order (with exponents larger than 2 in the exponents of the polynomials) in a similar way as Gauss did for the quadratic forms, and he even described these methods by means of a general theory. Together with Jonathan P. Hanke he also proved a number theoretic conjecture of John Horton Conway (*1937) dealing with a certain subclass of integral quadratic forms.

Bhargava also focused on one of the Millenium Problems, the conjecture of Birch and Swinnerton-Dyer which is a statement about elliptic curves, certain polynomials of degree 3. (They have little to do with ellipses, but are formulas that have been found when computing the length of ellipses.) Bhargava proved together with Arul Shankar among others that a positive measure of elliptic curves over the rational numbers satisfies the conjecture of Birch and Swinnerton-Dyer.