Science of chance

Mathematician S.R.S. Varadhan joins an illustrious line of scientists who have been awarded the Abel Prize.

THERE may or may not be any rejoicing at 7-E, Ezhumalai Street, West Tambaram, Chennai, but the mathematics community - not just Indian but of the world - is certainly celebrating the top award in mathematics, the Abel Prize, this year, which has gone to the Indian mathematician who hails from this address.

Sathamangalam Ranga Iyengar Srinivasa Varadhan, S.R.S. Varadhan for short and Raghu to his friends and colleagues, was chosen by the Norwegian Academy of Science and Letters on March 22 for the prestigious Abel Prize for, as the citation said, "fundamental contributions to probability theory and, in particular, for creating a unified theory of large deviations". He is at present a Professor of Mathematics and Frank J. Gould Professor of Science at New York University's Courant Institute for Mathematical Sciences (CIMS), one of the world's leading centres of applied mathematics. He has been working here ever since he landed at Courant in 1963 as a post-doctoral fellow after finishing his Ph.D in 1963 at the Indian Statistical Institute (ISI) in Kolkata under another famous Indian statistician, C.R. Rao.

Probability theory is the branch of mathematics for analysing chance events or random phenomena in the language of formal mathematics that uses concepts such as random variable (the results of measurement on chance phenomena like tossing a coin or measuring the height of waves over time), probability distribution (the mathematical function that describes the probability of different results of mutually exclusive measurements) and stochastic processes (phenomena that can be described in terms of well behaved probability distribution functions; the most common stochastic processes are time series events like noise, electromagnetic signals, random movements such as Brownian motion, exchange rate fluctuations, stock market behaviour and medical data of ECG, blood pressure and temperature).

The award is in some sense the Nobel equivalent for mathematics and carries a prize money of Norwegian Kroner (NOK) 6 million, roughly the same as what the Nobel Prize carries. It will be given away by King Harald V of Norway, in Oslo on May 22. This is the fifth Abel Prize being awarded since it was instituted in 2002 by the Norwegian Academy. It was given for the first time in 2003 to the French mathematician Jean-Pierre Serre, whose work has helped shape much of modern mathematics, in particular topology, algebraic geometry and number theory, and is one of the foremost living mathematicians (Frontline, April 5, 2003). The other recipients of the Abel Prize are Michael F. Atiyah of the University of Edinburgh and Isadore M. Singer of the Massachusetts Institute of Technology (2004), Peter D. Lax of the Courant Institute (2005) and Lennart Carleson of the Swedish Institute of Technology (2006). It is no exaggeration to say that Varadhan, with an extraordinary life in mathematics, belongs to this illustrious class.

Varadhan was born on January 2, 1940, in Chennai. Always a topper in his class, Varadhan got his B.A. (Honours) degree in statistics from Presidency College, Chennai, in 1959, scoring the highest marks in the history of Madras University, after which he joined the ISI as a research fellow. During the 1950s and the 1960s, you could possibly do good research in mathematics only at two places in the country, the Tata Institute of Fundamental Research (TIFR) in Mumbai and the ISI in Kolkata. While the former was the place for pure mathematics, statisticians preferred the latter, which was founded in the 1930s by the great Indian statistician Prasanta Chandra Mahalanobis and grew to be an extraordinary place for doing creative research in many fields.

It was a period when South India produced many students interested in research in mathematics, particularly from Presidency College (the only college where you could do an honours course in statistics at that time), Loyola College and Vivekananda College in Chennai and St. Joseph's College in Tiruchi. V.S. Varadarajan and J. Sethuraman from Presidency, K.R. Parthasarathy and R. Ranga Rao from Vivekananda, names that became well known in mathematics in later years, were already at the ISI from the South. Varadhan easily became one with that group.

"There was a tradition possibly [in the South] of grooming students for research in mathematics," says Parthasarathy, now Professor-Emeritus at the ISI in New Delhi. "This was mostly owing to individual teachers who themselves did not do any research but were enthusiastic about mathematics, like Father Racine at Loyola, Raghava Sastry at Vivekananda, Suryanarayana Iyer at St. Joseph's. Father Levy in Calcutta [Kolkata] too was doing the same," he adds. Besides, Varadhan's father, Ranga Iyengar, was a schoolteacher who taught mathematics and, according to Parthasarathy, always wanted his son to do the ultimate possible and was very proud of his son's achievements in college. Apparently, he used to taunt Varadarajan, three years senior to Varadhan in college, about how Varadhan had consistently outperformed him.

At the ISI, Varadhan started working on statistical quality control. "We had become interested in the application of formal mathematics to statistics. Varadhan met us and decided that along with us he too would study probability theory," recalls Parthasarathy. "Topics such as measure theory and theory of operators in Hilbert space had become popular but the ISI itself didn't have much tradition in probability theory and there was no professional mathematician to give us any systematic introduction to the subject. There were mathematics lectures by people like C.R. Rao, R.G. Raha and Raghu Raj Bahadur and also Jerzy Neyman, but these were only on topics related to `statistical inference'. Varadarajan had learnt topology entirely on his own and this gave us inspiration. We began learning mathematics by ourselves. We lived in the same hostel and we used to give seminars to each other. In this way we learnt topological groups and harmonic analysis from Ranga Rao," says Parthsarathy, describing the mileu into which Varadhan came.

"It is a tribute to C.R. Rao and Mahalanobis that we were not pressured in any way to do things we did not like to and so we were left pretty much to ourselves," points out Varadarajan, now a highly reputed mathematician at the University of California, Los Angeles. "If there was any discouragement it was only because there was no guidance at the ISI," adds Parthsarathy, "but otherwise we were pretty much left to do what we wanted to. We chose topics for research all by ourselves."

But the year Varadhan joined, Varadarajan left for the U.S. "In his absence, Ranga Rao, Varadhan and I embarked on extending `limit theorems' on the hitherto studied one-dimensional space of real numbers to an abstract n-dimensional coordinate space in which measurements of stochastic processes could be represented. Whatever [the Russian mathematician Andrey Nikolaevich] Kolmogorov had done [on standard limit theorems], we tried doing the same for stochastic processes. This formed the basis for obtaining limit theorems for abstract spaces that are also `groups' [in mathematics language]. We also wrote some joint papers, and this emboldened Varadhan to embark on his Ph.D thesis," says Parthsarathy. Parthasarathy later published the results of their joint research as a book in 1967.

Extending the programme further, Varadhan wanted to obtain the Central Limit Theorem - the basic mathematical principle underlying various statistical techniques - for infinite dimensional spaces. (The Central Limit Theorem essentially says that statistical properties that depend on many independent variables have a bell-shaped distribution, or `normal' or `Gaussian' distribution as it is called.) The only people who were looking at such problems were mathematicians like Alexander Kinchin, A.N. Kolmogorov and B.V. Gnedenko in Russia and Paul Pierre Levy in France. "Varadhan made the important jump to handle infinite dimensions," points out Parthasarathy. "The usual technique of using Fourier transforms does not work for infinite dimensions. He instead used the technique of `concentration functions', which provided a complete description of the limiting case for infinitely divisible distributions," explains Parthasarathy.

Varadhan's thesis was titled `Convolution Properties of Distributions on Topological Groups'. Formally, C.R. Rao was his thesis adviser, because one had to have an adviser. But he had nothing to do with Varadhan's path-breaking work. It had grown out of joint work with his colleagues at the ISI, on problems chosen by them, and Varadhan's incisive mind took it to a different plane.

For Varadhan's thesis, the foreign examiner, as was customary those days, was the famous Kolmogorov. According to Varadarajan, Kolmogorov's report was in Russian and he was one of the few at the ISI who was familiar with the language. "I still remember two sentences that stood out in that report. Kolmogorov wrote that this thesis was not that of a student but that of a mature master... [and] the thesis deserved the second degree in the Soviet Union." The first degree in the former Soviet Union was called the candidate's degree and is roughly equivalent to a Ph. D. elsewhere. The second degree, D. Sc., is given only for distinguished work, usually several years after the candidate's degree. The apocryphal story, which is included even in Varadhan's biography hosted on the Abel Prize's website, of Kolmogorov sitting as a stranger at his thesis defence at the ISI and asking probing questions much to Varadhan's surprise, does not seem to be true. It is, however, true that Kolmogorov did visit the ISI in February 1962 for a month on C.R. Rao's invitation.

In 1962, Varadarajan returned to Kolkata. He was inspired to work on quantum theory and Lie groups from his contact with the famous Indian mathematician Harish-Chandra at the Institute of Advanced Study in Princeton. He suggested to Varadhan that, now that his thesis work was over, he should also get into this exciting field. The two together undertook an extensive study of Lie groups, particularly `finite dimensional representations' of Lie groups, along the lines of Harish-Chandra and made fast progress. The two proved some important results but never wrote any paper. In 1963, Varadhan left Kolkata to join Courant and went back to his first love, namely, probability theory.

Parthasarathy, who was in Moscow during that time, came back to join Varadarajan and Ranga Rao to continue the work on quantum theory. The famous notes of their joint studies came out as a book, The Geometry of Quantum Mechanics, by Varadarajan much later. In 2001, reminiscing about that joint work with Varadhan, Varadarajan said: "I have always treasured that period of one year [1962-63] we worked together... we were really naive and optimistic, and when we were looking at things together, no mountain seemed too formidable to climb."

The period (1963-66) of Varadhan's post-doctoral fellowship at Courant proved to be very important. During this time, he came in contact with Daniel W. Stroock and Monroe D. Donsker. Collaboration with the two resulted in two remarkable and entirely independent streams of work, somewhat overlapping in time, which culminated into universal and enduring mathematical tools in the 1970s that could be applied to a variety of disciplines from physics to electrical engineering to finance.

"If you notice, practically all his work is done jointly with others, with Varadhan bringing in the key ideas," points out Rajendra Bhatia of ISI, New Delhi. "Varadhan is well known for that." As Varadarajan puts it, ""Varadhan is, of course, a great mathematician but what is unusual is the affection with which he is regarded by the members of the probability community. This is undoubtedly owing to his willingness to help and to share his insights with those who come to him, both students and colleagues, without any condescension and always with unbounded generosity."

Varadhan and Stroock began studying `second order partial differential equations of the elliptic type' to simulate solutions by statistical methods, what is known as the Monte Carlo Method. "The work by the Japanese mathematician [Kiyosi] Ito required the coefficients in stochastic partial differential equations to be very regular. But Varadhan and Stroock evolved some remarkable general methods for treating coefficients that were not regular or continuous and also they could be multi-dimensional, even infinite-dimensional, that arise in special stochastic processes called Markov processes," says Parthasarathy. "Their approach made it possible to treat the stochastic processes as problems of games of fair chance, what are known as martingales, and to arrive at solutions," adds Parthasarathy.

Describing the work, Tom Lindstrom writes, "Their approach unified, simplified and extended the previous results in the area substantially. The basic idea is that instead of looking for solutions to quite complicated problems of mathematical analysis, `all' one has to look for is a probability distribution, which turns certain processes into martingales." This new `Stroock-Varadhan' approach turned out to be an extremely powerful way of constructing new Markov processes, for example, infinite-dimensional diffusions arising in population genetics. The body of work done during 1968-71 led to the famous book Multi-dimensional Diffusion Processes by Stroock and Varadhan in 1979. According to Parthasarathy, like the earlier 1965 book on the subject by Ito and McKean, this book will not be outdated for the next 50 years.

The other stream (with Donsker, who is no more) led to what is perhaps Varadhan's greatest work, which finds special mention in the Abel citation, namely `the theory of large deviations'. What do we mean by large deviations? The law of large numbers in statistics, discovered by Jacob Bernoulli in the 18th century, says that the average (probability) of a long sequence of trials, say that of coin tosses, is usually close to the expected value, namely each for `heads' and `tails'. Yet there is a vanishingly small probability that the unexpected, say the proportion of `heads' will be instead of . That is, deviation greater than a given level goes to zero. The concept of `large deviations' is to be able to calculate the probability of such rare events. For practical applications it is important to know how fast it vanishes.

The subject has concrete applications in fields as diverse as physics, biology, economics, statistics, computer science and communications. For instance, what kind of contingency reserves should an insurance company maintain so that it does not get into a ruinous situation because of a largely deviant situation - one in which, say, all the cars insured are involved in accidents. In fact, analysing such situations, the Swedish mathematician and insurance statistician Harald Cramer (1893-1985) realised in 1937 that standard assessments based on the Central Limit Theorem (the bell curve principle) were actually misleading. It took 30 years before Varadhan discovered the underlying general principles and demonstrated their vast scope, far beyond the classical problems of independent trials with random variables.

His landmark paper of 1966, "Asymptotic Probabilities and Differential Equations", sets out the premise for a general theory of large deviations. It addressed the fundamental question: what is the qualitative behaviour of a stochastic system if it deviates from the behaviour predicted by the law of large numbers?

"Varadhan and Donsker wrote 300 pages of very hard analysis by studying the general principles of large deviations by reducing the problem to a powerful variational principle and studying its maxima and minima," points out Parthasarathy. "To any analysis of large deviations, they associated a rate function - the Donsker-Varadhan function - and studied its behaviour. The problem was reduced to finding an appropriate function whose fluctuating behaviour can be analysed," adds Parthasarathy.

The theory also gave rise to Varadhan's Integral Lemma, which is extensively used in quantum field theory, statistical mechanics, market finance, electrical engineering and traffic engineering. "It has also greatly facilitated the use of computers to stimulate and analyse rare events.

Over the last four decades, the theory of large deviations has become a cornerstone of modern probability theory, both pure and applied," notes the citation.

Another major area of Varadhan's work involves analysis of `hydrodynamical limits' describing macroscopic behaviour of very large systems of interacting particles. His breakthrough work, in association with Maozheng Guo and George C. Papanicolau, has greatly influenced analysis of `random walks' in a random environment. His name is now attached to the method of "viewing the environment from the travelling particle", one of the general tools in the field. There are, of course, many more significant achievements in Varadhan's career.

"He is a prolific scientist with deep insights and an impressive array of technical tools and he is highly regarded and esteemed in the probability community," says Lindstrom. "This not only has to do with his results but also his style - listening to a lecture by Varadhan, one is not only exposed to the best and most results in the subject, but one is also introduced to a way of thinking."

"His power, insight and willingness to think about anything mathematical," says Varadarajan, "are admirably balanced by his patience and inexhaustible good humour when dealing with people. He is a true master and I feel it is a great fortune of mine to have known him very closely. A great deal of what I know about doing mathematics has come from him."

The Abel citation perhaps sums up his work most evocatively: "Varadhan's work has great conceptual strength and ageless beauty." "As a person he is child-like; normally does not open his mouth. But in a small community of mathematicians he is quite jovial," says Parthsarathy. Apparently, he enjoys Carnatic music and likes listening to Thiruppavai, a collection of verses in Tamil written by Andal in praise of God. He has reportedly donated his Steele Prize money to a hospital in Tambaram, where his roots are. Currently he is on the governing council of the Chennai Mathematical Institute and visits Chennai regularly. Varadhan's only regret at this hour of glory is perhaps that his eldest son Gopalakrishnan Varadhan is not there to share it with him. He died while at work in the September 11, 2001, attack on the twin towers of New York's World Trade Centre.