Forty Minutes With a Fields Medallist

After the talk about “Poetry, Drumming and Mathematics”, we arrived at the appointed time at the Bose–Einstein Guest House to find our man, with his mother, standing transfixed by the deer grazing on the lawns. We managed to prise him away for a while — here’s what followed.

 

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Are you tired of all the media attention?

I have to say I’ve been pretty exhausted over the past year, because of the attention and…

And you were in Bangalore last month…

Yeah. It has been exhausting with all the travel.  This is my fifth trip to India this year. But I also appreciate that India’s been interested in scientists. That’s very nice. It’s not often that the media is interested in mathematics. So when it happens, you have to appreciate it and you have to go with it and encourage it. So I’m happy. But I’m exhausted.

What are you working on at the moment?

The truth is I haven’t done much math or music or poetry or anything in the past year. But I hope to. I hope to get back to it soon.

I can tell you what I’ve been working on. If you pick a random whole number…Do you know what a square-free number is? Square-free number means that when you factor it, no prime occurs more than once. So 6 is square-free: it’s 2 × 3. But 12 is not square-free: it’s 2 × 2 × 3. So 2 is repeated.

Suppose you pick a random whole number, what’s the probability that it’s square-free? The answer has been known for a long time. The answer is 6/π2. It’s unexpected, no, with the π? There are no circles here or anything. You’re asking for the probability that a whole number is square-free. And the answer is 6/π2. Here, π appears in this magical way in this number theory problem, not a geometry problem. So this is something that fascinated me.

One thing that I’ve been thinking about lately is: Often in number theory, you need to know about square-free numbers. If you have a polynomial with whole number coefficients and you look at its values when you plug in whole numbers, what’s the probability that the value of the polynomial is square-free? It depends on the polynomial, of course. For even a simple polynomial, x4 + 1, the answer is not known. What’s the probability that a random value of x4 + 1 is square-free? That’s one question that I work on.

 

Working with Collaborators

 

Do you collaborate with Indian mathematicians, who are based in Indian institutions?

Oh yeah, I do. Prof. Ramanan was there today. He’s someone I’ve learned a lot from. Then there’s Eknath Ghate at TIFR. We do a lot of work together. Arul Shankar is from the Chennai Mathematical Institute. He’s one of my collaborators. [Arul did his Ph.D. with Prof. Manjul Bhargava at Princeton.]

I believe you collaborated with him for your Fields Medal-winning work as well.

Yeah, some of it — some of the most recent parts of it.

Andrew Wiles, who proved Fermat’s last theorem, was your Ph.D. thesis adviser.

Yeah, it was a great experience. He’s a great teacher. I didn’t work in the area he works in. But he was great source of advice even though I didn’t work in his area. He had a great feel for what questions are important. As a young mathematician, you don’t always know what is important. Professor Wiles, he had a sense of the most influential or the most important mathematics.

During your Ph.D., did you actually work with him? Or he just guided you?

The questions were my own questions. Every once in a while, I’d go to him and show what I did and show him a couple of different things and he’d say: This is interesting but not as important as that. Then I’d go work on my own.

Is that the case with your Ph.D. students?

Some of them, yeah. I’ve got lots of Ph.D. students. So I have the whole range — some who like to work entirely on their own and come to see me once in a while and some whom I work with very regularly, as collaborators. The whole range. Every person is different — what they need or what their relationship with their advisers is.

 

“Ramanujan’s work has definitely been a huge inspiration to me”

 

Since we’re talking of number theory and we’re in Chennai, it’s inevitable that we talk about Ramanujan. Have you been inspired by his work?

Absolutely.

Some of his formulas are…they contain π, like you said…

Yes, in ways that you wouldn’t expect.

There’s one particular work that I did, that was inspired entirely by Ramanujan. It’s a question about quadratic forms. There’s an old, famous result by Lagrange that says every positive whole number can be expressed as the sum of four square numbers. It’s called Lagrange’s Four-Squares theorem. No matter how big a number you take, it will always be the sum of four square numbers. Another way to say that is, every positive whole number can be written as a2 + b2 +  c2 + d2.

So Ramanujan asked: Are there other such quadratic expressions that take every possible positive whole number value? He wrote a paper in which he collected 54 more such quadratic forms! Not only a2 + b2 +  c2 + d2 takes every whole number value, a2 + b2 +  c2 + 2 d2 also, a2 + b2 +  c2 + 3 d2 also. He listed 54 such things.

So that was something I worked on. Inspired by Professor John Conway, I asked the question: What are all the quadratic forms that take every positive integer value? With a collaborator, Jonathan Hanke, we’ve classified all of them, sort of inspired by Ramanujan’s work, building up on the techniques that he introduced to prove such things. We showed that a quadratic form takes every positive whole number value if and only if takes the values 1 through 290. So if you have some quadratic expression and you want to know whether it takes every positive whole number value, you just have to test the numbers 1, 2, 3, 4, up to 290. We call that the 290 Theorem and it was directly inspired by Ramanujan.

So yes, Ramanujan’s work has definitely been a huge inspiration to me.

There’s a biography of Ramanujan, The Man Who Knew Infinity, by Robert Kanigel. According to him, the way Ramanujan worked was that…he worked in flashes. He would have these sudden flashes of insight. Is that how a lot of the top mathematicians work? Does the proof come almost fully formed?

I think it’s very rare for people to have flashes like Ramanujan did. Those are pretty unique. But of course many of the ideas that I have, and scientists have, do come from flashes of inspiration.

So you’ll be toying with it for a while and it suddenly falls…

Yes, and sometimes when you’ve given up, one day in the shower you suddenly have this idea that solves the problem. It happens to a lot of mathematicians. It’s certainly happened to me.

 

Learning from the Ustad

 

You’ve said in one of your interviews that you sometimes play the tabla when you want a break and then you have some idea maybe…

Yeah, that happens a lot too. Sometimes I’ll be stuck on a problem and I’ll go and play tabla for a while, and then things will clear up in the mind and I figure out how to do it.

You are a student of Zakir Hussain. Were you famous at the time you met him?

Was he famous? [He genuinely misheard.]

No, were you famous? [All of us laugh.]

I met him when I was an undergraduate student at Harvard. He came to play there, and I was so excited and inspired. I used to play with one of his students there and he introduced us. I was already teaching as an undergraduate; I was teaching some of the math courses. He introduced me saying here’s an 18-year-old who already teaches mathematics courses at Harvard. Zakir ji said to come visit him in California. He and his family have been such a source of inspiration and encouragement and friendship to me ever since.

You still learn from him?

Sometimes. As I said, I haven’t done math or anything in a year. But I hope to return to it.

 

​Mathematics and Poetry in Ancient India

 

You’ve talked about mathematics in ancient India. Would you consider writing a book about it at some point?

Yes, I’d definitely like to sometime.  It probably won’t happen too soon, but I do hope to at some point.

I think there’s quite a bit of misinformation about what we’ve achieved all those years back and what we’ve not. There are people who’d like to twist it and claim that all of modern science was anticipated by ancient Indians.

Sometimes the debate in India is — everything came from ancient India versus nothing came from ancient India. The truth is of course something in between: lots of amazing things were discovered in ancient India — we sometimes forget them. I’ve certainly seen debates in the newspapers and TV that sound like the two sides are arguing whether nothing came, or everything came, from ancient India. It’s quite important for people to be informed about what actually was done. As with any academic conversation, it is important that the discussion be carried out scientifically and not emotionally.

That’s where we think that if someone like you could write a book about it…

Okay, that sounds like a good encouragement.

It’ll be authoritative. To say that this is not what we did but we definitely did do this.

Yeah, I hope to.

So what you talked about for Sanskrit poetry — the relation between mathematics and counting metres of Sanskrit poetry — is it valid for other languages over the world?

You mean is there such mathematics in other languages? No. As I explained in the talk, the long syllable is exactly double the length of the shorter one in Sanskrit. Because of that, mathematical patterns emerge when you arrange them in different ways. This is something very special about Sanskrit poetry and poetry that’s in other Indian languages. I think, because of that, there’s more mathematics here than most languages.

 

Prof. Mira Bhargava with her son.
Prof. Mira Bhargava with her son.

You had access to your grandparents’ collection of Sanskrit poetry…

That’s right. That’s why I was lucky. In my family, I had people who are experts in this field. I could ask them whenever I wanted, they had a lot of books on their bookshelves that I could just read and ask them anything I didn’t know.

Has the government contacted you for archiving these resources?

No, they haven’t. Maybe in the future.

It might be useful to have these available easily online, with commentary in accessible language.

They’re available in libraries if you’re willing to dig them up but there are so few people who are well-versed in Sanskrit, and in the traditional poetry and in the mathematics. There are lots of Sanskrit scholars who can understand the language but not the mathematics and on the other hand there are also many mathematicians who can appreciate the mathematics but not the language. I think the problem in India is that we don’t have enough people well-versed in both.

That’s why we think a book from you would be really useful.

But there’s so much, more than one person can do. I can write a book and get people interested but there have to be more people going into this in the future. There’s a lot there to be explored and I hope more people get interested. Nowadays, in India, if you’re strong at mathematics you just go into engineering and never study the artistic side of things. It’s important for creativity that you study both art and science. They work off each other, you should develop both sides of your brain and then you can appreciate these ancients’ work which can be a great source of learning.

Any unproved theorems and conjectures in the ancient Indian texts?

I don’t think there are many deep things that we don’t know… although I have seen some. Like, here is a rule to generate the number of rhythms that do the following things. I have collected some of these and hope to get around to proving some of them, so there are definitely some new things there. But I don’t think the important aspect of these ancient texts is that we’re going to learn many new theorems and conjectures; it is about the way of thinking and learning we’ve lost that should be brought back.

Kerala had a very active school of mathematics. There’s this theory that Jesuit priests transmitted calculus from Kerala to Europe.

I don’t know when and how these things were transmitted but certainly the seeds of calculus were discovered in India in the Kerala school. I don’t know if there is any evidence regarding this transmission.

It’s the same thing with Hemachandra and Fibonacci. We know Fibonacci read a lot of  ancient Indian texts but he never wrote that he got his ideas from there so we’ll never know. Fibonacci was the first one to introduce Indian numerals to Europe so it’s very likely he read about Fibonacci numbers in those texts but it’s hard to know definitively.

 

About  Teaching

 

 

Prof. Manjul Bhargava
Prof. Manjul Bhargava

 

We know that you love teaching. Does teaching help with your research? Do you get ideas when you teach?

Of course. When you teach you really have to go back to the basics and understand things from the very roots. Also, students ask such questions, they often ask things that you hadn’t ever quite thought of, about explaining that aspect. It makes you rethink the knowledge that you think you have. It makes you relearn it and reevaluate it. That can be very helpful.

Here in India there’s an unfortunate divide between teaching institutes and research institutes. Until recently we had exclusively postgraduate research institutions.

I think it’s unfortunate that scientists in India end up at purely research institutions and so they never get to pass on their knowledge to the next generation. And also they don’t get to teach. They don’t get that benefit. And students don’t get that benefit. That is unfortunate. Part of India’s future has to be to open institutes that involve both. Perhaps the IITs can become good research institutions as much as ICTS and TIFR. That is something that India definitely needs. That’s not to say that you shouldn’t have any research institutions. And it’s good to have purely teaching institutes too. There should be a place where top scientists get to teach undergraduates.

You’re associated with TIFR as well as the University of Hyderabad. Will you be spending some time in India teaching?

Yeah, I already do. This is my fifth time here this year. I come to India pretty much every year. I spend a lot of time at IIT Bombay, TIFR, teaching. I hope to keep that up.

So you teach small workshops?

Yeah, I haven’t done whole-semester courses yet. But I hope to.

What do you teach primarily? Do you span from very basic to very advanced courses?

I teach the whole range. I teach humanities majors, and also mathematics majors all the way up to Ph.D.

So it’s like Maths for Poets.

Yes, Maths for Poets all the way through advanced mathematics Ph.D. courses.

Do you get interesting questions from the poets?

Yeah, very much so. Some of the best questions actually come from the humanities students sometimes. They sometimes think in ways that mathematicians wouldn’t. I enjoy teaching of every kind.

Do you always use transparencies, like in your pre-convocation talk? It seemed very….

Very old-fashioned. I actually usually use blackboards. So even more old-fashioned than that! But since blackboard is hard to arrange in such a large talk, it’s the next best thing.

The reason I don’t like computer projection is it’s not very…it doesn’t look very personal. It’s sort of like the transition from handwritten letters to email. You lose the personal touch. Everyone’s font looks the same when you have a PowerPoint slide. You don’t see any of their personality in it. Here I get to do my own handwriting and my own colours, whatever notation I’m using.

The other thing is that computer projections are not well-suited to mathematics teaching. I don’t know how it is here. Do mathematics teachers here use computer projections?

No. They use the blackboard.

Even though the way people do presentations is modernized, mathematicians still stick to their old-fashioned devices. There’s no better way of teaching mathematics than with a blackboard in your own handwriting. And that formula you need will still stay on the board. In computer projections, when you go to the next slide, everything that was written before is lost. So any definitions you had, any formula you have, they disappear. Unless you decide to re-copy them.

I think also in computer projections the equations come fully-formed.

Exactly.

It’s really intimidating because the entire equation is…

Whereas if you write things gradually…

 

Fields Medal Work

 

About your Fields Medal work. I believe you considered computational evidence while working on the rank of the elliptic curves. Some mathematicians may disregard computational work and say it spoils the beauty. What’s the role of computational evidence in mathematics?

I love computational evidence. If you have a good prediction then you can bear it out in computation.

I believe in this case it was leading you in one direction, but it proved to be…

That’s right. Computations can also be misleading. Sometimes when you have a theoretical result and then you have computations that agree with that result, it’s very satisfying that you were able to predict it, just like for any scientist. A scientist builds a theory and computations bear out that theory…it’s the same for mathematicians. You have a theory that predicts that asymptotically this should behave this way and then computationally that’s actually happening. That’s very satisfying. It validates the theory and it’s another check that the proof was correct.

But you’re right, sometimes computations can be misleading. The ranks of elliptic curves are an example where the computations were actually misleading. So one has to be very careful in computations. Because sometimes the best computers working for hours still can’t provide enough data that will show the pattern. One has to always take computations with a grain of salt.

Do you use them along the way or only at the end when you have the entire theoretical idea ready?

Sometimes computations can help predict what the right theory should be. But sometimes they can be misleading. When you do computations you have to figure out which case you’re in. That’s part of the goal of theoretical science. Sometimes the theoretical heuristics tell you how much computation you have to do before a pattern emerges. And so you have to use those heuristics when you do computations. Just to know if the computations are telling you anything yet. They’re extremely valuable, just like they are in every science. They’re of value for mathematicians too, to suggest what the correct theory should be.

In physics, there are a lot of models which are deemed ugly. And then people say they’re probably not correct.

Good mathematics is beautiful mathematics, and in practice too the most applicable mathematics tends to be the simplest and the most beautiful mathematics. The simple ideas are the ones that tend to see the most applications and to be the ones that attract mathematicians.

Beauty has been a great guide for progress in mathematics and science. The simplest and the most beautiful ideas tend to be the ones that are true. History bears that out. So beauty is a great guide in telling what is the correct mathematics.

Could you explain the card trick that you did in the pre-convocation talk?

It’s magicians’ code, you know. [All of us laugh.]

It was invented by my adviser at Harvard, Persi Diaconis. He was a professional magician. Do you know his story? After being a magician over ten years, late in his 20s, he decided he wanted to a Ph.D in mathematics, because a lot of his tricks were based on statistics and combinatorics. He got so interested in mathematics he decided he wanted to get a Ph.D. And he applied to Harvard. [Laughs.] As a magician.

And he got a recommendation from Martin Gardner, the great magic and mathematics writer. Martin Gardner wrote a recommendation saying something like: Of the ten best tricks that have been invented in the last decade, two are due to Persi Diaconis. You should take him. And Harvard took him. And he got a Ph.D. in mathematics. He was a professor at Harvard a few years later. In mathematics. A great story.

Just like for me, I got into mathematics from poetry and music. For him, he got into mathematics through magic. So I learned a lot of magic from him. That’s why I now bring in magic.

So you have many more card tricks up your sleeve?

Yeah. [Laughs]

An earlier version of this post implied Arul Shankar did his Ph.D at CMI. In fact, he did his Ph.D with Prof. Manjul Bhargava at Princeton. The error is regretted. A few other minor corrections were also made, for which we thank Prof. Manjul Bhargava.

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  • Jenifer T.A.

    it is a very inspiring
    and interesting conversation…….

    excited to see Prof. Bhargava @ ISCA 2016