Israel Gelfand

Times obituary

Professor Israel Gelfand: mathematician

Mathematicians are often divided into theory-builders and theorem-provers. Professor Israel Gelfand was a theory-builder through and through, unlike his teacher Andrei Kolmogorov. Vladimir Arnold, who with Gelfand was one of Kolmogorov's most brilliant pupils, suggested that, if both men arrived in a mountainous country, Kolmogorov would start scaling the highest peaks, while Gelfand would start building roads.

Like Kolmogorov, however, Gelfand produced a large amount of extremely high quality work, across an exceptionally broad range of subjects. Both were also universalists, in the manner of the great French mathematician Henri Poincaré, whose famous motto was: "Il faut triompher par la pensée, pas par le calcul" -- one must triumph by thought, rather than by calculation. Poincaré's axiom, as well as encapsulating much of the essence of modern mathematics, also runs through the work of Professor Gelfand.

Israel Moiseevich Gelfand was born in 1913 in Okny (now Krasni Okny) in the southern Ukraine, into a Jewish family. His extraordinary ability was recognised early, and encouraged in a way that enabled him to avoid the usual route through school and university, where his being a Jew would have been an obstacle.

Gelfand grew up during a period when public life in the Soviet Union was disfigured by anti-Semitism but Gelfand was able to progress, at the early age of 19, to postgraduate study at Moscow State University.

Here he did research under Kolmogorov, one of the most eminent mathematicians of the last century, and a man noted for his refusal to be influenced by the anti-Semitism that surrounded him.

Gelfand's doctoral thesis followed in 1935, and his higher doctorate, the DSc, in 1938. The 1930s were a fruitful period in mathematics. Analysis -- the study of limiting processes which evolved out of calculus -- was developing into functional analysis, most notably in the book Théeorie des Opéerations Linéeaires (1932) by the Polish mathematician Stefan Banach.

Here the setting is typically infinite-dimensional, rather than finite-dimensional as in classical analysis, and the methods are a mixture of analysis, algebra and topology.

Meanwhile, one of the high points of classical analysis was the Wiener general Tauberian theory, also dating from 1932, undertaken by the American mathematician Norbert Wiener. This powerful theory, now widely used, studies how limiting behaviour of one kind of average of a function may be used to study that of other kinds of average.

Gelfand was able to both extend and simplify the Wiener Tauberian theory by systematically exploiting algebraic methods, specifically his theory of maximal ideals. This was recognised as the first spectacular triumph of modern functional analysis over the more traditional classical analysis.

Gelfand's ideas blossomed into the theory of Banach algebras (as they are now known -- the Russian term at that time was normed rings), in work from 1940 on. This led to the publication of an influential book in 1960 with D. A. Raikov and G. E. Shilov: the Gelfand-Raikov-Shilov work has since been translated into English and other languages.

Another area where functional analysis proved its worth was the broadening of the classical concept of a function to generalised functions (here, it is not the value of a function at a point that matters, but behaviour when multiplied by suitable "test functions" and integrated). The theory was promulgated by the French mathematician Laurent Schwartz in 1948. Gelfand's three-volume book with Shilov in 1958, in Russian, developed into a five-volume English version (1964-67), which has become a standard text.

Gelfand was a prolific mathematician, whose Collected Works, in three volumes, cover nearly 3,000 pages (1988-89). Another of Gelfand's continuing interests was representation theory, which has important applications in quantum physics.

While working for his DSc, Gelfand taught at the USSR Academy of Sciences from 1935 to 1941. He then became a professor at Moscow State University, where he taught for many years. Here he ran a near-legendary seminar on mathematical analysis, where the programme was improvised, and speakers were interrupted from the floor, not always kindly.

He subsequently started a second seminar, on mathematical biology, after the death of his son Aleksandr from leukaemia. In this context his work on integral geometry -- the subject of the fifth volume of his book on generalised functions -- was valuable, and led to advances in such areas as computerised tomography, used in the treatment of cancer.

Gelfand's Jewishness was a continuing source of problems for him in the Soviet Union, and he eventually left Russia in 1989 for the United States. After a year at Harvard and the Massachusetts Institute of Technology, he went to Rutgers University in New Brunswick, New Jersey, where he saw out his career.

Gelfand was widely honoured. He was awarded the Order of Lenin three times, became a Corresponding Member of the Soviet Academy of Sciences in 1953, and an Academician -- a full member -- in 1984. He was elected a Foreign Member of both the US National Academy of Sciences and the Royal Society, and of the American Mathematical Society and the London Mathematical Society, among others. He was awarded a number of honorary degrees, including one from Oxford, as well as the Kyoto Prize in 1989, and the Steele Prize in 2005.

Gelfand's first marriage, to Zorya Shapiro, ended in divorce. He is survived by his second wife, Tatiana, and their daughter, and two sons from his first marriage.

Israel Gelfand, mathematician, was born on September 2, 1913. He died on October 5, 2009, aged 96