Princeton University professor Manjul Bhargava Princeton University professor Manjul Bhargava

Manjul Bhargava, the first Indian-origin mathematician to win the coveted Fields Medal, says the inspiration behind his discoveries in number theory has been the classic works of ancient Indian mathematicians.

"Their works contain incredible mathematical discoveries, and were very inspirational to me as a young mathematician. The classic works of Pingala, Hemachandra, and Brahmagupta have been particularly influential in my own work," the 40-year-old Princeton University professor told IndiaToday.in in a detailed email interview.

In the interview, he also goes on the explain how the ancients derived elegant mathematical patterns from rhythms of Sanskrit poetry and how he managed to simplify and expand the work of 18th century German maths legend Carl Friedrich Gauss with the help of Hemchandra's Identity and a simple Rubik's Mini-Cube.

Bhargava also offers three suggestions to teachers and students in India to make maths learning fun.

Here is the full Q&A:

Q: Can you describe for our readers, in simple terms, the nature of the number theory work for which you have been awarded?

A: I work in the area of number theory, which studies the whole numbers ..., -2, -1, 0, 1, 2, ... and special whole numbers like the primes, and their properties and the equations that they satisfy.

My work was primarily about deciding when certain fundamental equations in mathematics (such as quadratic forms and elliptic curves) have solutions in the whole numbers. My work introduced, in particular, a new geometric method for answering such questions.

Manjul Bhargava receives the Fields Medal at the International Congress of Mathematicians (ICM) in Seoul on August 12. Photo: ICM Manjul Bhargava receives the Fields Medal at the International Congress of Mathematicians (ICM) in Seoul on August 12. Photo: ICM

A: In part because of the scientific nature of the Sanskrit language, many remarkable linguistic/poetic/mathematical works were written in ancient times in India. Growing up, I had a chance to read some of the works of the masters: the great linguists/poets such as Panini, Pingala, and Hemachandra, as well as the great mathematicians Aryabhata, Bhaskara, and of course Brahmagupta. Their works contain incredible mathematical discoveries, and were very inspirational to me as a young mathematician.

The classic works of Pingala, Hemachandra, and Brahmagupta have been particularly influential in my own work.

Q: You have said that you found inspiration to do maths through tabla playing and Sanskrit. How?

A: My greatest influences while growing up were my grandfather, a renowned scholar of Sanskrit and ancient Indian history, and my mother, a mathematician with strong interests also in music and linguistics. As a result, I also developed deep interests in language and literature, particularly Sanskrit poetry, and in classical Indian music. I learned to play a number of musical instruments, such as sitar, guitar, violin, and keyboard. But I always enjoyed rhythm and percussion the most! My favourite instrument was the tabla. I enjoyed thinking about the mathematics of rhythm in classic Hindustani and Carnatic music.

While growing up, I learned from my grandfather how much incredible mathematics was discovered in ancient times by scholars who considered themselves not mathematicians, but poets (or linguists). Linguists such as Panini, Pingala, Hemachandra, and Narayana discovered some wonderful and deep mathematical concepts while studying poetry. The stories that my grandfather told me about them were very inspirational to me.

Here is an example, originating in 500 B.C., that has been particularly fascinating to me as a drummer.

In the rhythms of Sanskrit poetry, there are two kinds of syllables - long and short. A long syllable lasts two beats, and a short syllable lasts one beat. A question that naturally arose for ancient poets was: how many rhythms can one construct with exactly (say) eight beats, consisting of long and short syllables? For instance, one can take long-long-long-long, or

short-short-short-long-long-short.

The answer was discovered by the ancients, and is contained in Pingala's classical work Chandashastra, which dates back to between 500 and 200 B.C.

Here is the elegant solution. We write down a sequence of numbers as follows. We first write down the numbers 1 and 2. And then each subsequent number is obtained by adding up the two previous numbers.

So, for example, we start with 1 and 2, and then 1+2 is 3, so we have so far 1 2 3. The next number is obtained by adding up the last two numbers 2 and 3, which is 5. So we have so far 1 2 3 5. The next number written is then 3+5 which is 8. In this way, we get a sequence of numbers 1 2 3 5 8 13 21 34 55 89... The n-th number written tells you the total number of rhythms, consisting of long and short syllables, having n beats. So for 8 beats, the answer is that there are 34 such rhythms in total.

This sequence of numbers is now ubiquitous in mathematics, as well as in a number of other arts and sciences! The numbers are known as the Hemachandra numbers, after the 11th century linguist who first documented and proved their method of generation -- called a "recurrence relation" in modern mathematics. The numbers are also known as the Fibonacci numbers in the West, after the famous Italian mathematician who wrote about them in the 12th century.

These numbers play an important role now in so many areas of mathematics (there is even an entire mathematical journal, the Fibonacci Quarterly, devoted to them!). They also arise in botany and biology. For example, the number of petals on a daisy tends to be one of these Hemachandra numbers, and similarly for the number of spirals on a pine cone (for mathematical reasons that are now essentially understood).

One of my favorite photographs, which I keep in my office, is of a vast field of daisies, in which every daisy has 34 petals! (Recall that 34 is the same number that appeared as the answer to our question about 8 beat rhythms, revealing a hidden connection that mathematicians now understand.)

This story inspired me when I was growing up because it is a wonderful example of a simple idea that grew into something so omnipresent, important, and deep- unraveling surprising and beautiful connections among different realms of thought. There are many examples of this phenomenon in mathematics and its sister areas; I grew up learning many examples stemming from Sanskrit poetry, and is one of those things that always makes a mathematician's eyes light up.

Manjul Bhargava (second from left) addresses a press conference with three other Fields Medalists Artur Avila (first from left), Martin Hairer (third) and Maryam Mirzakhani (fourth) at the International Congress of Mathematicians (ICM) in Seoul on August 13. Photo: ICM Manjul Bhargava (second from left) addresses a press conference with three other Fields Medalists Artur Avila (first from left), Martin Hairer (third) and Maryam Mirzakhani (fourth) at the International Congress of Mathematicians (ICM) in Seoul on August 13. Photo: ICM

A: I first started learning music from my mother, who sings and also plays the tabla. When I was maybe 3 years old, I used to hear my mother playing tabla often, and I asked her to teach me to play a little bit. She tried to teach me the basic sound "na".

She demonstrated the sound to me, and I tried to mimic her to reproduce the sound, but nothing came out. I was hooked! I always loved the beauty and the intricacy of the tabla sound and repertoire, and how it also perfectly complemented sounds on the sitar, or vocal, etc.

I learned with my mom first, and then with Pandit Prem Prakash Sharma in Jaipur whenever I visited there.

I met Zakir-ji when I was an undergraduate at Harvard. He came to perform there when I was a third year student. I had the exciting opportunity to meet him afterwards at a reception, and he invited me to visit him in California (where he lives). I have had the great pleasure and privilege of learning from him a bit off and on since then. He is not just an incredible artist, but also an amazing teacher.

More than that, he has been a wonderful and inspirational friend, and he and his whole family - in both California and Bombay - have been such a huge source of love, encouragement, and support to me for so long, and I am very grateful to them for that.

Q: You are known to use magic tricks in your class to explain points. As a professor, what would be your three suggestions to teachers in India to create interest in maths in the classroom, especially among students who fear the subject?

A: - Maths problems should be motivated not just through the sciences, but also through the arts: puzzles, toys, magic, poetry, music - these should all form a key part of the mathematics classroom.

- Students should not be taught to solve problems in a robotic way; instead, they should be guided to discover key mathematics ideas on their own. Maths should be a creative exciting process of discovery!

-Maths should be interactive and collaborative. Students should be encouraged to discover things together, and work together.

This is how mathematics research is, and so this should be reflected in the classroom!

Manjul Bhargava at a press conference at International Congress of Mathematicians (ICM) in Seoul on August 13. Photo: ICM Manjul Bhargava at a press conference at International Congress of Mathematicians (ICM) in Seoul on August 13. Photo: ICM

A: When people see mathematics done as described above, as a playful, creative, interactive subject, they see that it is not terrifying at all - it is beautiful! Among my favorite books as a child that taught me a lot about the fun of mathematics were those of Martin Gardner, e.g., his book "Mathematics, Magic, and Mystery".

Q: You are famous for finding a simpler way to express Gauss composition law through Bharmagupta's work and Rubik's Cube? Can you explain how you did it?

A: Gauss's law says that you can compose two quadratic forms (i.e., expressions of the shape ax^2+bxy+cy^2, where a, b, c are fixed whole numbers and x and y are the variables) to get a third such quadratic form.

I was in California in the summer of 1998, and I had a 2 x 2 x 2 mini-Rubik's cube in my room. I was visualizing 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, and realized that I had found a simple description of Gauss's law. It was a great day!

I can give more details if you are interested: A very classical question in mathematics is: what whole numbers can be expressed as the sum of two squares? (i.e., what whole number values are taken by the polynomial x^2 + y^2, for whole number values of x and y?) For example, 5 is such a number, because 5 = 1 + 4 = 1^2 + 2^2, but 7 is not.

One ancient theorem about such numbers is: if you take two whole numbers that are each a sum of two squares, and you multiply them, then you will always again get a number that is a sum of two squares! (Please try it!)

The reason for this is the following remarkable identity: (a^2 + b^2) (c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2.

This identity gives an explicit method for taking two numbers that are each a sum of two squares, and expressing the product again as a sum of two squares.

This identity was discovered by Diophantus in the third century, and is a first case of Gauss composition: We are taking an expression of the form x^2 + y^2, and multiplying it with another expression of the form x^2 + y^2, and we are ending up with a third quadratic expression, also of the form x^2+y^2!

More generally, for any number n, if you take two whole numbers that are each a square plus n times a square, and you multiply them, you will always end up again with a number that is a square plus n times a square. This is because of a generalization of the above identity, discovered by Brahmagupta in 628.



It was not until Gauss that this phenomenon was fully understood. Gauss found all triples Q, R, S of quadratic expressions such that: if a is any value taken by Q, and b is any value taken by R, then the product ab will always be a value taken by S. In that case, we say that "the composition of Q and R is S" (in Gauss's law of composition). For example, the composition of Q = x^2 + ny^2 and R = x^2 + ny^2 is again S = x^2 + ny^2, due to Brahmagupta's identity.

Gauss showed that, in general, S is essentially uniquely determined by Q and R, so it makes sense to say that S is the composition of Q and R. (By "quadratic expression", we mean here an expression of the form ax^2 + bxy +cy^2, where a, b, c are fixed whole numbers, and x and y are the variables.)

Gauss discovered this law of composition in 1801 in his famous work Disquisitiones Arithmeticae, but it took about 20 pages of complicated calculation to describe the composition law in general.

In my Ph.D. thesis, inspired by Rubik's cubes as you mentioned, I found a very simple way to formulate Gauss composition, which took only a few lines to describe. (Please see, e.g., pages 10, 11, and 12 of this link).

This new perspective on Gauss composition also led me in my Ph.D. thesis to 13 new laws of composition, which applied to higher degree polynomials and to polynomials in more than 2 variables (Gauss composition applies to binary quadratic forms, i.e., degree 2 polynomials in two variables).

Previously, it was thought that Gauss composition is an isolated phenomenon that applies only to quadratic expressions in two variables.

My Ph.D. thesis work showed that it was in fact part of a much larger theory. That's probably more than you wanted to know! :)

Q: Do you think maths is a natural gift or can it be inculcated in students by proper guidance?

A: I think it can be a combination of both, but proper guidance/encouragement/inspiration is perhaps the more important aspect. A student's talent and interest has to be fostered and cultivated in order to truly bloom.