THE HEIRS TO RAMANUJAN’S GENIUS

Dilip D’Souza

Writing about mathematics and its practitioners is a double-edged sword. I can and do look forward to the delights that glimpses of esoteric mathematics afford a dabbler like me. Yet even with glimpses, I can and do sink quickly into the often dizzy feeling that I’m entirely beyond my level of competence.

Really no way around that sword, so I will leave it there.

When I set out to “profile” some outstanding young Indian mathematicians, I started by writing to some (older) mathematician friends, asking for names. This gave me a shortlist. I sent messages to everyone on it asking if they would let me write about them, and to tell me a little bit about their work and interests. A few did not reply, a few others declined. (One of my friends remarked that some of the folks he had suggested had “somewhat austere views on publicity in the media”.)

But others wrote back, and this is the result. As they will themselves agree, this is by no means a definitive list, not even in the areas of mathematics it covers (number theorists are disproportionately present, as are modular forms, and so on). This introduction to these six people is instead meant to give you just a taste of the work and personalities of some bright, thoughtful Indians.

And oh yes, they happen to be mathematicians.

So, with no further ado:

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Soumya Das (33), Indian Institute of Science, Bengaluru

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Possibly one of the least-known and yet most charming characteristics of mathematicians is their search for beauty in their work. It can be hard for a non-mathematician to understand why some apparently obscure and opaque collection of text, symbols and concepts qualifies as “beautiful”, or “elegant”, or even “nice”. Yet you will often see such adjectives used.

And certainly, you will see them used in discussions of the field in which Soumya Das works—modular forms. Consider the terms I found in just five minutes reading about them: “enchanting”, “neat”, “beautiful”, “nice”, “spectacular” “unreal”, “shiny”, “mysterious”. I get the feeling one or two of these have specific mathematical meaning, but then that itself tells a story. (One page also had these lines: “Why are modular forms interesting? We don’t quite know… that’s why they are so interesting!”)

And Das’s own passion for them comes through in the phrases he used in his mail to me: “some very basic questions that have intrigued me”, “favourite aspect of my work”, “interesting objects… worth studying”.

So, what are these interesting objects? Part of number theory, modular forms are certain carefully-defined mathematical functions with wide uses. While they are hard to explain in a space like this, here’s one way of getting a little closer to understanding their use.

One of the oldest problems in mathematics concerns partitioning a number—splitting it into parts that add up to the whole. Imagine, for example, that you are a teacher of a small class and you want to assign projects to your students. They are not all equally bright or diligent, so you probably want to put some kids in teams and have others work alone. In general, you want to know how many ways there are to divide up, or partition, your class—and with that understood, how many projects you need to have ready.

So, if you have four kids, you have these partitioning options: 4 itself (i.e., one part that is really the whole), 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. That’s 5 possibilities, which is to say, there are 5 partitions of the number 4. The more the students, the greater the number of partitions, and so the harder it is to simply enumerate them. (After all, my school class of 24 can be partitioned in no less than 1,575 ways, and had we been 100, we would have had 190,569,292 ways.) So, naturally, mathematicians search for formulae, or functions, to do such calculations. And as Das told me, such functions are “closely related to a modular form”.

The remarkable Srinavasa Ramanujan was interested in these themes too. For example, he proved that the number of partitions of every fifth number starting from 4—i.e., 9, 14, 19, 24, and so on—was divisible by 5. He had similar results for 7 and 11 too. And the proof that certain mathematical objects, called elliptic curves, are really just disguised modular forms was a major stepping stone for Andrew Wiles. In 1995, of course, Wiles proved a problem that had bedevilled mathematicians for several hundred years: Fermat’s Last Theorem.

Das’ research seeks to answer various questions in this field. At the Indian Institute of Science, he can regularly be found playing table tennis—that is, when he takes time away from caring for his newborn son. In the way that small things tell you a lot, it seems just right to me that Das’ Google Plus profile picture features Tintin. That spirit of fearlessly venturing into the unknown that Tintin is known for is no doubt one reason that the Indian National Science Academy awarded Das their Young Scientist Medal in 2014.

One of his papers says this: “We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.”

No, I don’t understand. But in those lines, there’s almost poetry. Even beautiful poetry.

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Kaneenika Sinha (34), Indian Institute of Science Education and Research, Pune

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Ninety-nine years ago, Ramanujan published a paper in a Cambridge University journal. It had the deceptively innocuous title On Certain Arithmetical Functions, and it caused waves then that still resonate. In the paper, Ramanujan, with the mathematical ease he was known for, analysed a particular function that—I am hardly ashamed to admit—I still don’t fully comprehend. But others did. In 1927, his Cambridge colleague and mentor G.H. Hardy wrote of this function that it “is important in the theory of the representation of a number as a sum of 24 squares”.

Here’s what’s delightful about mathematics: I bet you, like me, didn’t even know there was a theory of the representation of a number as a sum of 24 squares, much less that at least two major mathematicians of the 20th century spent time studying it. But that paper greatly influenced number theory. And that function, arguably, ended up shaping Kaneenika Sinha’s research.

Ramanujan proposed three conjectures based on this one function, one of which wasn’t proved until 1974. But the intervening decades saw the birth and expansion of a whole new area of mathematical interest, modular forms. Ramanujan’s conjectures can be understood as special cases of some facets of modular forms, and Sinha’s work focuses on those. It involves asking questions about aspects of irrational numbers, meaning numbers (like pi) that cannot be expressed as a regular fraction (like 3/4). In particular, if you manipulate irrational numbers in particular ways, you end up with numbers between 0 and 1. But are they bunched around 0? 1? Somewhere else? Or are they evenly distributed? In another famous 1916 paper, German mathematician Hermann Weyl asked some more refined questions about this, laying the ground for the theory of uniform distribution. By her own admission, these two seminal papers from nearly a century ago have greatly influenced Sinha’s research.

The questions they ask are not just exercises in pie-in-the-sky futility, nor mere number manipulations. As Sinha points out, what we learn from those irrational numbers is “connected to deep and beautiful facets of various other areas of mathematics”. What’s more, it has profound implications for quantum physics. Grasping for at least a flavour of what she does, I began to understand why a mathematician friend wrote to me that Sinha is “well regarded in her work”.

But Sinha’s interests stretch beyond mathematics as well. For two years, she used to blog anonymously, writing in detail about her search for an academic position in India, and then about her life at the institute she joined, IISER Kolkata. In 2013, she gave up the anonymity, though she still posts regularly. Surprisingly, there is nearly no mathematics in the blog. Instead, she muses about student supervision, collaborators, the Booker Prize-winning novel she is reading, moving from IISER Kolkata to IISER Pune, a conversation with her father, parenting and plenty more. It’s a startlingly introspective, always honest blog, one that offers a rare glimpse into the academic life in this country.

One post, from April this year, is about running into a relative (“who wrote motivational psychology books”). He says to her: “I don’t understand your job. Do you study mathematics for the sake of beauty or as a duty?”

Sinha was flummoxed by the question and remains so even now. She writes: “I am still not sure I have an adequate answer. Sure, it is beauty that attracts many to science, but is it entirely what keeps one on it, day after day, year after year?”

A question worth asking, really. It’s easy for a layman like me to talk about beauty in mathematics and earn brownie points for doing so with readers and mathematicians alike. After all, I’m not doing the hard work that produces the beauty.

But do professional mathematicians think about it all the time?

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U.K. Anandavardhanan (39), Indian Institute of Technology Bombay

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U.K. Anandavardhanan is the only person here whom I have actually met. I first got to know him several years ago, but not as a mathematician. He was the man behind a thoughtful, erudite blog on subjects ranging from politics to literature, with the occasional mathematical diversion thrown in. Intrigued, I wrote to him and we started corresponding, then worked together for a while on a collaborative blog. It was also Anand who first introduced me to Sudoku. In those days, he was a post-doctoral scholar at Tata Institute of Fundamental Research. He has since moved to IIT Bombay as a professor.

Anand’s mathematical interests lie in number theory, under the broad rubric of what’s now known as the Langlands programme. In the 1960s, Robert Langlands proposed an overarching theory, or framework, for mathematics. It linked fields—number theory, geometry and more—that until then were considered unrelated. This was really a new way to look at these fields—for that matter, at mathematics itself—and as invariably happens with a change in perspective, it offered new insight into old problems. Perhaps the most celebrated triumph of the Langlands programme was Andrew Wiles’ proof of Fermat’s Last Theorem.

Anand particularly seeks to understand the behaviour of mathematical objects called groups. Think of a group as a set of objects, along with an operation that combines two or more of those objects to form another member of the set. For example, take the set of all books published by authors from Telangana. Suppose you then picked out all the works by a particular author and put them together as an anthology, and you do this for every author represented. The anthologies are, by definition, also in the set. So these books, along with this one operation, form a group.

Mathematically, of course, groups are more precisely defined. But think of the set of all integers, and the operation of addition: there’s a group. The set {1, -1} and the operation of multiplication: another group.

So, why study groups? One major area where they are useful is in cross-checking transmitted data—like a credit-card transaction or an online college application. Groups are useful in pinpointing errors in transmission, and even correcting them.

Anand focuses on ways to represent groups—itself a long-studied field of mathematics. Such representations make it possible to look at problems related to groups through other mathematical lenses that offer new insights. The great mathematician Carl Friedrich Gauss struggled with one aspect of all this for four years. When, in 1805, he finally found the proof he was looking for, he wrote to a friend that “as lightning strikes [so] was the puzzle solved”.

Some of Anand’s work builds on Gauss’ famous result. It has already brought him recognition—he has won Young Scientist awards from both The National Academy of Sciences, India, and the Indian National Science Academy; INSA has also made him one of the founding members of the National Young Academy of Science.

But something else about Anand is even more remarkable. “He is also a great teacher”, another mathematician friend told me, “[which] is unusual for a young mathematician”. And one of his students, evaluating Anand, wrote, “I can scarcely remember a moment [in his classes] when I felt disinterested, or even bored.”

No wonder IIT Bombay gave him their Excellence in Teaching award in 2010. Knowing him as I do, I suspect it is the prize he is proudest of winning.

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Amritanshu Prasad (40), Institute of Mathematical Sciences, Chennai

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Amritanshu Prasad tried to explain his research interests to me via, of all things, necklaces. Let’s say you have a supply of beads of three different colours—red, blue and green—and you want to make a necklace with five beads. How many possible necklaces are there to be made?

At first glance, this is a classic problem in basic combinatorics, which many college students learn early on. For each of the five bead positions in the necklace, there are three possible colours. So you would think you can make 3 x 3 x 3 x 3 x 3 = 3^5 = 243 different necklaces.

But remember that this is a necklace, therefore circular. This means that the bead pattern “rgbbg” is the same as “bgrgb”, by virtue of moving the last two beads around the string. Meaning, “rgbbg” can be trivially transformed into “bgrgb” without breaking the necklace. So, those two are identical. However, “rgbbg” is not the same as “gbrgb”—because there’s no way to transform one into the other without breaking and restringing. Thus “rgbbg” is a different necklace from “gbrgb”.

As you can tell, 243 is clearly too many necklaces. The correct answer is given by a mathematical result known as the orbit counting theorem, and it is 51.

Knowing how to accurately count such structures is important in chemistry, for example, because a necklace is a good way to think of the molecular structures of certain compounds. Similarly, a double helix is a good descriptor of a DNA molecule. So, we might want to know how many compounds can be formed with the same atoms, moving them around much as we do the beads. Or, with DNA, how proteins fit on the double helix can decide how tall you are, or what colour eyes you have. And there might well be other constraints—some atoms or proteins may not take well to being juxtaposed with others, for example.

One of Prasad’s research interests is how such constraints affect the counting. He’s also interested in symmetries in such objects, meaning transformations that make no difference to the structure. There are ways to represent such symmetries as mathematical groups—the same kinds of mathematical structures that Anandavardhanan is interested in—and that kind of representation, it turns out, can help chemists understand the properties of those compounds.

It also turns out Prasad and I share an admiration for a man I once wrote about for Mint (“The choices we make”)—V. Krishnamurthy, who taught me combinatorics at Birla Institute of Technology and Science, Pilani. Prasad has not met VK, but treasures his books. He even shared with me a slide presentation—with Tamil subtitles, no less—that draws on VK’s Culture, Excitement and Relevance of Mathematics. Titled “Counting and Symmetry”, it starts with the necklace example and moves from there to symmetries and their implications for chemical compounds—and all in terms that a non-mathematician like me can grasp.

His work, another mathematician told me, “is highly regarded”. But his ability to explain it to layfolk is itself worth regarding highly.

Beyond research, Prasad also participates actively in his Institute’s outreach programmes, speaking to high-school students about different mathematical ideas. One of these lectures explained the ancient Greek master Euclid’s method to find the greatest common divisor of two numbers, involving two sticks of different sizes. This is interesting not just because it’s a fresh approach to a concept that so many kids agonize over, but also because Prasad takes it further, into a discussion of irrational numbers. For few things demystify mathematics so much as when you draw links between apparently disparate ideas. Another lecture was about the so-called Platonic solids. Here, he showed his audience how to use origami to make models of these five objects.

In fact, Prasad has a tremendous enthusiasm for origami, particularly as a tool for teaching mathematics to children. He told me he believes all mathematicians—maybe even all humans—can benefit from learning it.

A mathematician learning and teaching Japanese paper-folding techniques—who would have thought it?

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Ritabrata Munshi (38), Tata Institute of Fundamental Research, Mumbai

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There’s a million-dollar reward (the Clay Millennium Prize) if you can solve the Riemann Hypothesis. It suggests that the real part of the non-trivial zeros of the Riemann zeta function is always 1/2. If you are anything like me, you are looking for a substantial reward for even comprehending what that means. In any case, mathematicians are interested in the Riemann Hypothesis because it tells us things about prime numbers. If you want to know how many prime numbers there

are below a million, or a billion, or any given number—and you don’t care to enumerate them, the Riemann Hypothesis is your friend.

And this has connections to Ritabrata Munshi’s research interests. He works with L-functions, which can be thought of as generalizations of the Riemann zeta function. If the Riemann Hypothesis considers the places where the zeta function becomes zero, Munshi is interested in the conjecture that such functions don’t take large values anyway. In fact, a Finnish mathematician called Ernst Lindelöf suggested that the values are, says Munshi, “quite small”.

To an outsider, what’s interesting about this area of mathematics is that it is a veritable nest of conjecture and hypothesis. Meaning, there’s plenty in here that remains to be proved, and thus plenty of material to excite mathematicians. Lindelöf’s hypothesis actually goes further than “quite small”: it says a particular function’s value is precisely zero. But nobody has managed to prove that yet, and that’s typical for L-function theory.

I realize this is getting somewhat opaque, but there’s an interesting history here. If Lindelöf’s hypothesis remains unproved, we do know from some other work that that function’s value is no more than 1/4. Around 1910, G.H. Hardy and J.E. Littlewood managed to lower that limit to 1/6, or about 0.166667. Since then, a series of mathematicians have taken it steadily lower. Last year, Jean Bourgain nailed it at 53/342, or about 0.154907.

If you are looking for a quick demonstration of how difficult mathematics can be, I might offer you this nugget: the work of several diligent mathematicians has taken that Lindelöf limit all the way from 0.166667 to 0.154907. A 7% drop in a century of hard work, and that in pursuit of the really difficult task—proving that it is actually zero. Talk about progress.

Munshi told me that making even more progress with L-functions is—understatement alert coming up—“a challenging problem as the well-known methods do not work”. I like to think of that in the context of the other things he told me about himself. He was a “late starter” as a child, he said, and faced difficulties learning languages. But he was always good at mathematics and physics. Ramanujan and Albert Einstein were his heroes. By the end of high school, he had become fascinated by analytic number theory. And from an early age, he also said, he worked out his own style of doing mathematics.

For those reasons, I tend to believe Munshi when he says he thinks he has a new approach to the L-function problem “which provides the only breakthroughs”. His mathematical peers think so as well: one described his work to me as “really outstanding”.

Obviously, I’m incompetent to judge where all this will lead. But I would like to place it on record: if you win that Clay Millennium Prize for solving the Riemann Hypothesis, Munshi, I’m your friend.

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Nikhil Srivastava (31), University of California, Berkeley

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In an earlier life as a computer scientist, I worked with three others building a system we called PlaneText. It allowed you to link documents and use those links to leap from one to another: yes, a precursor to the hypertext of today’s World Wide Web. My part of the project was a software program that, once you created a network of linked documents, would draw you a picture of that network. That made it easier to visualize what you had done. Of course, we were only dealing with tiny networks then—I shudder at the prospect of setting my creation to work on the immense tangle that is today’s Web. I mean, it would make no sense to even try that.

But memories of that project came flooding back as I learned about Nikhil Srivastava’s work. Given a network—one we created with PlaneText, or a roadmap of a large city—there are various questions to ask about it. What is the shortest distance between any two nodes? How many links are redundant? If you want to visit every node, can you do it without repeating a node, and what’s the minimum distance you must travel?

Some of these are hard questions that mathematicians and computer scientists have grappled with for years. This is partly because networks are generally densely connected, and therefore algorithms designed to answer these questions can take very long indeed to run their course. But Srivastava has been working for a while on what’s known as “sparsification”. That is, he searches for a less dense (sparser) approximation to a given network. He uses that to get an approximate, but relatively accurate, answer to those questions that is still adequate for most purposes. The best approximations are known as—that genius yet again!—Ramanujan graphs.

You can think of all kinds of applications for sparsification. Here’s one that occurred to me after a recent brush with crime: say you are worried about burglars and you install a grid of iron bars on your windows. But the result is that you find no outside light penetrates the mesh. As you mope about in the dark, the question arises: how many (and which) bars will you remove so that you are still safe from burglars, but enough light gets through? Srivastava’s sparsification can offer answers.

He does so by translating graph problems into equivalents in another area of mathematics. That process itself produces unexpected connections to still other fields. For example, sparsification is “intimately related” (his words) to the L-functions that Ritabrata Munshi studies. Such connections are, he wrote to me, “extremely fruitful and reveal hidden, beautiful structure”.

It’s this work that led him and two colleagues, Adam Marcus and Daniel Spielman, to the proof of a half-century-old conundrum, the Kadison-Singer problem. He told the journal Asian Scientist: “It implies that it is possible to ‘approximate’ a broad class of networks by networks with very few edges.” Sparsification, in other words.

The proof brought them a major mathematics award last year, the George Pólya Prize. (Pólya was a distinguished Hungarian mathematician who also wrote a small classic on solving problems, How To Solve It).

It also inspired the science journalist Dana Mackenzie to almost poetic heights in SIAM News: “What is the best kind of mathematical problem? Does it resemble a distant mountaintop, beckoning people from far and wide to attack the summit…? Or is it like a vast subterranean river, connecting different realms of mathematics, mysteriously disappearing below the surface and reappearing where you least expect it? [In 2014], it was the turn of one of the great subterranean rivers of mathematics to emerge into the limelight. [A] three-member team of mathematicians and computer scientists posted a proof online for the Kadison-Singer conjecture.”

Connections between seemingly unrelated themes, the “hidden, beautiful structure” they reveal; this is why Srivastava told me: “There seems to be a deep underlying unity in math, and it is thrilling to make contact with it.”

What would it do to our kids if it was taught in that spirit of thrill and wonder? But let Srivastava have the last word here. Speaking toMan’s World after he won the Pólya Prize, he said, “The main reason I like to do math is because it’s beautiful… I actually think of mathematics as magic.”

Maybe that even answers Kaneenika Sinha’s dilemma.

http://mintonsunday.livemint.com/news/meet-the-heirs-to-ramanujans-genius/2.3.3593071363.html

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