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Like a Bach fugue, precise yet beautiful

V.S. Varadarajan

Brownian motion and the methods of dealing with it, a substantial part of them created by Srinivasa Varadhan, the winner of the 2007 Abel Prize, allow one to have a remarkably clear view of an immense number of areas of mathematics, physics, and, in recent years, economics and finance.

Professor Srinivasa Varadhan.

THE NEWSWIRES have been buzzing since the announcement a couple of weeks ago from the Norwegian Academy of Sciences that Srinivasa Varadhan, an American mathematician of Indian origin, is the winner of the prestigious Abel Prize for the year 2007. Since mathematics is not in the public eye and its highest reaches are very difficult, if not impossible, to comprehend even for the scientifically informed layperson, there has been tremendous curiosity in India and among Indians all over the world about this achievement by an Indian (ignoring the distinction expressed above): who is he? what is he like? what has he done that has propelled him to this tier of greatness? and for that matter, what is the meaning of creativity in mathematics?

I cannot answer all of these questions in a short article written for a non-technical audience. But I shall try to go some distance towards answering them, so that one who is interested can start where I leave off. At the end of this piece, if the reader wants a description of Varadhan's work in one sentence, I cannot do much better than referring him to the title of my article, which I have taken from a review of one of Varadhan's books by a fellow mathematician. The pursuit of science has always been admired in cultured societies and the highest degree of veneration has been reserved for those special persons who have reached extraordinary levels in their achievements. Thus names like Pythagoras, Galileo, Newton, Einstein, and Ramanujan, have percolated to universal consciousness, to which, one may add perhaps such names as Euler, Gauss, von Neumann, Pauling, Heisenberg, Dirac, Hawking, and a host of others.

But as modern science continues to evolve at a dizzying pace, the general public no longer understands much of what newspapers and journals describe as happening at its frontiers. This is so already in physics where every day we see an announcement of yet another discovery in string theory and a news correspondent, in a manner clearly reminiscent of blockbuster sequels at the movie houses, is desperately trying to convince himself and the readers that the newest discovery is the beginning of a new horizon.

What I shall do is to tell the reader something about Probability Theory and its evolution as a science and what the role of Varadhan has been in its modern development. Probability is the science of predicting events which are uncertain. In ordinary life we all have sufficient experience with such events. If I bet on a tennis singles match, if the players are closely matched the odds for either to win are 50-50. More refined bets are made depending on the form of the contestants, as in horse racing or in calculating insurance premiums. I am sure a lot of people must have lost a good deal of money when the Indian team made an unexpected exit in the first round of the recent World Cup Cricket tournament.

The scientific way of working out such bets goes all the way back to the seventeenth century when two French mathematicians, Pierre Fermat and Blaise Pascal, responding to questions posed by some of their friends who routinely gambled in casinos, worked out in their letters to each other some of the basic odds. Later on, Jakob Bernoulli, a scion of the famous Bernoulli family in Switzerland, published one of the great early treatises on probability, Ars Conjectandi, in which he discovered and proved the famous law of large numbers, a work which was further continued by many others. These discoveries led to the formulation of general laws governing the probabilistic evolution, but the laws governing phenomena that do not conform to the general behaviour, because their probabilities are extremely small, did not begin to be analysed till Harald Cramer, a Swedish mathematician and actuarial analyst, began a new theory of such phenomena.

Varadhan, in the 1960s, went far beyond Cramer and created the definitive theory of these large deviations (from normal behaviour) especially in the context of brownian motion. Brownian motion, first observed in 1827 by the English Botanist Robert Brown, is the ceaseless but small motion of small particles of matter (pollen, dust), say in a glass of water, even when the water appears to be absolutely still. Some years later, when the atomic hypothesis began to be accepted more widely, it was shown that this motion is due to the bombardment of the piece of matter by the molecules of water moving about randomly even though, at the level where we are observing, the molecules (hence their motions) are invisible.

The physicist Jean Perrin made the first experiments cataloguing this motion, in response to the theoretical calculations on brownian motion by Einstein and Smouluchowski. Perrin won the Nobel Prize in 1926 for his work and the Einstein paper is regarded as one of his great trio in the annus mirabili, the miraculous year, 1905. These discoveries played a decisive role in the universal acceptance of the atomic hypothesis. Then in the middle of the twentieth century, the mathematical model for brownian motion was created by the great American mathematician Norbert Wiener. Wiener's work was continued by the Russian mathematician Andrei Kolmogorov, widely regarded as one of the greatest mathematicians of the twentieth century, and Paul Levy, a great French mathematician.

How can a layperson envision brownian motion? Imagine a great ballroom where hundreds of people are dancing, randomly and with no pattern, jostling against each other. If you follow the movement of the woman in this corner with the red dress, you will soon begin to be interested in what she is doing: will she get into the far left corner? will she do this without meeting her husband who is dancing on the far right corner? And so on. If you think of the dancers as molecules and the ballroom as a quantity of liquid, you will begin to have a small appreciation of the type of questions asked in the theory of brownian motion.

Kolmogorov and Wiener were profound mathematicians who were at the highest levels of their science. Levy had great intuition on what the brownian particle is likely to do under almost any circumstance. Remarkably, Varadhan combines the qualities of both, great mathematical power with profound intuition. It is no surprise therefore that he created his monumental theory of large deviations in the 1960s, in a series of seminal papers in collaboration with the American mathematician Monroe Donsker. Donsker was always quite open about the crucial role of Varadhan in this collaboration, once telling a close friend that Varadhan was the greatest problem-solver he has ever come across. After this Varadhan worked on the theory of diffusion processes originally created by the Japanese mathematician Kiyosi Ito, and carried it, with the help of Daniel Stroock, an American mathematician, to levels undreamt of. In recent years he has been exploring probabilistic systems in fluid dynamics with extraordinary success.

My brief remarks may suggest that the contributions of Varadhan are somewhat specialised, arcane even, and of great interest to only a small group of believers, even within mathematics. But it would be a mistake to think so. Brownian motion and the methods of dealing with it, a substantial part of them created by Varadhan himself, allow one to have a remarkably clear view of an immense number of areas of mathematics, physics, and, in recent years, economics and finance. For instance, the epoch-making work of Michael Atiyah and Isadore Singer, Abel Prize winners two years ago, dealing with the geometric shape of manifolds of arbitrary dimension, can be understood by imagining brownian particles speeding through the manifold and tracking them, after the work of the French mathematician Jean-Michel Bismut.

Quantum field theory, especially in modern high energy physics, based on the methods created by the great American physicist Richard Feynman can be understood to a large extent from the point of view of brownian motion; all one has to do is to make the time variable imaginary! This science fiction idea, due to the Polish-American mathematician Mark Kac and the American physicist Julian Schwinger, reduces many parts of quantum field theory to probability and brownian motion. However, overarching all of these developments is the theory of Varadhan, a structure of ageless beauty, to quote from the citation of the Norwegian Academy. These achievements have pushed Probability Theory to an extraordinary position within mathematics, as may be seen from the fact that two of the winners of the Fields medals in the past year are probabilists or those whose work has very close contacts with Probability Theory.

Varadhan is an alumnus of the Indian Statistical Institute, Kolkata. This institute, created by Professor P.C. Mahalanobis in the early 1930s, has been at the forefront of research in Statistics and Probability ever since its inception, and has produced several outstanding mathematicians and statisticians. Varadhan was one of the first to get his Ph.D. degree from the ISI. He has won many other awards and is a member of the Royal Society of London as well as a foreign member of the U.S. National Academy of Sciences. He has been the Director of the Courant Institute on two different occasions for substantial periods of time. He has had a number of distinguished students and collaborators. In him are combined transcendent achievement and a rare modesty, an acute awareness of the incremental nature of science and the necessity of standing on the shoulders of others to advance our knowledge.

If one combines these attributes with those of a man to whom doing mathematics is effortless, whose creativity and insight always pick out the simplest path to the solution of even the most tangled questions, we begin to get a glimpse of the unique personality of Varadhan. Even though he has lived in America for close to 45 years, he has never forgotten the milieu in which he grew up. A teacher and colleague of exceptional patience, he is regarded with pride and affection by all his friends.

(The author is a Professor of Mathematics at the University of California at Los Angeles. He and Varadhan were both alumni of the Indian Statistical Institute at Calcutta and worked together till they both left for the U.S. during the mid 1960s.)

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