Manjul Bhargava was awarded the prestigious Fields Medal, considered to be the Nobel Prize of mathematics, at a grand ceremony held in Seoul, Korea, on August 13, 2014. Officially called the International Medal for Outstanding Discoveries in Mathematics, it is awarded once every four years and regarded as the greatest honour a mathematician can receive.
His award citation described Bhargava's groundbreaking work in number theory as "based both on a deep understanding of the representations of arithmetic groups and a unique blend of algebraic and analytic expertise."
Bhargava, a mathematics professor at Princeton University, is driven by his search for artistic truth and beauty and is credited with some of the most profound recent discoveries in number theory, the branch of mathematics that studies the relationships between whole numbers.
In the past few years, he has made great strides toward understanding the range of possible solutions to equations known as elliptic curves, which have bedeviled number theorists for more than a century.
The award citation also mentiones that he was inspired to extend Gauss's law of composition in an unusual way which he explained to the New Scientist as follows:
Gauss's law says that you can compose two quadratic forms, which you can think of as a square of numbers, to get a third square. I was in California in the summer of 1998, and I had a 2 x 2 x 2 mini Rubik's cube. I was just visualising putting numbers on each of the corners, and I saw these binary quadratic forms coming out, three of them. I just sat down and wrote out the relations between them. It was a great day!
But before Rubik's cube, there was an Indian connection too. Erica Klarreich describes Bhargava's formative years and experiences in the Quanta Magazine: The Musical, Magical Number Theorist:
Every few years, Bhargava’s mother took him to visit his grandparents in Jaipur, India. His grandfather, Purushottam Lal Bhargava, was the head of the Sanskrit department of the University of Rajasthan, and Manjul Bhargava grew up reading ancient mathematics and Sanskrit poetry texts.
To his delight, he discovered that the rhythms of Sanskrit poetry are highly mathematical. Bhargava is fond of explaining to his students that the ancient Sanskrit poets figured out the number of different rhythms with a given number of beats that can be constructed using combinations of long and short syllables: It’s the corresponding number in what Western mathematicians call the Fibonacci sequence. Even the Sanskrit alphabet has an inherent mathematical structure, Bhargava discovered: Its first 25 consonants form a 5 by 5 array in which one dimension specifies the bodily organ where the sound originates and the other dimension specifies a quality of modulation. “The mathematical aspect excited me,” he said.
Sanskrit poems feature a mix of short and long syllables that last for one or two beats, respectively. As a child, Bhargava was fascinated by the question of how many different rhythms it is possible to construct with a given number of beats. A four-beat phrase, for example, could be short-long-short or short-short-short-short (or one of three other possibilities).
The answer, Bhargava discovered, was given in “Chandahsastra,” a treatise on poetic rhythms written by Pingala more than two millennia ago. There’s a simple formula: The number of different rhythms with, say, nine beats is the sum of the number of rhythms with seven beats and the number of rhythms with eight beats. That’s because each nine-beat rhythm can be constructed by adding either a long syllable to a seven-beat rhythm or a short syllable to an eight-beat rhythm.
This rule generates the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on —in which each number is the sum of the preceding two. These are known as the Hemachandra numbers — after 11th-century scholar Acharya Hemachandra, who wrote about poetic rhythms — or the Fibonacci numbers, to Western mathematicians. Bhargava enjoys showing his students that these numbers arise not only in poetic rhythms but also in natural settings, such as in the number of spirals on a pine cone or petals on a daisy.
"If two numbers that are each the sum of a perfect square and a given whole number times a perfect square are multiplied together, the product will again be the sum of a perfect square and that whole number times another perfect square."
That is what Bhargava had read in one of his grandfather’s Sanskrit manuscripts about a generalisation developed by the great Indian mathematician Brahmagupta in 628 AD.
As a graduate student at Princeton he learnt that in 1801, the German mathematical giant Carl Friedrich Gauss had grappled with a similar concept, and had reduced it as binary quadratic forms: expressions with two variables and only quadratic terms, such as x2 + y2 (the sum of two squares), x2 + 7y2, or 3x2 + 4xy + 9y2. Multiply two such expressions together, and Gauss’ “composition law” tells you which quadratic form you will end up with.
But Gauss’ law was a mathematical behemoth, which took him about 20 pages to describe. Bhargava took it upon himself to find a simple way to describe what was going on and whether there were analogous laws for expressions involving higher exponents.
As Erica Klarreich describes in in the Quanta Magazine: The Musical, Magical Number Theorist:
He has always been drawn, he said, to questions like this one — “problems that are easy to state, and when you hear them, you think they’re somehow so fundamental that we have to know the answer.”
The answer came to him late one night as he was pondering the problem in his room, which was strewn with Rubik’s Cubes and related puzzles, including the Rubik’s mini-cube, which has only four squares on each face. Bhargava — who used to be able to solve the Rubik’s Cube in about a minute — realized that if he were to place numbers on each corner of the mini-cube and then cut the cube in half, the eight corner numbers could be combined in a natural way to produce a binary quadratic form.
There are three ways to cut a cube in half — making a front-back, left-right or top-bottom division — so the cube generated three quadratic forms. These three forms, Bhargava discovered, add up to zero — not with respect to normal addition, but with respect to Gauss’ method for composing quadratic forms. Bhargava’s cube-slicing method gave a new and elegant reformulation of Gauss’ 20-page law.
Additionally, Bhargava realized that if he arranged numbers on a Rubik’s Domino — a 2x3x3 puzzle — he could produce a composition law for cubic forms, ones whose exponents are three. Over the next few years, Bhargava discovered 12 more composition laws, which formed the core of his Ph.D. thesis. These laws are not just idle curiosities: They connect to a fundamental object in modern number theory called an ideal class group, which measures how many ways a number can be factored into primes in more complicated number systems than the whole numbers.
Also hear this old NPR programme:
And an interview last year: