Raghavan Narasimhan

Raghavan Narasimhan: The Man Who Closed the Levi Problem

In 1962 the International Congress of Mathematicians invited a young Indian analyst to deliver one of its addresses. He was twenty-five and would not receive his doctorate for another year. The invitation was not a courtesy: two years earlier Raghavan Narasimhan had settled the Levi problem for complex spaces, a question that had stood open for half a century and had defeated a generation of the best analysts in Europe. He spent the following five decades in Chicago, writing the textbooks from which everybody else learned the subject.

Madras and Father Racine

He was born on 31 August 1937 in Madras. His mathematical formation began at Loyola College there, where he took his bachelor's degree in 1957 and studied under Father Racine, the French Jesuit who carried continental analysis into a country that had produced Ramanujan but little institutional infrastructure. Narasimhan went on to Bombay, into the orbit of the Tata Institute, and took his doctorate from Bombay University in 1963 under the number theorist K. Chandrasekharan.

The chronology is worth pausing on. The doctorate came in 1963. The theorems came in 1960 and 1961.

The Levi Problem

Complex analysis in one variable is a subject of beautiful rigidity: a holomorphic function on a disc is determined by remarkably little data. In several variables the rigidity becomes strange. Some regions of complex space have the property that every holomorphic function on them extends automatically to a larger region — the region is not, so to speak, the natural home of its own functions. Regions that are their own natural homes are called domains of holomorphy, and in 1910 E. E. Levi asked which ones they are, conjecturing that a geometric condition on the boundary called pseudoconvexity is the answer.

The question turned out to be brutally hard. Kiyoshi Oka finally settled it for domains in complex Euclidean space in the 1940s and 1950s. Hans Grauert then handled complex manifolds. What remained was the general and most delicate case: complex spaces, which are allowed to have singularities — points where the space is not locally a nice patch of C^n but something pinched or crossed. Singularities break the tools. Narasimhan's papers, "The Levi problem for complex spaces" in *Mathematische Annalen* in 1961 and its sequel, closed that case. The result is still cited as Grauert–Narasimhan, which is the discipline's way of recording that the last and hardest step was his.

Embedding

His other named theorem arrived first, in 1960, in a paper in the *American Journal of Mathematics* titled "Imbedding of Holomorphically Complete Complex Spaces." Stein manifolds are the complex-analytic objects that carry enough holomorphic functions to separate their own points — the analytic counterpart of affine algebraic varieties, and the natural setting for the entire subject. Defined abstractly, they are hard to picture. Narasimhan proved that every Stein manifold of complex dimension n admits an injective proper holomorphic immersion into C^{2n+1}.

That sentence does a great deal of work. It says an abstract Stein manifold is never exotic: it can always be realised concretely as a closed subset of an ordinary complex Euclidean space of bounded dimension, with the bound depending only on n. It is the complex-analytic analogue of Whitney's embedding theorem in differential topology, and it converted a class of abstractly defined objects into things one can hold. The result carries his name.

Chicago

He was a visiting scholar at the Institute for Advanced Study in Princeton in 1966 and joined the University of Chicago mathematics faculty in 1969, where he remained for more than four decades and finished as professor emeritus. His interests widened rather than narrowed: alongside several complex variables he worked in analytic number theory, and in the 1990s he collaborated with Charles Fefferman of Princeton, a Fields Medallist, on papers at the junction of analysis and real algebraic geometry — a pairing of two of the most technically formidable analysts alive. The University of Geneva awarded him an honorary doctorate in 1986.

The Books

Narasimhan wrote six books, and their influence may in the end exceed that of the theorems. *Analysis on Real and Complex Manifolds* (1968), the Chicago Lectures volume *Several Complex Variables* (1971) and *Complex Analysis in One Variable* (2001) are compressed, exact and famously unforgiving — books that assume the reader wants the shortest correct route rather than reassurance. Generations of graduate students learned several complex variables from them. Colleagues remembered a man who was equally serious about classical music and wine, and who, one recalled, "had a truly astounding insight in analysis."

He died on 3 October 2015, aged seventy-eight, after a brief illness.

Why Raghavan Is Called a Genius

Be concrete, because that is where the case lives. Two results, both obtained before he was twenty-five and before he had a doctorate, each of which closed a problem rather than advanced it.

The embedding theorem is the cleaner. It answers a question of the form "how wild can these objects be?" with "not wild at all" — every Stein manifold of dimension n sits properly inside C^{2n+1}. Results of that shape are rare and disproportionately useful, because they let everyone else stop worrying about a whole category of possible pathology.

The Levi problem is harder, and harder to explain. Oka and Grauert had done the smooth cases; the residue was complex spaces with singularities, exactly where the standard machinery becomes treacherous. Finishing a fifty-year-old problem at its most degenerate case, as a graduate student, against the best analysts in Europe, is what the word "genius" was coined to describe. The ICM's invitation to a twenty-five-year-old without a doctorate is the profession's own verdict, delivered at the time.

The counter-case is one of scale rather than quality. Narasimhan did not found a field or reorient a discipline; he solved two hard problems inside a field others had defined, then spent fifty productive but less spectacular years working within it. He won no Fields Medal, no Wolf, no Abel; his major distinction was an honorary degree from Geneva. Set beside the mathematicians who changed what the subject is about, his record is that of an exceptionally powerful analyst with two magnificent results early and a long, distinguished plateau after — a genius in the strict, local sense, who saw at twenty-three what nobody else could see, rather than a mind that redrew the map.

Legacy

The Narasimhan embedding theorem and the Grauert–Narasimhan solution of the Levi problem are permanent fixtures of several complex variables, taught wherever the subject is taught. His textbooks are the second legacy, and the one that has touched more people: generations of mathematicians first met complex analysis in his exacting prose. The third is institutional and easy to miss — he was part of the cohort that Father Racine and K. Chandrasekharan produced in the 1950s and 1960s, the generation that made Indian mathematics a permanent presence in the international literature rather than an occasional prodigy.

Achievements

• Held posts at University of Chicago
• Fields: mathematics