Dear
Reader,
Manjul
Bhargava, a Canadian mathematician of Indian origin, has been awarded the
prestigious 2014 Fields Medal at the International Mathematical Union’s (IMU)
International Congress of Mathematicians held in Seoul. Bhargava, 40, is the
first person of Indian origin to receive the prestigious medal, which has been
awarded since 1936 and recognizes outstanding mathematical achievement for
existing work and the “promise of future achievement”. Known to colleagues as a
gifted mathematician who would devise new and simple methods to prove
centuries-old theorems in number theory, Bhargava’s work has been described by
IMU as based both on a deep understanding of the representations of arithmetic
groups and a unique blend of algebraic and analytic expertise. Bhargava was
awarded the medal “for developing powerful new methods in geometry of numbers,
which he applied to count rings of small rank and to bound the average rank of
elliptic curves”, IMU said. One of Bhargava’s most prominent discoveries was a
thesis that threw a new light on Gauss’s Law for the composition of two binary
quadratic forms. When Bhargava was a graduate student, he read Disquisitiones
Arithmeticae, where Gauss developed his composition law which gives a method
for composing two binary quadratic forms to obtain a third one. According to
IMU’s description of his work, Bhargava one day came up with the idea that if
he cut off the Rubik’s mini-cube into half and labelled each corner of the cube
with two sets of numbers, that would lead to a cubic analogue of the Gauss
Composition. After being cut, a Rubic’s mini-cube could generate three
quadratic forms. From the three ways of slicing the cube, three binary forms
emerged, and so Bhargava had discovered a simple way to prove the law. “It’s
all about contributing to the understanding of ourselves and the world around
us. One of the keys to understanding mathematical problems that people have
been thinking about for years is to think about them in a totally different
way,” said Bhargava in an interview with Simons Foundation and IMU. He went on
to discover 12 other analogues of Gauss compositions over the years. He
received his Ph.D in 2001 from Princeton University for a thesis that has been
described as “stunning” and a “breakthrough”. It generalized the classical
Gauss composition law for quadratic forms to many other situations, according
to the Tata Institute of Fundamental Research. By 2003, Bhargava had become a
professor at Princeton . Born in 1974 in Canada, Manjul Bhargava grew up in the
US and often visited Jaipur to stay with his grandparents. His grandfather was
a professor in Sanskrit from whom Bhargava learnt Sanskrit poetry, while his
mother was a mathematics professor who allowed Bhargava to attend her
college-level classes as a child, according to interviews. “He is an extremely
original mathematician, who has developed a style which is distinctly his own,”
said Benedict Gross, professor of mathematics at Harvard University, who has
known Bhargava since his days as a Harvard undergraduate. “His method—combining
the arithmetic interpretation of integral orbits with their enumeration, using
the geometry of numbers—has led to the solution of a large number of problems
in number theory. I think his mathematical work is tremendously important, and
has changed the face of the subject,” added Gross, who himself is a renowned
mathematician for his work on the Gross-Zagier Theorem that won him the Cole
Prize along with his collaborators. Bhargava has also collaborated with
mathematicians to increase the understanding of elliptic curves, which are one
of the fundamental objects in number theory with applications in the fields of
cryptography and data security. Describing the uniqueness of his work, Gross
said, “He obtains results ‘on average’ that we can’t obtain in individual
cases. For example, for a cubic equation (an elliptic curve), it is quite
difficult to determine the size of the group of rational solutions. Bhargava
determines an upper bound on the average size, over all cubic equations with
rational coefficients.” Bhargava is an accomplished tabla player, having
trained under Ustad Zakir Hussain, one of the greatest exponents of the
percussion instrument. He is also an impressive and lucid orator. “He is a very
kind person, who shares his ideas freely with others. He is also a talented
tabla player, who loves music and other arts,” said Gross. Peter Sarnak,
professor in the department of mathematics at Princeton University, said,
“Bhargava has developed fundamental new tools geometric and diophantine, to
count objects in algebraic number theory, that before his work seemed hopeless.
His work reminds me of the great German mathematician Hermann Minkowski’s work
in number theory: it is brilliant, clear and beautiful and has reshaped the
landscape.” Three others received the Fields medal—Maryam Mirzakhani, the first
Iranian and the first woman to be awarded the medal; Artur Avila, the first
Brazilian to be awarded, and Martin Hairer, the first Austrian to win the
medal. Subhash Khot, an Indian-origin scientist from New York University, was
awarded the Nevanlinna Prize in Seoul for his prescient definition of the
“Unique Games” problem, and his efforts to understand its role in the study of
efficient approximation of optimization problems, which have produced
breakthroughs in algorithmic design and approximation hardness, according to
ICM.
