S.R.S. Varadhan Is 2007 Abel Laureate

May 1, 2007

Shortly after the announcement on March 22 that Srinivasa S.R. Varadhan had won the 2007 Abel Prize, I began receiving requests for biographical essays about him. It seems that, having written two biographical pieces about him in the past, I have become his official biographer. Unfortunately, I am not capable of producing a new essay each time, especially under the time constraints imposed. Thus, I have been recycling, with minor variations, the same essay. Moreover, the essay I have been circulating is really just a slightly expurgated version of an article I wrote ten years ago for the Notices of the AMS when Varadhan ran for the presidency of that august society, an article to which, with all due modesty, I attribute the sound trouncing that he received in the subsequent election.

Varadhan, whom everyone else calls Raghu, came to America from his native India in the fall of 1963. Arriving at Idlewild Airport (renamed Kennedy Airport shortly afterward), he proceeded to Manhattan by bus, past twenty miles of uninterrupted cemeteries. For a young Hindu from Madras accustomed to tidier procedures for disposing of the dead, this introduction to his future home must have been less than auspicious: Was he entering some sort of necropolis? His destination in Manhattan was that famous institution The Courant Institute of Mathematical Sciences, where, at the behest of Monroe Donsker, he had been given a postdoctoral fellowship. At the time, CIMS had not yet moved out of the hat factories to which NYU had originally consigned Richard Courant's reincarnation of Göttingen. Thus, when I, a humble graduate student from the opulent Rockefeller Institute, first met Varadhan, he was sequestered in one of the many dingy, windowless offices from which flowed a remarkably large fraction of the postwar mathematics of which America (or at least the American mathematical community) is justly proud.

Varadhan had completed his PhD at the Indian Statistical Institute in Calcutta. (As much as any other institution, ISI is responsible for the (apparently incorrect) rumor that the Indian term for statistician is "Rao.") Thus, it was a surprise to no one that Varadhan came equipped with a superb grounding in statistics (a subject about which few other probabilists know anything at all). But CIMS was hoping for more. Varadhan's own arrival at CIMS had been preceded by that of V.S. Varadarajan, another renowned graduate of ISI, whose extraordinary mathematical erudition was already evident in the much coveted set of notes that he produced during his sojourn there.

Within a year or so, Varadhan demonstrated that he certainly could and probably would fulfill or exceed any of the hopes that Donsker and the rest of CIMS might have for him---and this without publishing the research from his first year at CIMS, most of the results having been found slightly earlier by no less a figure than K. Itô. Rather than pine over his misfortune, Varadhan dropped the project on which he had spent a year and took up, mastered, and brought to fruition an idea of Donsker's that had made its first appearance in the beautiful thesis of Donsker's student M. Schilder. The general idea in Schilder's thesis was that one should attempt Laplace-type methods to develop asymptotics for the evaluation of Wiener integrals. Although physicists, thinking in terms of Feynman integral representations for solutions to Schrödinger's equation, had made somewhat casual reference to related ideas in order to justify Ehrenfest's "theorem" (the one asserting that quantum mechanics becomes classical mechanics as Planck's constant goes to 0), Schilder seems to have been the first mathematician to come to grips with the challenge presented by carrying out Laplace asymptotics in an infinite-dimensional setting. However, Schilder's treatment was somewhat primitive and its applicability was severely limited. In particular, only after Varadhan took up the problem did it became clear that Schilder had been studying a very special example of what statisticians call the theory of large deviations.

The study of large deviations goes back to the work of Khinchine and Cramér, but the term "theory" is not an accurate description of what they had produced. In fact, if there is, even now, something that deserves the name, the theory of large deviations was born in Varadhan's famous 1966 article on the subject (in Vol. XIX, No. 3, of the CIMS journal CPAM). It was in that article that he clarified the analogy between large deviations and the theory of weak convergence of measures, an analogy on which he based his formulation of the large deviation principle in terms of an upper bound for closed sets and a lower bound for open sets. Of course, a formulation does not a theory make. But Varadhan provided the theory as well. Namely, as summarized to me by a Japanese friend, the theory of large deviations consists of two steps: The first requires you to prove either the upper or lower bound yourself; the second step is to get on the telephone and ask Varadhan how to prove the other bound.

As anyone who has followed his career will confirm, large deviations have been a recurring theme in Varadhan's mathematics. For one thing, he has had an uncanny ability to understand that large deviations are manifest in all sorts of situations in which nobody else even suspected their presence. To me, the most spectacular example of his special insight lies in his realization that M. Kac's old formula for the first eigenvalue of a Schrödinger operator can be interpreted in terms of the large deviations. Like those in Schilder's thesis, the large deviations here involve Wiener measure. However, whereas Schilder dealt with large deviations of Brownian paths over a very short time interval, the explanation for Kac's formula must be sought in the large deviations of Brownian paths from ergodic behavior over very long time intervals. So far as I know (and I was one of his students), Kac himself, much less anyone else, had never guessed that such an interpretation might exist. Further-more, I suspect that not even Varadhan anticipated the wealth of results to which systematic exploitation of his insight has led over the last 20 years. His insight not only underlies the profound applications that appear in his own famous work with Donsker, but also accounts for the subsequent (possibly over-abundant) effusion of articles by others (including myself) on the topic.

Toward the end of the period when Varadhan was polishing off the program initiated in Schilder's thesis, he and I began the discussions that eventually led to our formulation of diffusion theory in terms of what we called the martingale problem. Those discussions took place more than 35 years ago, but they remain in my mind as the single experience that makes me most grateful to have entered mathematics. Of course, the pleasure of participating in what turned out to be a successful enterprise was great. But I think that I am being honest when I assert that the ultimate success of our collaboration was only part of the pleasure I derived from it. The other part was getting to know Varadhan. I was a young man who had been afforded every advantage: I had educated, prosperous parents who paid my passage through the best schools in America. Here was a man my own age whose parents, though superbly educated, were far from prosperous. He had won his passage by outperforming all but a handful of the literally millions of Indians his age. Perhaps more impressive to me was that, unlike most of the people I knew who had succeeded in the face of adversity, Varadhan had emerged unscathed. Unlike the majority of gifted people whom I have encountered, Varadhan never has used his gifts as a weapon to bludgeon his less gifted colleagues. He was then, and remains, a true gentleman.---Daniel W. Stroock, Simons Professor, Department of Mathematics, MIT.

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