Manjul Bhargava

Native spelling: मंजुल भार्गव

fl. 1974, Hamilton, Ontario, Canada

Arithmetic and number theoryAlgebraFields Medal 2014

Canadian-American number theorist who generalized Gauss's composition law and used the geometry of numbers to count rings and bound the average rank of elliptic curves; Fields Medal 2014.

Manjul Bhargava was born in Hamilton, Ontario, in 1974 to an Indian family and grew up largely on Long Island, New York. A gifted tabla player as well as a mathematician, he studied at Harvard University and received his PhD at Princeton University in 2001 under Andrew Wiles. After two years at the Institute for Advanced Study he joined Princeton's faculty, becoming a full professor in 2003 — one of the youngest in the university's history.

Bhargava's doctoral thesis produced what are now called the Bhargava composition laws: fourteen higher analogues of Gauss's 1801 composition of binary quadratic forms, arising from counting integer orbits of group actions on lattices. This 'geometry-of-numbers meets representation theory' program grew into a general theory of arithmetic statistics. By counting number fields, rings, and Selmer elements via their lattice points, he determined average sizes of Selmer groups of elliptic curves, deducing that the average rank is bounded — famously less than 1 for a suitable ordering — and that a positive proportion of curves have rank 0. With collaborators he also proved that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.

He received the Fields Medal in 2014 in Seoul, together with Artur Avila, Martin Hairer and Maryam Mirzakhani. Besides research, Bhargava is known for public lectures connecting mathematics to Sanskrit prosody, music and magic squares, and he holds an adjunct professorship at the Tata Institute of Fundamental Research.

Workplaces: Princeton University

Canada, United States, India

Award-winning work

Fields Medal 2014The medal cites his development of powerful new methods in the geometry of numbers, which he used to count rings of small rank and to bound the average rank of elliptic curves. Bhargava began by finding fourteen new composition laws generalizing Gauss's composition of binary quadratic forms, discovered through systematic counting of lattice orbits. He then built a formidable machinery for counting rings and arithmetic objects by asymptotic density, determining, for example, the average size of 2-, 3-, 4- and 5-Selmer groups of elliptic curves — implying that the average rank of an elliptic curve is bounded (less than one) and that a positive proportion of elliptic curves have rank zero. These results give the sharpest unconditional evidence for the Birch–Swinnerton-Dyer conjecture's predictions about ranks.

Contributions, linked to the library

Great problemBirch and Swinnerton-Dyer conjectureProved the average rank of elliptic curves is bounded and that a positive proportion of curves satisfy the conjecture.Library topicElliptic curvesDetermined average sizes of Selmer groups, giving the strongest unconditional results on ranks of elliptic curves.Library topicArithmetic geometryCreated a systematic geometry-of-numbers machinery for counting rings, number fields and arithmetic objects by density.