Home / Opinion / Online-views /  Numbers and ancient symphonies

Eknath Ghate’s eyes light up as he explains to your columnist the intricate, chaotic beauty of Fantasie Impromtu, a piece of piano music composed 179 years ago by Frédéric Chopin, a Polish prodigy. Ghate, 44, a pianist schooled in Western classical music, talks of the popular composition’s time signature, a notational convention that signifies tempo.

Every 12 beats, says Ghate, the four-time tempo—more conventionally used as musical substructure in rock music—used by Chopin for the right hand, as well as by most musicians for both hands, merges with a waltz-like three-time tempo for the left hand.

“There’s a sort of periodical chaos," says Ghate, as we sit on two worn chairs under a grove of coconut trees by a promenade at the Tata Institute of Fundamental Research, a whisper-quiet corner of teeming Mumbai that most of the city has never seen. “There’s beauty, there is also an intense mathematical structure in this piece of music."

Ghate is a mathematician, one of India’s finest, and mathematicians are given to seeing order where others see beauty or chaos. “Everything is number," the ancient Greek mathematician Pythagoras is supposed to have said, after he found numerical patterns in musical harmonies.

“Ever since this discovery scientists have searched for the mathematical rules which appear to govern every singly physical process and have found that numbers crop in all manner of natural phenomena," writes Simon Singh, a British physicist, in a book called Fermat’s Last Theorem. For example, Singh explains how a Cambridge scientist found the length of rivers more than three times (3.4 approximately) the distance as the crow flies. Albert Einstein suggested a mathematical order to the seemingly natural loopiness of a river. Isaac Newton used calculus to understand gravity.

Mathematics has always inspired and advanced itself, a great melding between the world of beauty—natural and artificial—and the power of numbers. “Mathematics is an art form," says Ghate, a fit, sharp-nosed, sharply dressed man, more banker than mathematician in buttoned-down shirt and polished shoes. “It’s a bit like architecture, both art and science. It should be stable, stand the test of time, it should look beautiful—not like a D-type government flat—and make people happy."

In one presentation made to a Mumbai audience of theatre aficionados, Ghate—the only mathematician in a list of eight scientists awarded India’s national prize for scientists under 45—once explained how math is fun like Sudoku, challenging like the Himalayas (he was an avid trekker) and beautiful like Chopin’s music. When the presentation started at Prithvi theatre in the tony suburb of Juhu, he counted 11 people in the darkened hall; when the lights came on, there were nearly 200, among them lawyers, doctors and bankers. “One guy said, ‘dude, if you ever need to get out of jail, let me know,’" recalls Ghate.

A full-time father to a year-old toddler and a five-year-old (his wife, a biologist, works full time at a pharmaceutical company), Ghate may compare quadratic reciprocity to Fantasie Impromptu and bring much-needed romance to a field that is dense and distant to the layperson, but when he sits down to his math, what he really requires is a pen, paper, perhaps a computer—and solitude. The numbers are then in his head, not in music, not in the wind, not in the ripples disturbing the winter, morning calm of the Arabian sea nearby.

“My main inspirations come from scribbling on paper," confesses Ghate. “I can’t say, like Ramanujan, that a devi comes to me in a dream."

Interpreting the abstract reality of mathematicians for the general public is, in general, a difficult job, and Ghate’s field of interest over the last 20 years, modular forms, an area pushed by the great Indian mathematician , Srinivasa Ramanujan, is particularly arcane.

A modular form is a smooth mathematical function, not unlike the function y = x2 we study in school. Modular forms are, as Ghate, says, studied for their own sake, not for an immediate application. The two problems that Ghate has been working on are in the area of Galois representations and modular endomorphism algebras.

Both are a part of number theory, one of the oldest branches of mathematics. It studies whole numbers (1,2,3…) and the relationships between them.

Purity was once the essence of number theory, meaning it was supposed to have no real-world applications. That changed with the advent of computing, where it is specifically used in cryptography, security and coding. “What used to be an esoteric playground for mathematicians has become applicable research," write Cornell University computer science professors David Gries and Fred Schneider in A Logical Approach to Discrete Math. Nikolai Lobachevsky, a Russian mathematician, contended, about two centuries ago, that there is no branch of mathematics, however abstract, “which may not some day be applied to phenomena of the real world".

Ghate’s fields are still esoteric, numbers presently investigated for no other reason than understanding them better, and there’s no easy way of dumbing down his work. Here’s one definition of Galois representations that I could find: “Let G be a Galois group and let k be a topological field. By an n-dimensional Galois representation of G we mean a continuous homomorphism of groups. ρ : G → GL (n,k)."

A search entitled “modular forms for dummies" reveals a Harvard professor explaining them thus: “Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist."

Indeed, they do, allowing bright minds like Ghate to make a living off them. Like many leading scientists at TIFR, Ghate travels abroad for two or three months a year, spending the rest in his sylvan corner of Mumbai. “The best of both worlds," he confesses. As we stroll down the promenade, Ghate explains the freedom that mathematicians like him now enjoy, to live well and to think. He points to a scruffy, unshaven man, rubber slippers discarded, lying untidily on a rock, staring into space, notebook and pen in hand. “That," whispers Ghate, “is one of India’s best young mathematicians."

How does a mathematician work on a problem? First, says Ghate, he takes time to read research papers, up to a year, then lots of calculations to see if there are principles that emerge and see if they are connected. You make conjectures. Then you try and prove your or other people’s conjectures. You get credit mostly only for proving theorems. “It’s a lot of fun," says Ghate. Sometimes this can take quite a lot of time. It took 300 years, for instance, to solve what is known as Fermat’s Last Theorem, a simple equation with no solution. It was proposed in 1637 (by Pierre de Fermat, a French lawyer and amateur mathematician) as a scribble and solved in 1994 by an Oxford professor called Andrew Wiles when Ghate was in graduate school in the US, cementing his career choice. “It was like an earthquake in mathematics," he says. “It certainly was in my life."

Ghate is conscious that he must do his bit for mathematics in a country where rote-memorisation of formulae obstructs its growth. So, he speaks at schools and colleges, although, as he acknowledges, it takes away from his research. To understand how rarefied advanced mathematics is in India, you must know there are a handful Indians who even understand Wiles’ proof of Fermat’s last theorem. Ghate is one of them.

Samar Halarnkar is a Bangalore-based journalist. This is a fortnightly column that explores the cutting edge of science and technology. Comments are welcome at To read Samar Halarnkar’s previous columns, go to

Catch all the Business News, Market News, Breaking News Events and Latest News Updates on Live Mint. Download The Mint News App to get Daily Market Updates.
More Less
Subscribe to Mint Newsletters
* Enter a valid email
* Thank you for subscribing to our newsletter.

Recommended For You

Edit Profile
Get alerts on WhatsApp
Set Preferences My ReadsWatchlistFeedbackRedeem a Gift CardLogout