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Juggling numbers, musical notes with aplomb

Love for music and a facility for mathematics often go together. Einstein played the violin, Feynman the bongo. Manjul Bhargava, Fields medallist and R Brandon Fradd Professor of mathematics at Princeton, is a tabla maestro. Trained by Zakir Hussain, Bhargava has various concert performances to his credit.

His mother, Mira, was a mathematics professor (and his first tabla teacher). She often took her prodigious eight-year-old son to college lectures. Bhargava is fluent in Sanskrit, unusual for someone born in Canada and brought up in the US. Taught by his grandfather, noted scholar P L Bhargava, he has been known to deconstruct the metres of Sanskrit poetry in impromptu lectures.

Often, he also deconstructs the mathematical basis of games and puzzles during lectures. He has a penchant for performing sleight-of-hand magic tricks.

Bhargava’s major work is in number theory. While studying for a master’s degree, he found a radical extension of the 200-year-old Gauss Composition Law, drawing upon his love for math puzzles and knowledge of Sanskrit arcana.

Around the 6th century, noted mathematician and astronomer Brahmagupta had said the product of two numbers that were the sums of perfect squares would also be the sum of two perfect squares. For example, if 20 is multiplied by 34, the result is 680 (20=4+16; 34=9+25 and 680=484 (22 squared) + 196 (14 squared). German mathematician Carl Friedrich Gauss found many such relationships involving quadratics (squares and variations of squares), writing a 20-page “composition law” for such relationships.

Bhargava visualised placing such numbers on the edges of Rubik’s Cube and, subsequently, cutting cubes in various ways (to add and multiply numbers). Using a 2×2 cube and describing “cuts” mathematically, he found a much simpler way to restate Gauss. Using 3×3 cubes, he extended the composition laws to cubed numbers. His insights revolutionised the understanding of factorising in multiple number systems. If that sounds obscure, it’s a central problem in public-key encryption (PKE): it is much more difficult to arrive at the factors of a given number than to multiply two numbers. Much PKE depends on this, as one number is publicly released.

Bhargava’s later work on elliptical curves also has cryptographic applications. Elliptic curves arise in equations of cubed and squared relationships (the typical form is y^2=x^3+ax+b). Ellipticals are not ellipses; at times, they might be doughnut-shaped (mathematically called “torus”).

Some ellipticals have infinite rational solutions; others have no rational solution (a rational number is one that can be expressed as a fraction). These are very important in topology and in number theory. The solution to Fermat’s last theorem, for instance, involves ellipticals.

Ellipticals are also used in PKE systems in which one set of coordinates is publicly released. Bhargava and his collaborators have discovered many important properties. Along with his colleague Chris Skinner and collaborator Wei Zhang (of Columbia), he has attacked the Birch & Swinnerton-Dyer conjecture. If they can prove/disprove this definitively, they stand to win one of the Millennium million-dollar prizes. Bhargava is said to find similarity between Indian classical music and maths – both involve rapid mental improvisation and a search for the truth. His ability to find relatively simple solutions to very complex problems has helped open entirely new vistas of study.

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