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Atiyah began his career at a time when mathematics was becoming ever
more fragmented. There seemed little to be gained from applying ideas from
one area of research to another. The twisting thread of Atiyah’s own research
subsequently sewed together half-a-dozen distinct areas during what mathematicians
consider a historic period of reunification.

From the mid 1970s on, his linking went a stage further. In collaboration
with the American physicist Edward Witten, he built a new bridge between
maths and theoretical physics for the first time since Einstein. The bridge
gave the world string theory – the influential idea which regards the fundamental
building blocks of matter as being lengths rather than points.

‘He is a kind of universalist,’ says Christopher Zeeman, master of Hertford
College, Oxford, ‘distinguished by taking part in so many different fields.’
Zeeman says Atiyah has stayed at the top of world mathematics for 30 years.

The constant theme of Atiyah’s work is topology. Topology is often described
as ‘rubber sheet geometry’, all stretching and smoothing. The phrase catches
the formlessness of the study – the way a coffee cup is the same as a doughnut
to a topoligist because they both have one hole – but not the point of it.
Topology is holistic. It tells you about the global properties of a system,
not a corner of it. In the past 30 years, with Atiyah’s help, it has reinvigorated
vast tracts of mathematics which have become bogged down in minutiae of
ever-decreasing significance.

He started in the 1960s by reintroducing topology to a long-lost relation,
algebraic geometry. The marriage produced something rare a new topological
invariant. Holes are one kind of invariant because so long as you only stretch
a shape and do not cut it, the shape will keep the same number of holes.
Atiyah’s invariant was the basis of a new branch of maths called K-theory.

K-theory was the launch pad for Atiyah’s co-discovery with Isadore Singer
of the index theorem which wom him the Fields Medal in 1966. The theorem
compares pairs of elliptic differential equations. Under specified conditions,
it calculates their index, the difference between the number of solutions
of the two equations.

Atiyah remembers the origin of the theorem. ‘Singer was visiting me
in Oxford and we got to talking. We got to looking at the Dirac equation
and I talked to (Robert) Smale who happened to be passing through. OVer
a matter of a few months we got to formulate the problem we wanted to solve.
Formulating the problem is three-quarters of the work. First you’ve got
to realise there is a problem, then formulate it, then you’ve got to solve
it. That last bit of solving it is really a technical matter.

‘We faced a large problem of how to understand the solution of various
elliptic differential equations. Coming into it from the direction that
we did we were able to see already the shape of the answer and we were able
to use all the K-theory machinery to solve that problem. ‘Of course it took
a while to work it all out.’ In fact it turned into a 10-year problem.

The index theorem counts led Atiyah to theoretical physics in the 1970s.
Physicists were just getting to grips with parity violation, the discovery
that right-handed particles do not behave in the same way as left-handed
one. The index theorem turned out to be a way of quantifying difference
between the number of particles of each type. ‘It is rather an subtle thing,
measuring the extent to which things are asymmetrical,’ he says.

By 1980 Atiyah was 51, old enough for traditional wisdom to write off
a mathematician. In fact, he was in the middle of his most influential work.
‘The biggest thing in physics in the 20th century has been quantum mechanics.
It has only dawned on people recently that quantum mechanics is not really
local in a lot of ways. It has global effects,’ he said. Topology, with
its global outlook, turned out to be the natural new setting for quantum
mechanics.

Atiyah’s application of topology led Witten to string theory which,
though loathed by Richard Feynman and others for being led by mathematics
rather than observation, became the most influential theory of the 1980s.
The work was proof to Atiyah that physicists found maths worth watching
again, particularly since experiments have become so expensive and rare.
‘They are often not being guided by experimental results but by mathematical
consistency,’ he says. As the physical theories became better understood
mathematics benefited, too. Simon Donaldson, now at Oxford, grabbed Atiyah’s
ideas for his ground-break work on the topology of surfaces in four dimensions.

This interplay between maths and physics has become the most vivrant
area of maths. All four winners of this year’s Fields Medal, including Witten,
worked in the area. The usefulness of the mathematics is an example of the
unreasonable reasonableness of the Universe which Atiyah finds perfectly
reasonable. ‘Maths is not just an abstract logical system. It is something
people can understand. On the other hand, sicience is exploring the world
via the human mind. So I am not as surprised by the Connections as other
people. To me it is a natural process.’

It is a humane, local view of scientific enquiry which is buttressed
by a belief in the value of simply sitting around and talking. ‘I like talking
with other people. You could not do this sort of work if you were not a
gregarious person. There is not time in life to learn everything you need
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