Tag Archives: Manjul Bhargava

Motivation: Zakir Hussainji on tabla; horse running

This comes from real “sadhana”. I think I had read somewhere that Zakir Hussain used to get up at 2:00 am in the morning to do “riyaaz” with his father Ustad Allah Rakha.

The same kind of passion, spirit and “sadhana”/tapasya are required to do real mathematics or to acquire “siddhi”.

I was wondering if Prof. Manjul Bhargava too could play such stuff on his tabla. 🙂 🙂 🙂

Hats off to “the” tabla maestro and another a “Fields” medalist-cim-tabla player.

-Nalin Pithwa.

Major Mathematics awards to two Indian origin scientists


The Hindu Aug 14 2014

Manjul Bhargava and Subhash Khot, are among the eight winners of the prestigious International Mathematical Union awards

Two mathematicians of Indian origin, Manjul Bhargava and Subhash Khot, are among the eight winners of the prestigious awards of the International Mathematical Union (IMU) that were announced at the inaugural of the 9-day International Congress of Mathematicians (ICM) which began today at Seoul, Republic of Korea. The President of Korea, Park Geun-hye, gave away the awards.

The ICM is held every four years and, traditionally, the IMU awards are presented at this quadrennial event. The awards include the Fields Medal, the highest award in mathematics, the Rolf Nevanlinna Prize and the Carl Friedrich Gauss Prize. At the last ICM held at Hyderabad, India, two new awards, the Chern Medal and the Leelavati Prize, were added to the existing three awards.

The 40 year-old Canadian-American Manjul Bhargava, a number theorist from Princeton University, is one of the four Fields Medalists chosen for the ICM2014 awards. The Fields Medal is awarded “to recognize outstanding mathematical achievement for existing work and for the promise of future achievement”. A minimum of two and a maximum of four Fields Medals are given to mathematicians under the age of 40 on January 1 of the year of the Congress.

“Manjul Bhargava has developed powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves,” said the IMU citation for the award.

The other three Fields Medalists are:

The Brazilian mathematician Arthur Avila (35) of the Paris Diderot University-Paris 7 and Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, who has been awarded “for his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalizations as a unifying principle”; the British mathematician Martin Hairer (39) of the University of Warwick “for his outstanding contributions to stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations”; and, the Iranian mathematician Maryam Mirzakhani (37) of Stanford University “for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their modulii spaces”.

The 36 year-old IIT Bombay alumnus Subhash Khot, an Indian-American theoretical computer scientist at the Courant Institute of Mathematical Sciences of New York University has been chosen for the ICM2014 Rolf Nevanlinna Prize. The Nevanlinna Prize is awarded “for outstanding contributions in mathematical aspects of information sciences”.

“Subhash Khot’s prescient definition of the “Unique Games” problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry,” the award citation said.

The Gauss Prize in Applied Mathematics is awarded “to honor scientists whose mathematical research has had an impact outside mathematics – either in technology, in business, or simply in people’s everyday lives”.

The winner of the ICM2014 Gauss Prize is Stanley Osher (72) of University of California, Los Angeles, who has been awarded the Prize “for his influential contributions to several fields in applied mathematics, and for far reaching inventions that have changed our conception of physical, perceptual and mathematical concepts, giving us new tools to apprehend the world.”

The Chern Medal is given “to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics”.

The Chern Medal this time goes to the American algebraic geometer Phillip Griffiths (76) “for his groundbreaking and transformative development of transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties”.

Unlike the other awards, the Leelavati Prize is not given for achievements in mathematics research but for outstanding public outreach work in mathematics. Proposed by India, it was originally intended as a one-time award using the grant from the Norwegian Abel Foundation. Thanks to the efforts by Indian mathematicians in finding a sponsor to make it a regular affair, it has now been instituted as a recurring four-yearly award under the IMU charter to be given away at the closing ceremony of the ICM. The award is now being sponsored by Infosys, the Indian IT major.

The ICM2014 Leelavati Prize has been given to the Argentine Adrián Paenza (65) “for his decisive contributions to changing the mind of a whole country about the way it perceives mathematics in daily life, and in particular for his books, his TV programmes, and his unique gift of enthusiasm and passion in communicating the beauty and joy of mathematics”.

Hope Indian youth take up research in sciences: Fields Medal winner


The Hindu Aug 14 2014.

Manjul Bhargava, one of the recipients of the Fields Medal, speaks about mathematics, music and more.

How does it feel to have won the Fields Medal? You are the first person of Indian origin to be getting it…

It is of course a great honour; beyond that, it is a great source of inspiration and encouragement – not just for me, but for my students, collaborators, and colleagues who work with me. Hopefully, it is also be a source of inspiration for young people in India to take up research in the sciences!

You have grown up in Canada… did you have any cultural identity questions? Do you think of yourself as a Canadian, American, Indian or none of these or all of these?

I was born in Canada, but grew up mostly in the U.S. in a very Indian home. I learned Hindi and Sanskrit, read Indian literature, and learned classical Indian music. I ate mostly Indian food! On the other hand, I grew up playing with American kids and went to school mostly in the U.S. I liked growing up in two cultures like that because it allowed me to pick and choose the best of both worlds. My Indian upbringing was very important to me.

I also spent a lot of time in India growing up. Every three or four years, I would take off six months of school to spend it in India — mostly in our hometown Jaipur — with my grandparents. There I had the opportunity to truly live in India for extended periods of time, go to school there, brush up on my Hindi and Sanskrit, and learn tabla (as well as some sitar and vocal music). I particularly enjoyed celebrating all the Indian holidays as a child, and flying kites on Makar Sankranti.

I feel very much at home in all three countries. So I definitely think of myself as all three – Canadian, American, and of course Indian.

How did you get interested in Tabla playing? You have learnt the Tabla from Ustad Zakir Husain… Can you tell us how this came about and what it means to you?

I first started learning from my mother, who also plays the tabla. When I was maybe 3 years old, I used to hear my mother playing 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. 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.

Do you collaborate with mathematicians in India? Do you have contacts with the institutes in India?

For many years now, I have been an adjunct professor at TIFR-Mumbai (Tata Institute for Fundamental Research), IIT-Bombay, and the University of Hyderabad. I’ve spent a lot of time at these three institutes, especially at TIFR and IIT-B, over many years. I’ve lectured extensively to students at these institutes, as well as collaborated a lot with mathematicians there, such as with Eknath Ghate at TIFR (who recently won the Shanti Swarup Bhatnagar Prize for mathematical sciences).

I’ve also been involved in starting a new institute in Bangalore called “ICTS” (International Center for Theoretical Sciences). It will be inaugurated next year, and we hope it will be a great success. The director is Professor Spenta Wadia of TIFR, and the head of the International Advisory Board is Nobel Prize Winner Professor David Gross. So hopefully I will spend even more time in India after the inauguration next year!

Recently you have won prizes for your work on the Birch and Swinnerton-Dyer conjecture which was listed as one of the seven millennium prize problems. Can you explain the significance of this work?

In joint work with Christopher Skinner and Wei Zhang, we have shown that the Birch and Swinnerton-Dyer Conjecture is true “most” of the time (more precisely, for more than 66.48 per cent of elliptic curves!). Previously, it was not known that it was true for more than 0 per cent. So that is significant progress, but it is still “not” a complete solution!

Finishing a proof of the Birch and Swinnerton-Dyer Conjecture would be a momentous achievement, and it is one of my favorite problems!, but it is not solved yet.

Do you believe that this is the best time to study math – for instance, number theory is now being applied in cryptography and so on? What does it take to do great mathematics?

It is interesting that pure mathematicians, like myself, rarely think directly about applications. We are instead guided primarily by what directions we find most beautiful, elegant, or most promising. We tend to treat our discipline more as an art than as a science! And indeed, this is the attitude that allows us to be the most creative and productive.

On the other hand, it is also true, historically, that the mathematics that has been the most applicable and important to society over the years has been the mathematics that scientists found while searching for beauty; and eventually all beautiful and elegant mathematics tends to find applications.

That is why it is very important to fund basic science research. When science funding is only application-driven, it does not allow full freedom and creativity. Funding basic science allows a large interconnected database of scientific techniques and knowledge to accumulate, so that when a societal need arises, the science is ready to be applied and adapted to the purpose.

Elliptic curves (and the related Birch and Swinnerton-Dyer Conjecture) are indeed a good example! They were first studied by pure mathematicians, but are now one of the most important mathematical objects in cryptography. So that is indeed exciting, but I just want to emphasize that they were exciting and central to number theory well before these applications were found; but it was inevitable that they would be found, given their fundamental nature.

That is why elliptic curves have fascinated me! They are so fundamental in both pure and applied mathematics. Beyond advancing the subject of number theory in general, a heightened understanding of elliptic curves also has important implications in coding theory and cryptography. Encryption schemes, such as those used to protect our privacy when transmitting information online, often centrally involve the use of elliptic curves.

Math is generally considered a difficult subject but you have been enjoying math since your childhood. What aspect of your education could have contributed to this enjoyment?

I’ve always enjoyed mathematics as far back as I can remember, since I was two or three years old. Since my mother was a mathematician, I always had her as a resource – I would always go and ask her questions and so I learned a lot from her. She was also a great source of encouragement – she always answered my questions enthusiastically, and always encouraged me to pursue whatever I was interested in – and that probably single-handedly contributed the most to my enjoyment of mathematics (and of all my interests)!

An Indian origin mathematician with Midas Touch Manjul Bhargava

Midas Touch Mathematician Manjul Bhargava

The Hindu Aug 14 2014


Number theorist Manjul Bhargava wins Fields Medal

Manjul Bhargava, the Canadian-American number theorist from Princeton University, is one of the four who have been chosen for the highest award in mathematics, the Fields Medal, which is given once every four years by the International Mathematical Union (IMU) during the quadrennial International Congress of Mathematicians (ICM). The ICM2014 got underway on August 13 at Seoul, Republic of Korea.

Fields medal

Awarded in recognition of “outstanding mathematical achievement for existing work and for the promise of future achievement”, the Fields Medal is given to mathematicians of age less than 40 on January 1 of the year of the Congress. Born of Indian parents who migrated from Jaipur in the late 1950s, Bhargava, who turned 40 just last week, could not have hoped for a better birthday gift.

“Bhargava”, says the IMU citation, has been awarded the Fields Medal “for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.” (See Box for definitions of italicized terms)

  • In ‘geometry of numbers’ one imagines a plane or a 3-dimensional space populated by a lattice whose grid points have integer co-ordinates.• A ‘ring’ is an algebraic structure with two binary operations, commonly called addition and multiplication, which are generalizations of the familiar arithmetic operations with integers applied to algebraic objects. Examples of rings are polynomials of one variable with real coefficients, or square matrices of a given dimension. Algebraic number theory is the study of this and other algebraic structures.• ‘Rank’ refers to the minimum number of objects required to generate the entire set of algebraic objects being studied; the dimension of a vector space, for example. The familiar 3-d vector space is of rank 3.• ‘Elliptic curves’ are graphs generated by equations of the form y2= a polynomial of degree 3, such as x3+ ax + b, where a and b are rational numbers.

A large body of work in number theory relates to the study of how numbers of interest, such as prime numbers, are distributed among the entire set of integers. Bhargava developed novel techniques to count objects in algebraic number theory that were previously considered completely inaccessible. His work has completely revolutionized the way in which fundamental arithmetic objects in algebraic number theory, such as number fields and elliptic curves, are now understood and studied, and this has given rise to wonderful applications.

About 200 years ago the German mathematician Carl Friedrich Gauss, one of the historical greats, had discovered a remarkable ‘composition law’ for binary quadratic forms, which are polynomials of the form ax2 + bxy + cy2, where a, b and c are integers. Using this law two binary quadratic forms could be combined to give a third one. Gauss’s law is a central tool in algebraic number theory. Bhargava discovered an ingenious and simpler geometrical technique to derive it and the technique allowed him to obtain composition laws for higher-degree polynomials as well.

Geometry of numbers

The technique reportedly dawned upon Bhargava one day while he was playing with Rubik’s cube. Implicit in Gauss’s method was the use of ‘geometry of numbers’ and it is this realization that enabled Bhargava to extend it to higher degrees. He then discovered 13 new composition laws for higher-degree polynomials. Until then, Gauss’s law was thought to be accidental and unique to binary quadratics. Nobody had even imagined that higher composition laws existed until Bhargava showed that Gauss’s law is part of a bigger theory applicable to polynomials of arbitrary degree. His approach has also broadened the canvas of applying geometry of numbers to address outstanding problems of algebraic number theory.

This work immediately led Bhargava to tackle a related problem, which was the counting of ‘number fields of fixed degree by discriminant’.


A number field is obtained by extending the rational numbers to include non-rational roots of a polynomial equation; if the polynomial equation is quadratic, such ax2+bx+c = 0, whose roots are given by the well-known formula [– b/2a ± √(b2 – 4ac)/2a], then one obtains a quadratic number field. The expression under the square root sign is called the ‘discriminant’ (defined appropriately for polynomials of different degrees). Higher degree number fields — cubic, quartic, quintic etc. — are correspondingly generated by higher degree polynomials.

The degree of the polynomial and its discriminant are two fundamental quantities associated with a polynomial. Despite number fields being one of the fundamental objects in algebraic number theory, answers to questions like how many number fields there are for a given degree n and a given determinant D were not known. If one has a quadratic polynomial, counting the number of lattice points in a certain region of 3-d space gives information about the associated quadratic number field. For example, using the geometry of numbers it can be shown that, for discriminant with absolute value less than D, there are approximately D quadratic number fields. The case of cubic number fields had been solved 40 years ago by Harold Davenport and Hans Heilbronn but since then the higher degree cases saw little progress until Bhargava came on the scene.

Quintic number fields

Armed with his new technique, Bhargava was able to solve the case of quartic and quintic number fields. The new composition laws and his new technique in using the geometry of numbers have together extended the reach and power of counting number fields. The cases of degrees greater than 5 still remain open as Bhargava’s composition laws alone seem inadequate to resolve these higher cases at present.

While the above work were al carried out between 2004 and 2008, more recently, Bhargava has employed his improved geometry of numbers technique to obtain striking results about ‘hyperellpitic curves’, which are graphs of equations of the form y2 = a polynomial with rational coefficients, the case where the degree of the polynomial is 3 being called the ‘elliptic curve’.

Elliptic curves have important applications in pure as well as applied mathematics. Even though Fermat’s Last Theorem seems to be not even remotely connected with elliptic curves, it was key to its proof in 1995 by Andrew Wiles, who, incidentally, was also Bhargava’s thesis advisor. Operations using elliptic curves have become a core component of many of the cryptographic protocols that encode credit card numbers in online transactions. “Intellectual stimulation, beautiful structure, applications – elliptic curves have it all,” Bhargava has said.

An outstanding problem in algebraic number theory has been how to count the number of points on ‘hyperelliptic curves’ that have rational coordinates, which is the same as asking how many rational solutions does a hyperellptic equation have? The answer, it turns out, following Bhargava’s work, depends on the degree of the curve.

One can easily see that the number of rational solutions of a polynomial equation of degree 1, such as y = 9x + 4, is infinite: any rational value for x produces a rational value for y, and vice versa. Quadratics, such as, such as y2 = 2x2 + 5x – 3, have either no rational solutions or infinitely many. For curves of degree 1 and 2, there is an effective way of finding all the rational points. In 1983, Gerd Faltings, director of Max Planck Institute for Mathematics, Bonn, showed that for degree 5 and more there are only finitely many rational points. That left unresolved the cases of degree 3 – the elliptic curves – and of degree 4.

Finding rational points for elliptic curves is, however, not an easy matter. They can have zero, finitely many, or infinitely many rational solutions. When does a cubic equation have infinitely many solutions has been a central question in number theory since Pierre de Fermat in the 17th Century. In the recent past mathematicians have attempted to devise algorithms to decide whether a given elliptic curve has finitely many or infinite rational points but that route took them nowhere. They have only been able to guess how often these different possibilities arise.

But once you have found some rational points on an elliptic curve, it becomes possible to generate more by using the simple connecting-the-dots method. For example (see fig.), if you draw a line through two rational points, it usually intersects the elliptic at exactly one more point, which is again a rational point. But the opposite, namely given one rational point finding the two rational points that would generate it. This is what underlies the use of elliptic curves in cyber security.

connecting the dots method

Connecting-the dots method: Given two rational points of an elliptic curve y2 = x3 + 2x + 3, the point at which the line through those points intersects the curve at one more point is guaranteed to be a rational point. This connect-the-dots procedure is a means to generate all of an elliptic curve’s rational points starting from a small finite number. (Credit: Quanta, illustration by Manjul Bhargava)

Curve’s rank

When the number of rational points of an elliptic curve is infinite, the smallest number of rational points that can generate essentially all the rational points is called the curve’s rank. When the infinite set of rational points can be generated essentially from just one point, the curve has rank 1, and so on. When the number of rational points is finite or none at all, the rank is 0.

In 1992 Armand Brumer showed that a 1965 conjecture made by Birch and Swinnerton-Dyer (BSD) implied that the average rank of the group of rational points of an elliptic curve defined over rational numbers is bounded. Later in 1979 Dorian Goldfeld conjectured that the bound is, in fact, is equal to ½. That is, in a statistical sense, half of all elliptic curves have rank 0 and half have rank 1. Previously, however, mathematicians did not even know that the average rank was finite (let alone ½).

The conjecture, of course, does not mean that curves of higher rank – 2, 3 and so on – do not exist, or even that there are only finitely many such. Indeed, computationally mathematicians have found such curves, the highest known rank till date is 28! But as the number of elliptic curves asymptotically becomes infinitely large, the curves with higher ranks approach a vanishingly small percentage of the whole.

Enter Bhargava and his collaborators, his doctoral student Arul Shankar (a 2007 Chennai Mathematical Institute graduate) in particular. Instead treading the beaten track of algorithms, they asked the question: what could be said about rational points on a typical curve? From this perspective they first showed that a sizeable fraction of elliptic curves has only one rational point (rank 0) and another sizeable proportion has infinitely many rational points (rank > 0). Using newly developed techniques, they were able to show that the average rank is, in fact, bounded. They have been further able to show that the bound is also less than 1, indicating that the conjecture is perhaps true.

“Bhargava introduced dramatically new ideas ​to study the average number of solutions and proved that the average rank of elliptic curves is bounded, and that the BSD Conjecture is true on the average, making it one of the most spectacular successes in number theory in recent years,” says Deependra Prasad, a number theorist from Tata Institute of Fundamental Research (TIFR).

Analogously, for the case of degree 4 too Bhargava and Shankar showed that a significant chunk of such curves has no rational points and another significant chunk positive proportion has infinitely many rational points. Using his expanded geometry of numbers technique Bhargava has also explored higher-degree curves in general.

While Faltings Theorem tells us that for curves of degree greater than 5, there are only finitely many rational points, it does not give a way to determine how many exactly there are. For the even degree case, Bhargava showed that the “typical” hyperelliptic curve had no rational points at all. The joint work of Bhargava and Benedict Gross, followed up by that of Bjorn Poonen and Michael Stoll, established the same result for the odd degree case as well. Bhargava’s work has thus clearly shown that the number of curves having rational points decreases rapidly as the degree increases. For example, for a typical 10 degree polynomial, there is a greater than 99 per cent chance that the curve has no rational points.

Bhargava’s work in number theory has had profound influence in the field. “A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insights,” said IMU’s information sheet on his work. “With his keen intuition, immense insight and great technical mastery, he seems to bring a ‘Midas touch’ to everything he works on,” it added.

Tabla player

Besides being one of the world’s leading mathematicians, Bhargava is also an accomplished Tabla player and plays at the concert level. He learnt the art initially from his mother and later came under the tutelage of the well-known tabla maestros Pandit Prem Prakash Sharma and Ustad Zakir Hussain. “Classical Indian music,” Bhargava told Princeton Weekly Bulletin when he was featured, “is very mathematical, but consciously thinking of the math would interfere with the improvisation and emotion of the playing. But somehow the connection is there. I often use music as a break, and many times I come back to the math later and things have cleared up.” Indeed, Bhargava thinks of mathematics art. He is also keenly interested in linguistics in which he has published research work. It was his grandfather, a linguistics scholar, who taught him Sanskrit and developed his interest in linguistics.