Atiyah: Algebraist or Geometer Is Like Deaf or Blind

In June 2000, the year UNESCO had declared World Mathematical Year, the Fields Institute in Toronto asked Michael Atiyah to look back over the century. He was 71, a Fields Medallist, a former President of the Royal Society, and the man whose index theorem had tied analysis, topology and geometry into one statement. He spoke without a written text, and the lecture was later printed from the recording.

He organised the century as a series of shifts: from local to global, from low dimensions to high, from commutative to non-commutative, from linear to non-linear. Then he came to the oldest division of all, geometry against algebra, and said it was not a shift but something “very, very fundamental”. Geometry is about space, the thing you take in at a glance. Algebra is about time: an algorithm is a sequence of steps performed one after another, and a computer makes that plain. So asking a mathematician to choose is asking which sense to give up.

A few sentences later he gave the quarrel a darker name. Algebra, he said, is the offer the devil makes to the mathematician: here is a machine that answers any question, in return for your soul, which is geometry. We would all like to cheat the devil and keep both. But once you pass over into calculation you stop thinking about what the symbols mean, and that, he said, is the real price.

The remark is sharper for who made it. Atiyah was a geometer by temperament, but with Ian Macdonald he wrote the thin 1969 textbook that has taught commutative algebra to every generation since. His own theorem could not have been found by either faculty alone. In the same lecture he gave the century's topology to Poincaré, whose definition of mathematics is the natural companion to this one, and noted how little of it had made Hilbert's famous list of problems.

Michael Francis Atiyah was born in London to a Lebanese father and a Scottish mother and grew up in Khartoum, where his father worked for the Sudanese civil service. He went to Victoria College in Alexandria and Manchester Grammar School, then to Trinity College, Cambridge, where he took his doctorate under W. V. D. Hodge.

With Isadore Singer he proved the Atiyah–Singer index theorem in 1963, which links the number of solutions of a differential equation to the shape of the space it lives on. It earned him the Fields Medal in 1966 and, with Singer, the Abel Prize in 2004. He held the Savilian chair of geometry at Oxford, spent three years at the Institute for Advanced Study in Princeton, was President of the Royal Society from 1990 to 1995, Master of Trinity College, and founding director of the Isaac Newton Institute. He was knighted in 1983 and appointed to the Order of Merit in 1992. He died in Edinburgh in January 2019, at 89.