Israel Gelfand

Israel Gelfand: The Man Who Built Roads Through Mathematics

He never finished high school and never sat a single undergraduate course. At sixteen he was working as a door keeper at the Lenin Library in Moscow, teaching evening classes in elementary arithmetic to pay for himself, and slipping into university lectures when he could. At nineteen Moscow State University admitted him directly to graduate study, on Kolmogorov's judgement. Vladimir Arnold later put the difference between the two men this way: Kolmogorov would climb the highest mountain, and Gelfand would start to build roads.

No School, No Degree, Moscow Anyway

Israil Moiseevic Gelfand was born on 2 September 1913 in Okny, then in the Russian Empire and now in Ukraine, near Odessa. In 1930, aged sixteen, he left for Moscow without a completed secondary education. What followed was two years of improvised survival and self-instruction: odd jobs, the library door, and a ladder of teaching posts in Moscow's evening institutes that took him from elementary mathematics up to advanced courses while he was still learning the material himself.

He attended lectures at Moscow University as an outsider — his first course was complex analysis under Lavrent'ev. In 1932 Kolmogorov took him on as a research student, dropping him into a functional analysis school that also contained A. E. Plessner and L. A. Lyusternik. Three years later, at twenty-two, he defended a thesis on abstract functions and linear operators.

The Trick With Maximal Ideals

The 1935 thesis already showed the move that would define him. Faced with functions living on an abstract normed space, where none of the ordinary tools of calculus apply, Gelfand's tactic was to apply linear functionals to them, turning the intractable object into an ordinary function of a real or complex variable — and then bring the full weight of classical analysis to bear on that.

His D.Sc. thesis of 1938, on commutative normed rings, made this into a general method. The idea that carried it was the maximal ideal. An algebra of functions and a topological space had, until then, looked like different kinds of object. Gelfand showed that the points of the space could be recovered from the algebra alone, as its maximal ideals — meaning an abstract algebra secretly *is* an algebra of functions on a space you can reconstruct from the inside. Colleagues credited him precisely: it was Gelfand who brought to light the fundamental concept of a maximal ideal, and in doing so united a set of previously disconnected facts into one structure.

That construction is now called the Gelfand representation, and the objects are called Banach algebras. Its most spectacular payoff came in the early 1940s with Mark Naimark, when the two of them handled non-commutative normed rings with involution and showed that these too can always be represented concretely, as algebras of linear operators on a Hilbert space. The Gelfand–Naimark theorem is one of the load-bearing results of twentieth-century analysis, and it is the reason the algebraic formulation of quantum mechanics works at all: the abstract algebra of observables can always be realised as operators.

A Career in Several Mathematics at Once

Gelfand worked at the USSR Academy of Sciences from 1935 to 1941 and then became professor at Moscow State University. What he did afterwards resists summary because it does not stay in one field.

He took representation theory — Frobenius and Schur's work on finite groups, Weyl's on compact ones — and extended it to non-compact groups, where the machinery is far harder and where the physics lives. He worked on the inverse Sturm–Liouville problem in differential equations: not "given this system, how does it vibrate?" but the reverse, "given how it vibrates, what is the system?" With Georgi Shilov he built up the theory of generalized functions, following Sobolev and Schwartz, which is how physicists get to use objects like the Dirac delta without the mathematics collapsing.

And he developed integral geometry, including work on the Radon transform — the mathematics of reconstructing an object from a family of its cross-sections or projections. That turned out to be the theory underneath medical imaging: CAT scans and MRI produce three-dimensional pictures of a living body by solving exactly this problem. Between 1968 and 1972 he added a body of work on the cohomology of infinite-dimensional Lie algebras. Over his lifetime he published more than five hundred papers.

The Seminar

The Moscow seminar was an institution in its own right, held weekly at Moscow State University and remembered by everyone who survived it. It ran closer to improvisation than to a lecture programme — attendees frequently did not know the topic until it began. Gelfand's characteristic contribution was the reframing: the formulation of new problems, the unexpected question, the tendency to look at even well-known things from an angle nobody had tried. He also built a correspondence school that carried serious mathematics to students across the Soviet Union who had no access to a research centre.

He was not a comfortable man to be around. Andrei Zelevinsky's assessment was blunt: he was not the most delicate, polite person in the world. Alexander Goncharov, who found him unpredictable, still called him the most interesting person he had ever met.

Biology, and a Reason for It

From 1958 Gelfand moved a substantial part of his attention into biology and medicine, co-founding the Institute of Biological Physics with Fomin in 1960. The trigger was personal: his son developed leukemia, and Gelfand started a biology seminar. The work that came out of it was characteristically structural rather than descriptive — general principles for how control is organised in complex multi-cellular systems, with applications reaching into X-ray analysis and pattern recognition.

Why Israel Is Called a Genius

The word is defensible here for a specific reason: Gelfand's gift was for finding the level of abstraction at which two unrelated things become the same thing. The maximal ideal insight is the cleanest example — an algebraic object and a geometric space turn out to be two views of one structure, and once you see it, an entire literature of separate results collapses into a single theorem. He did that repeatedly, across analysis, representation theory, differential equations, integral geometry and Lie theory, and then across the border into biology.

Arnold's mountain-and-roads distinction is the sharpest thing anyone has said about him, and it is not entirely flattering. Gelfand was not primarily a solver of famous standing problems. He was an infrastructure builder — he made territory passable for other people. His influence ran heavily through the seminar and through students, which means a good deal of the work bearing his imprint is not, strictly, his. That he never received a Fields Medal, in a career of this reach, is worth noting honestly.

Against that: the self-education is not a romantic footnote but evidence. A boy with no secondary schooling reached, unsupervised, the point where Kolmogorov admitted him to graduate work, and by twenty-five had produced a structural idea that reorganised functional analysis. Whatever that faculty is, very few people have it.

Legacy

The honours arrived steadily and from every direction: president of the Moscow Mathematical Society from 1968 to 1970, the Order of Lenin, Fellow of the Royal Society in 1977, the Wolf Prize in 1978, the Kyoto Prize in 1989, a MacArthur Fellowship in 1994, and the Leroy P. Steele Prize in 2005 for profoundly influencing many fields of research. He taught at Harvard in 1989–90, emigrated to the United States in 1990, and became Distinguished Visiting Professor at Rutgers, where he founded the Gelfand Outreach Program for high school students — the same instinct as the Soviet correspondence school, transplanted.

He died on 5 October 2009 at Robert Wood Johnson University Hospital in New Brunswick, New Jersey, aged ninety-six, survived by his wife Tatiana, his sons Sergei and Vladimir, his daughter Tatiana, four grandchildren and three great-grandchildren. Anyone who has been inside an MRI scanner has been through his mathematics.

Achievements

• Wolf Prize in Mathematics — 1978
• Kyoto Prize in Basic Sciences — 1989
• Notable work: Gelfand pair
• Notable work: Gelfand representation
• Notable work: Gelfand ring
• Notable work: Gelfand–Mazur theorem
• Held posts at Academy of Sciences of the USSR, Harvard University and Keldysh Institute of Applied Mathematics
• Educated at Lomonosov Moscow State University
• Fields of research: algebra, biology, functional analysis and group theory