On Manjul Bhargav's achivement.
The ‘Midas Touch’ mathematician
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.
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: 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)
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.
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.
Finance More: Math Rubik's Cube A Winner Of Math's Most Prestigious Award Solved A 200-Year-Old Problem By Using A Rubik's Cube Elena Holodny
Aug. 13, 2014, 10:40 AM
On Tuesday the winners of the most prestigious award in mathematics — the Fields Medal — were announced. The award is given to two to four mathematicians under age 40 every four years "to recognize outstanding mathematical achievement for existing work and for the promise of future achievement."
This award is typically given for mathematical research that solves or extends complicated problems that mathematicians have struggled with for decades or even centuries.And, believe it or not, one of the winners, Manjul Bhargava, reformulated a 200-year-old number theory problem in a rather unconventional way: by using a Rubik's Cube and Sanskrit texts
But before we get to his idea, let's go over some background.
Here's an interesting mathematical idea: If two numbers that are each the sum of two perfect squares are multiplied together, the resulting number will also be the sum of two perfect squares.
Let's take the numbers 25 (a perfect square of 5) and 36 (a perfect square of 6) and add them together as an example. The resulting equation would read:
(25 + 25)(36 + 36) = 3600, where 3600 equals ((36*36) + (48*48)), or (1296 + 2304).
You can keep trying this with all different number combinations — as long as the four numbers in the brackets are perfect squares — and the output will always also be a perfect square. (Yes, math is awesome.)But let's get back to Bhargava. His grandfather, Purushottam Lal Bhargava, was the head of the Sanskrit department of the University of Rajasthan — and so "Bhargava grew up reading ancient mathematics and Sanskrit poetry texts," according to Quanta Magazine.
In one of these manuscripts, he discovered a "generalization [similar to the above math] developed in the year 628 by the great Indian mathematician Brahmagupta" that stated:
If two numbers that are each the sum of a perfect square and a given whole number times a perfect square are multiplied together, the product will again be the sum of a perfect square and that whole number times another perfect square.
Later on, when he was a graduate student, Bhargava learned that Carl Gauss — a leading mathematical figure of the 18th and 19th century "came up with a complete description of these kinds of relationships" — provided that the expressions had only two variables, and only quadratic forms (meaning x2, but not x3).
Essentially, Gauss came up with a "composition law" that would tell you which quadratic form you'd get if you multiplied two of these kinds of expressions together.
But here's the downside: Gauss' law took him approximately 20 pages to write out
. Ouch. No one wants to read through all of that.
Bhargava was interested in uncovering an easier way to describe Gauss' law — and perhaps if he could tackle higher exponents, according to Quanta Magazine.
Then one day, Bhargava thought of a solution when he was sitting in his room, looking at a Rubik's Cube. He took a mini-cube (a 2x2 Rubik's Cube) and "realized that if he were to place numbers on each corner of the mini-cube and then cut the cube in half, the eight corner numbers could be combined in a natural way to produce a binary quadratic form," according to Quanta Magazine.
If you think about a Rubik's Cube, you'll note that there are three ways to cut the cube in half. You can split the top from the bottom; you can split the right from the left; you can split the front from the back.
In fancy math terms, that means that by cutting a Rubik's cube in three ways, you can generate "three quadratic forms."
The interesting thing about this, is that Bhargava discovered that these quadratic forms "add up to zero" — not when using normal addition, but "with respect to Gauss' method for composing quadratic forms."
And ta-da, by simply dividing a Rubik's Cube, he "gave a new and elegant reformulation of Gauss' 20-age law."
Bhargava said about this moment to the New Scientist:
Gauss's law says that you can compose two quadratic forms, which you can think of as a square of numbers, to get a third square. I was in California in the summer of 1998, and I had a 2 x 2 x 2 mini Rubik's cube. I was just visualising 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. It was a great day!
Afterward, Bhargava worked with a Rubik's Domino (a 2x3x3 shape) — and he realized that he could go beyond the limited quadratic forms of Gauss' law. Eventually, he went on to discover 12 more compositional laws, which later became his Ph.D. thesis
According to Quanta Magazine, Barghava said that he had always been interested in ideas like this — the “problems that are easy to state, and when you hear them, you think they’re somehow so fundamental that we have to know the answer.”
Read more: http://www.businessinsider.com/fields-m ... z3AJ46sKzO
Clearly his genius is in using simplicity to unravel complexity.