Manjul Bhargava

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This is a collection of articles archived for the excellence of their content.


The authors of this article are...

i) Indian-origin wizard wins 'Nobel Prize' of Mathematics Chidanand Rajghatta,TNN | Aug 13, 2014 The Times of India/ ii) Math Union / iii) Infosys-Science-Foundation/ iv) Deanna Haunsperger, A Break for Mathematics: An Interview with Joe Gallian

Biographical details

Manjul Bhargava is the R. Brandon Fradd Professor of Mathematics at Princeton University, U.S.A. He earned his A.B. in Mathematics from Harvard University in 1996 and Ph.D. from Princeton University in 2001. He joined Princeton University as a Professor of Mathematics in 2003. He was the first five-year Research Fellow of the Clay Mathematics Institute during 2000-05. His primary research interests lie in number theory, representation theory, and algebraic geometry. He is a member of the National Academy of Sciences and has received numerous awards, including three Derek Bok Awards for Excellence in Teaching (1993-95), the Hoopes Prize for Excellence in Scholarly Work and Research from Harvard University (1996), the AMS-MAA-SIAM Morgan Prize for Outstanding Undergraduate Research in Mathematics (1997), the MAA Merten M. Hasse Prize for Exposition (2003), a Packard Foundation Fellowship in Science and Engineering (2004), the Blumenthal Award (2005), the SASTRA Ramanujan Prize (2005), which he shared with Prof. Kannan Soundararajan, the AMS Cole Prize for number theory (2008), and the Prix Fermat (2011). He was also the 2011 Simons Lecturer at MIT.

Joseph A. Gallian on Bhargava’s diligence

Joe Gallian, President of the Mathematical Association of America, recalls, Manjul Bhargava is an interesting story. He agreed to come to my REU, but he had very close ties to a grandfather in India. There was to be a family reunion in India the first week of the program, and he asked if he could he go there before coming to Duluth. I said, “Sure, fine.” He also said that if I gave him a problem in advance he’d work on it. When he arrived in Duluth, he had completely finished it except for the write-up, so he asked whether I had another problem for him. I gave him a number theory problem, and it turned out that that was beautiful, it worked out so well, because he ended up with a spectacular, completely novel approach. It was in number theory, but he had this ingenious ring-theoretic approach. He won a Morgan prize for that’’

The pinnacle: The Fields Medal 2014

Manjul Bhargava was among the four winners of the 2014 Fields Medal, widely considered the Nobel Prize for Mathematics that has been mostly dominated by white men (Americans, Russians, French, and Britons: 38 medals between them) since it was instituted in 1936.

The award going to Princeton University's Manjul Bhargava, a Canadian-American maths wizard was no surprise. Although he is the first person of Indian origin, he was the hot favourite in pre-award polls among peers. Which is not surprising for someone who became a tenured full professor within two years of finishing graduate school, an Ivy League record, and the second youngest full professor in Princeton's history.

In early August 2014 as speculation heated up about possible 2014 winners of Fields Medal, an online poll put Bhargava on top with 516 votes

The blog Jost a Mon wrote three months before the announcement of the prize, ‘’A lot of people are talking up Manjul Bhargava. It is his last chance to win - by the time the next ICM comes around he will be older than 40.’’

Scope and impact of his work

Prof. Bhargava is an eminent algebraic number theorist. He has developed novel techniques to count objects that were previously considered completely inaccessible. An important theme in number theory has been how various objects of interest are distributed. An archetypal example is the prime number theorem, which tells us how the prime numbers are distributed among all integers. The theorem states that the number of prime numbers less than x is asymptotically x/log x.

Prof. Bhargava’s research focused on the counting of number fields of fixed degree by discriminant. Results for cubic fields had been obtained about 40 years ago in the classical work of Davenport and Heilbronn, but no progress was made on higher degree number fields until Prof. Bhargava’s work opened up the subject again. For example, using elementary but ingenious generalizations of Gauss' composition law for binary quadratic forms, which had been missed for 200 years, he was able to count the number of quartic and quintic number fields with absolute discriminant less than x, as x tends to infinity. He showed that about 83% of quartic fields had Galois group the full symmetric group, but that 100% of quintic fields had for its Galois group the full symmetric group.

Recently, he extended his research methods to provide information about the average number of rational points on certain curves. Elliptic curves (cubic equations in two variables) have been a major subject of study by number theorists for over 300 years. The basic measure of the number of rational points is called the rank. Empirically, it is conjectured that “most” elliptic curves have a rank of either 0 or 1, but nothing definite was known until recently. Stunningly, Prof. Bhargava and his student Arul Shankar showed that for a positive proportion of elliptic curves the rank is 0. They also showed that the average rank of all elliptic curves is bounded and in fact is less than 1. In addition, Professors Bhargava and Benedict Gross showed that a positive proportion of hyper-elliptic curves of genus greater than 1 with a rational Weierstrass point have at most 3 rational points.

The Fields Medal Citation

Manjul Bhargava

is awarded a Fields Medal for developing powerful 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.

Bhargava’s thesis provided a reformulation of Gauss’s law for the composition of two binary quadratic forms. He showed that the orbits of the group SL(2, Z)3 on the tensor product of three copies of the standard integral representation correspond to quadratic rings (rings of rank 2 over Z) together with three ideal classes whose product is trivial. This recovers Gauss’s composition law in an original and computationally effective manner. He then studied orbits in more complicated integral representations, which correspond to cubic, quartic, and quintic rings, and counted the number of such rings with bounded discriminant.

Bhargava next turned to the study of representations with a polynomial ring of invariants. The simplest such representation is given by the action of PGL(2, Z) on the space of binary quartic forms. This has two independent invariants, which are related to the moduli of elliptic curves. Together with his student Arul Shankar, Bhargava used delicate estimates on the number of integral orbits of bounded height to bound the average rank of elliptic curves. Generalizing these methods to curves of higher genus, he recently showed that most hyperelliptic curves of genus at least two have no rational points.

Bhargava’s work is based both on a deep understanding of the representations of arithmetic groups and a unique blend of algebraic and analytic expertise.

The award

One to four Fields Medals are awarded once every four years to mathematicians under the age of 40 years at the International Congres of the International Mathematical Union (IMU), which meets every four years. (Hyderabad hosted it in 2010). Seoul is the venue of the 2014 presentation

The award is the highest recognition in the world of mathematics.

Manjul Bhargava’s work

Mathematics offers solutions to everyday issues from airline scheduling to Internet security, even though many practitioners pursue esoteric problems described in dense language incomprehensible to the layman. Bhargava's PhD thesis is said to have helped in the "determination of the asymptotic density of discriminants of quartic and quintic number fields."

Previous prizes and honours

Recognition came early. He has often recounted how in Grade 3, he became curious about how many oranges it takes to make a pyramid.

Manjul has also collaborated with many Indian mathematicians, and his work with fellow Princeton scholar Arul Shankar, his PhD student, won them the Fermat Prize in 2011. Manjul's own PhD advisor was Andrew Wiles, famous for proving Fermat's last theorem.

Bhargava was awarded the 2012 Infosys Prize in mathematics for his "extraordinarily original work in algebraic number theory, which has revolutionized the way in which number fields and elliptic curves are counted." That came on top of almost every other top prize in maths, from the SASTRA Ramanujan Prize in 2005 to the American mathematical Society's Cole Prize in 2008. So the Fields Medal comes as no great surprise to the mathematical community in the US or in India.


Although a Canadian-American who was born in Hamilton, Ontario, Bhargava is no stranger to India or to Indian mathematicians. Indeed, his mother, Mira Bhargava, is herself a rare female mathematician, teaching at Hofstra University (another well-known female Indian-American mathematician is Bhama Srinivasan at the University of Chicago).

His mathematician mother and chemist father were well-to-do: they indulged him with oranges till he figured out the pyramid answer, which was not long coming.

Other interests

That's not all. Before you think all he does is crunch numbers, Bhargava is also an accomplished tabla player (tutored by Zakir Hussain) and has the number on Sanskrit, which he learned from his grandfather Purushottam Lal Bhargava, was the head of the Sanskrit department of the University of Rajasthan, during family visits to Jaipur. He sees close links between his three loves noting how beats of tabla and rhythms of Sanskrit poetry are highly mathematical.

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