Manjul Bhargava’s Fields Medal and beyond

Manjul Bhargava, the Canadian-American number theorist from Princeton University.

Manjul Bhargava, the Canadian-American number theorist from Princeton University.

The award of the 2014 Fields Medal to Professor Manjul Bhargava of Princeton Uniersity is a crowning recognition for his path-breaking contributions to some of the most famous and difficult problems in Number Theory. The Fields Medal, known as the Nobel Prize of Mathematics, has an age limit of 40, and therefore recognizes outstanding contributions by young mathematicians who are expected to influence the development of the subject in the years ahead. Manjul Bhargava was awarded the Fields Medal for his pioneering work on several long standing and important problems in number theory, and for the methods he has introduced to achieve this phenomenal progress — methods that will influence research for years to come.

Bhargava's PhD thesis concerns a problem going back to Carl Friedrich Gauss, one of the greatest mathematicians in history. In the nineteenth century, Gauss had discovered a fundamental composition law for binary quadratic forms which are homogeneous polynomial functions of degree two in two variables. No formula or law of the Gauss type was known for cubic or higher degree forms. Bhargava broke the impasse of 200 years by producing a composition law for cubic and higher degree forms.

His elegant solution to the cubic case was inspired by studying slicings of the Rubik's cube. In his PhD thesis of 2001 written under the direction of Professor Andrew Wiles of Princeton University, and in his post-doctoral work, Bhargava established the composition law for cubic forms and extended this to the quartic and the quintic cases as well. He also used these composition laws to settle a number of classical problems in number theory, including some cases of the Cohen-Lenstra-Martinet conjectures on class groups, as well as the determination of the asymptotic number of quartic and quintic _fields having bounded discriminant. Around this time he also got interested in the fundamental problem of determining all universal integral quadratic forms, namely quadratic forms that represent all positive integers. This is a problem that stems from the work of Ramanujan who wrote down 54 examples of such universal quadratic forms. Bhargava solved two fundamental problems on universal quadratic forms (the second jointly with Jonathan Hanke), and announced these for the first time in his Ramanujan Commemmoration Lecture at SASTRA University, Kumbakonam, on December 22, 2005 (Ramanujan's birthday) when he was awarded the First SASTRA Ramanujan Prize. This prize has a stricter age limit of 32 because Ramanujan achieved so much in his brief life of 32 years.

I should point out that only in 2005 were two SASTRA Ramanujan Prizes awarded — one to Bhargava and another to Kannan Soundararajan; they were two full prizes, and not shared.

For more details about this early work of Bhargava, see my article in The Hindu , Dec. 23, 2005.

Bhargava's work on quadratic and higher degree forms brought a resurgence of activity worldwide on these topics. In view of this, three major conferences were organised in 2009 funded by the National Science Foundation : A conference on quadratic forms and another on higher degree forms at the University of Florida in March and May, and The Arizona Winter School on Quadratic Forms in March. Bhargava acted as a co-organiser for the Florida conferences with me and my colleagues, and was a lead speaker at the Arizona conference.

Selected lectures from these conferences have appeared recently as a volume in the book series Developments in Mathematics published by Springer with Bhargava co-editing this volume with me and David Savitt and Pham Tiep of the University of Arizona.

For his seminal contributions on composition laws and on quadratic and higher degree forms, Bhargava received several awards in addition to the SASTRA Ramanujan Prize, such as the Clay Research Award in 2005, and the Frank Nelson Cole Prize of the American Mathematical Society in 2008. As Professor Peter Sarnak of Princeton University said: “For a guy so young, I can't remember anybody so decorated at his age.” But Bhargava did not rest on his laurels. He continued to pursue fundamental questions with unidiminished zeal.

His next assault was on the important problem concerning ranks of elliptic curves and the famous Birch{ Swinnerton-Dyer (BSD) Conjecture, which is one of the Millennium Prize Problems.

Bhargava's work is making deep inroads in algebraic number theory. The Fields Medal to Bhargava recognises all these pathbreaking contributions to various celebrated problems in number theory, as well the fundamental new methods that he has introduced.

He is the second SASTRA Ramanujan Prize Winner to also receive the Fields Medal, the other being Terence Tao in 2006.

In addition to guiding PhD students and mentoring post-docs in Princeton, Bhargava visits India regularly, especially the Tata Institute, IIT Bombay, and the University of Hyderabad, where he holds Adjunct Professorships. He has a deep regard for Indian culture and is proud of his Indian heritage. He enjoys working with mathematicians in India.

For example, he has an active collaboration with Eknath Ghate of the Tata Institute who recently received the Bhatnagar Award. Bhargava's profound influence on mathematics is not only due to his own fundamental contributions, but also due to the students and post- docs he has groomed, and the impact he has had on the work of the present generation of mathematicians. This influence will continue strongly in the years ahead and consequently will lead to more breakthroughs in the future. Perhaps we will see the resolution of the BSD Conjecture!

The author is Professor from University of Florida.

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