Sir Michael Atiyah (1929–2019) is one of the most influential British mathematicians of the twentieth century. His work sits at the intersection of topology, differential geometry, and mathematical physics, and the theorem most closely associated with his name — the Atiyah–Singer index theorem, proved jointly with Isadore Singer in 1963 — is routinely cited as one of the most important results of the era. He was awarded both the Fields Medal (1966) and the Abel Prize (2004), placing him among a small group of mathematicians to win the two most prestigious prizes in the subject.
He was also, throughout his career, one of the most prominent public voices in British mathematics: a president of the Royal Society, master of Trinity College Cambridge, founder of the Isaac Newton Institute, and a vocal advocate for international scientific cooperation.
Life
Atiyah was born in London in 1929 to a Lebanese father and a Scottish mother. He spent his childhood in Khartoum, Cairo, and Alexandria, before returning to England for secondary school. At Trinity College Cambridge he took mathematics, and he stayed at Cambridge for his doctorate under W. V. D. Hodge — a supervisor whose own work would ripple through much of Atiyah’s later career.
He held positions at Cambridge, Oxford, and the Institute for Advanced Study in Princeton, where he collaborated with Isadore Singer and was part of the remarkable mid-century community that included Hermann Weyl, Kurt Gödel, and Albert Einstein before him. From 1990 to 1995 he served as president of the Royal Society, and from 1997 until his death he was honorary professor at the University of Edinburgh.
He was knighted in 1983 and awarded the Order of Merit in 1992 — the highest honour the British monarch can bestow on a non-royal subject. He died in January 2019 at the age of 89.
Scientific contribution
K-theory
Atiyah’s earliest major work, in the late 1950s and early 1960s, was on topological K-theory — a way of studying spaces by looking at the vector bundles over them, and organising those bundles into an algebraic structure. Building on earlier work by Alexander Grothendieck in algebraic geometry, Atiyah and Friedrich Hirzebruch developed the topological version into a powerful tool.
The resulting theory gave clean proofs of classical results (like the non-existence of higher-dimensional analogues of the complex numbers and quaternions) and opened new ones. K-theory is now a standard subject in topology and has seen applications in C*-algebras, operator algebras, and condensed matter physics. Topological insulators — the materials honoured by the 2016 Nobel Prize in Physics — are classified using K-theoretic invariants.
The index theorem
The theorem that most secured Atiyah’s reputation is the result he proved with Isadore Singer in 1963, and extended and reproved in several different ways over the following fifteen years: the Atiyah–Singer index theorem.
Suppose you have a compact manifold and an elliptic differential operator acting on sections of vector bundles over it. The operator has a kernel (solutions of ) and a cokernel (obstructions to solving ). The analytical index is the difference of their dimensions:
This is a number whose definition is purely analytical — it’s about solving differential equations.
On the other hand, there is a completely different number you can attach to the same setup: the topological index, built from characteristic classes of the manifold and the operator. This number is defined by combinatorial-topological data and has no obvious connection to solving PDEs.
The Atiyah–Singer theorem says these two numbers are always equal:
That a purely analytical quantity equals a purely topological one is surprising on its face. The theorem unifies three major areas of mathematics — topology, geometry, and analysis — and contains as special cases the Riemann–Roch theorem, the Hirzebruch signature theorem, and a long list of classical results. It is the single result most often used to justify the phrase “deep theorem.”
Atiyah and Singer were awarded the Abel Prize in 2004 for this work — more than four decades after the original proof, by which point the theorem had become one of the most-cited results in mathematics.
Mathematics and physics
From the late 1970s onwards Atiyah became increasingly interested in physics — specifically in the mathematical structures underlying gauge theory and the path integral. He recognised that the ideas physicists were using (instantons, Yang–Mills equations, monopoles) were deeply connected to topology and differential geometry in ways that neither community fully understood.
His collaboration with Edward Witten from the 1980s onwards helped establish the field of mathematical physics in its modern sense. Witten’s topological quantum field theories, and the use of path integrals to produce topological invariants, drew heavily on Atiyah’s ideas. Simon Donaldson’s Fields Medal-winning work on exotic smooth structures on 4-manifolds used Yang–Mills gauge theory as its central tool — a direction Atiyah helped steer.
Atiyah later wrote several influential articles attempting to articulate why mathematicians and physicists had become collaborators in the way they had, and what the long-term relationship between the two disciplines might look like.
Public role
Few mathematicians have spent as much of their career in positions of public responsibility. Atiyah served as:
- Savilian Professor of Geometry at Oxford
- Founding director of the Isaac Newton Institute for Mathematical Sciences in Cambridge (1990)
- Master of Trinity College Cambridge (1990–1997)
- President of the Royal Society (1990–1995)
- President of the Royal Society of Edinburgh (2005–2008)
- President of the Pugwash Conferences on Science and World Affairs
Throughout his career he was outspoken on questions of scientific ethics, international cooperation, and nuclear disarmament. He campaigned against weapons research and, as president of Pugwash, continued work begun by Bertrand Russell and Albert Einstein.
The final years
In 2018, at age 89, Atiyah announced a purported proof of the Riemann Hypothesis at a conference in Heidelberg. The argument was short and rested on a number of steps that the wider community did not consider adequately supported. The consensus among number theorists was that the proof was not valid.
This episode is worth treating compassionately. A lifetime of rigorous, important work sits on one side of the balance; a single late-career announcement that did not hold up on the other. The mathematical record of Michael Atiyah is determined by the former, not the latter. He passed away a few months later, in January 2019.
Legacy
The index theorem alone would have secured Atiyah’s place in mathematical history. Combined with K-theory, his work in gauge theory, and his role in bringing mathematicians and physicists into productive collaboration, his legacy is unusually broad.
A generation of geometers, topologists, and mathematical physicists trained in his orbit. The list of his students and collaborators reads like a directory of leading mid-to-late-twentieth-century geometers — Graeme Segal, Nigel Hitchin, Simon Donaldson, Frances Kirwan, among many others.
He was a reminder, for his whole career, of a certain kind of mathematical style: breadth rather than narrow specialisation, willingness to learn languages from neighbouring disciplines, and a conviction that the most beautiful theorems are almost always the ones that unify two things that didn’t look related.
Known for
- Atiyah–Singer index theorem (1963)
- K-theory in topology
- Fields Medal (1966)
- Abel Prize (2004, jointly with Singer)
- Pioneering the geometry behind gauge theory and physics
Frequently asked
What is the Atiyah–Singer index theorem?
It connects two initially unrelated quantities on a compact manifold: the analytical index of an elliptic differential operator (roughly, the difference in dimensions of its kernel and cokernel) and a topological invariant built from the manifold's and the operator's topology. The theorem says these two numbers are always equal. It unifies huge swathes of mathematics and has deep applications in physics.
Why did Atiyah help bring mathematicians and physicists together?
From the late 1970s onward, Atiyah recognised that the mathematical structures physicists were using — gauge fields, instantons, Yang–Mills equations — were deeply connected to topology and geometry. He worked with Edward Witten and others to translate between the two languages. That work helped launch modern mathematical physics and eventually contributed to string theory's mathematical side.
Did he believe he had proven the Riemann Hypothesis?
In 2018, at age 89, Atiyah announced a purported proof of the Riemann Hypothesis at a conference. The argument was not accepted by specialists — it rested on steps that the wider community considered inadequately justified. He passed away the following year. This episode is best treated compassionately, set against a lifetime of rigorous work.
More on: en.wikipedia.org