Israel Gelfand, who has died aged 96, was a major figure in mathematics for seven decades. His research ranged over most of pure maths, including algebra, analysis, and geometry. He also worked in mathematical biology, opening up the field of integral geometry, a topic that is fundamental to medical scanners. He was an incomparable teacher and made significant advances in every field that he touched.
Gelfand was born to Jewish parents in the small town of Okny (now Krasni Okny) to the north of Odessa in southern Ukraine, which was then a part of the Russian empire. In 1930 he moved to Moscow to complete his secondary education. However, he was not permitted to enrol as an undergraduate, having (according to some sources) been expelled from school because his father, a miller, was considered to be a capitalist. Israel took a part-time job as doorkeeper at the Lenin Library and taught evening classes on mathematics. The work made it possible for him to attend mathematics courses at Moscow State University.
He showed such talent that Andrei Kolmogorov, the leading Soviet mathematician of the period, took him on as a postgraduate student. His 1935 PhD thesis was in the relatively new area of functional analysis, where the ideas of calculus are extended from finitely many variables to infinitely many. One practical application is to partial differential equations, the mathematical physicist's favourite tool for describing the natural world. Another is the mathematical formulation of quantum mechanics.
Gelfand was appointed to the Steklov Mathematical Institute and taught at the university, but lost both positions temporarily through antisemitism. He was elected a corresponding (low-status) member of the Soviet Academy of Sciences, but it was more than 30 years before he was made a full member. His seminar series, run independently of the university and open to anybody, ran for nearly 50 years and is famous throughout the mathematical world. He moved to America in 1989, first to Harvard University, Cambridge, Massachusetts, and then Massachusetts Institute of Technology, then settling at Rutgers University, New Jersey.
The heart of Gelfand's research was representation theory, a formal setting for symmetry, a concept of central importance in mathematics and physics. A symmetry of an object is a transformation that preserves its structure, and the collection of all such transformations is the object's symmetry group. The physical world, at subatomic level, is highly symmetric: if you change an electron's direction of spin, or its electric charge, the laws of physics still work the same way. Representation theory studies all the contexts in which a particular symmetry group can arise. Its applications include subatomic particles and pattern formation – why snowflakes are six-sided, and why tigers have stripes but leopards have spots.
The most important types of symmetry are the "classical groups", a typical example being the group of all rotations of space. Gelfand solved many fundamental questions about classical groups, using a mixture of algebraic and geometric methods. His interests went beyond mathematics into theoretical and experimental science. In 1958, when his son, Aleksandr, contracted leukaemia, he started applying mathematics to cell biology, setting up the Institute of Biological Physics of the Russian Academy of Sciences.
Some of his discoveries have applications that are important for everyone: medical scanners. Doctors routinely use several different kinds of scanner. CT scanners, for example, use beams of x-rays to obtain a three-dimensional image of the body's internal organs. This is a bit like holding a semi-transparent object up to the light and using the resulting shadows to work out its true shape. The first steps in this area were taken in 1917 by Johann Radon. Gelfand developed Radon's ideas extensively, founding an entire field of mathematics, now called integral geometry. His ideas are vital to today's medical imaging methods.
I first came across Gelfand in 1973, early in my academic career. Oxford University was awarding him an honorary degree, and – unusually for that time – he had been allowed to leave the Soviet Union to receive it. So this was a rare opportunity to see the great man in action. Several of us piled into a car and drove to Oxford's Mathematical Institute. I still remember the lecture, which was about a remarkable geometrical phenomenon, the "five subspace" theorem. Today it is interpreted as a deep phenomenon in representation theory, placing limits on what is theoretically possible. Gelfand had a reputation for clear, well-organised lectures, and this one was no exception. It was aimed at professionals, and quite technical, but he developed the ideas systematically, explaining their significance as he went along. By the end of the talk, he had made a very surprising result seem natural and inevitable – a sure sign of high-quality mathematics.
Gelfand received many awards. The Soviet Union awarded him the Order of Lenin three times. He won the Wolf prize (comparable to a Nobel) in 1978, and the Kyoto prize (for "significant contributions to the progress of science, the development of civilisation, and the enrichment and elevation of the human spirit") in 1989. He was elected to innumerable academic bodies, including the Royal Society and the US National Academy of Science.
He was also a great teacher. He set up a distance-learning school for mathematics in the Soviet Union, and a similar one in the US in 1992. He considered teaching and research to be inseparable, and was equally comfortable talking to schoolchildren or his research colleagues. He supervised 22 PhD students, several of them now outstanding mathematicians in their own right.
Gelfand is survived by his second wife, Tatiana, two sons, a daughter, four grandchildren and three great-grandchildren.
• Israel Moiseevich Gelfand, mathematician, born 2 September 1913; died 5 October 2009
