Michael F. Atiyah

Mathematical bridge-builder

Print edition : March 01, 2019

Michael Atiyah. Photo: TIFR Archives

Michael Atiyah, on the occasion of the International Colloquium on Vector Bundles on Algebraic Varieties at the TIFR, Bombay (now Mumbai), in 1984. M.S. Narasimhan is also in the picture. Photo: TIFR Archives

Michael F. Atiyah (1929-2019) was a prominent member of the scientific establishment in Britain, where he dominated the mathematical scene and established Oxford as a major centre for geometry.

SIR Michael Francis Atiyah was one of the major figures of 20th century mathematics. He was a master of modern geometry and a crucial agent in its rapprochement with theoretical physics. He passed away on January 11, a few months short of his 90th birthday.

Michael Atiyah’s father, Edward Selim Atiyah, was Lebanese and a civil servant in Khartoum, the capital of Sudan, then a British colony. His mother, Jean Levens, was Scottish. Of his father, Atiyah said: “My father’s main dream was to go to Oxford. He wanted to convert himself into an Englishman. It didn’t quite work out. When he came back to Sudan, he found he wasn’t part of the English class structure; he was regarded as one of the lower classes although he was Oxford-educated and regarded himself as culturally English. That turned him over a bit. He became an Arab nationalist to some extent. All his life was divided between wanting passionately to be English and yet sympathising with the Arab political position within the British Empire.” (I have relied on the incomparable biography available at http://www-history.mcs.st-andrews.ac.uk/Biographies/Atiyah.html for information on Atiyah’s life.)

Those who met Atiyah in his prime would have recognised a certain brisk British type: always impeccably dressed and who spoke in confident, clear, declarative sentences, with a twinkling sense of humour that put the listener at ease. His was a short, slightly stocky figure, but through force of personality he dominated any room he was in. Although he was an internationalist by temperament and held for three years a prestigious professorship at the Institute for Advanced Study in Princeton, New Jersey, United States, it was clear that he felt most at home in Britain, where he was a prominent member of the scientific establishment and where he dominated the mathematical scene and eventually established Oxford as a major centre for geometry. After retirement from Oxford, he moved to Edinburgh and back to his Scottish roots.

It is often said that there are two tribes of mathematicians: the problem-solvers and the theory-builders. There is in fact a third group, the bridge-builders, of whom Atiyah was the archetypical example. It is this aspect of the man that I would like to convey in this article.

(An aside: In the 1990s, Atiyah gave a public lecture at the Tata Institute of Fundamental Research (TIFR) in the Homi Bhabha Auditorium, a majestic structure perched on the rocky south Bombay shore facing the Arabian Sea. Unfortunately, I do not have access to the text, but I remember well his description of the roles of the theoretical physicist and mathematician in the exploration of the mathematical universe, or at least those parts of it that were of interest to him at that time. A theoretical physicist is an explorer who jumps from island to island and reports back on the flora and fauna that she has seen; a mathematician then builds the bridges linking the islands. The physicist roams far and wide and gets impatient with the mathematician who is in it for the bridge-building—a punchline, somewhat self-deprecating and clearly only half-meant.)

To set the stage, one first needs to understand how mathematics, like the natural sciences, took its modern form in a process that began in the 19th century and culminated in the emergence of some grand narratives in the 20th century. Most of this happened in Europe.

Euclid’s construction of geometry was concerned with the deduction of mathematical truths by systematic reasoning from a set of axioms. In spite of this, for much of the ensuing two millennia, mathematics proceeded on the basis of ingenious calculations based on intuitive justifications. As these became more and more baroque, the practitioners began to feel the need to moor their reasoning in solid ground. This resulted in careful constructions of mathematical objects and subtle definitions.

By the early decades of the last century, two great branches of mathematics had been cleared of obscuring overgrowth and given a recognisably modern form: these were analysis (which grew out of the study of calculus and infinite sums) and algebra (which grew out of the study of number theory, polynomial equations and symmetries). Much of the foundational work happened in Germany.

The third great branch of mathematics, geometry, was late to this spring-cleaning. This was partly because, notwithstanding Euclid, the subject itself was a late developer. Its component parts—topology (“rubber-sheet” geometry), Riemannian geometry (the study of distances on curved spaces), algebraic geometry (the grown-up version of “coordinate geometry”) and algebraic topology (which deals with puzzles such as those involving the Konigsberg Bridge and much more sophisticated problems that are attacked by attaching algebraic structures to topological objects)—came into focus only in the latter half of the 19th century and the early decades of the 20th century.

Although the British have to their credit many scientific revolutionaries—Isaac Newton, Charles Darwin, James Clerk Maxwell, Michael Faraday, Paul Dirac and Francis Crick among them—their mathematics shied away from grand theorising, leaving this to the Continentals, in particular Germany and France. It is fair to say that at the turn of the century, British mathematics was overshadowed by developments on the Continent. Even when this changed with the advent of J.E. Littlewood and G.H. Hardy, who revitalised analysis and analytic number theory, British work in algebra and geometry tended to be not of the same order as that on the Continent, albeit with some exceptions, notably the work of William Hodge, F.S. Macaulay and John Todd.

All this changed with Atiyah.

Atiyah’s doctoral thesis was written under the supervision of Hodge. Atiyah’s first substantial work was in collaboration with Hodge. With its combination of differential geometry and algebraic geometry, it foreshadowed the bridge-building instincts that would later characterise his work. In this and in later work, he quickly assimilated modern formulations of geometry and topology and tools that were largely developed in France (by Henri Cartan, Elie Cartan, Jean Leray, Alexandre Grothendieck and Jean-Pierre Serre) with important contributions from the Japanese mathematician Kunihiko Kodaira.

Then began the first of the major collaborations that would be a feature of his work. Together with the German mathematician Friedrich Hirzebruch, Atiyah developed a major technique in topology called “topological K-theory” that succeeded in answering many outstanding questions in geometry.

Index theorems

Then followed two decades devoted to deep questions regarding partial differential equations and geometry, which culminated in the “index theorems” and their cousins. This was the product of collaborations with Isadore M. Singer, Raoul Bott, Graeme Segal and the Indian mathematician Vijay Kumar Patodi.

The index theorem of Atiyah and Singer is undoubtedly a high-water mark of 20th century mathematics. It brings together differential geometry, topology and analysis and subsumes and unifies many similar theorems that went before. It is only fair that I try to give a flavour of its content. (Believe me that it is in fact easier to understand than the Big Bang or the double helix, which are routinely referred to in the lay press!)

The theorem concerns itself with “systems of elliptic linear partial differential equations on compact manifolds”. A manifold is a “space” on which calculus can be done; it is compact if it is bounded in a suitable sense. Maxwell’s equations governing the propagation of electromagnetic waves are examples of partial differential equations. When a partial differential equation is “elliptic and linear”, it has finitely many solutions out of which any solution can be built. A system of elliptic partial differential equations is a collection of such equations and has an “analytical index”, essentially built out of the numbers of solutions of the equations in the system. The analytic index does not change if the system is subject to small changes, and the Atiyah-Singer index theorem gives a formula for the index in terms of some geometric and topological data associated with the system.

The mid 1970s saw a simultaneous realisation among theoretical physicists and mathematicians when the equations (due to Chen-Ning Yang and Robert Mills) underlying modern theories of the fundamental forces of nature at the subatomic level are put in the proper mathematical language, they involve subtle geometric objects, and that the nonlinear partial differential equations which encapsulate these physical laws are pregnant with geometric information. This was the start of an intense period of communication between physicists (primarily those interested in theories of the fundamental interactions) and mathematicians (primarily geometers). Atiyah was at the heart of this interaction, as were physicists such as Richard Ward, Roger Penrose, Gerard ’t Hooft and Alexander Polyakov. A particularly fruitful partnership began between Atiyah and the incomparable Edward Witten, who combines within himself the qualities of a physicist of the first rank and a Fields Medal-winning mathematician. In the decades that followed, the public dialogue between Atiyah and Witten set much of the agenda of the field that came to be known as “geometry and physics”.

During the 1980s and 1990s. Atiyah together with Bott, Witten, Nigel Hitchin and many others used insights and techniques from quantum field theory and in particular the Yang-Mills theory and the Seiberg-Witten theory to shed light on many questions in mathematics. (The most spectacular and unexpected applications of these ideas were due to Simon Donaldson, Clifford Taubes and Witten.) In particular, Atiyah was one of the inventors of “topological field theory”. As the century drew to a close and the attention of many theoretical physicists moved to string theory, Atiyah was one of its most prominent supporters in the mathematical community.

Electrifying lecturer

Atiyah was an electrifying lecturer who communicated mathematics with clarity and passion. In his research papers and expository writings, he cultivated a light touch that combined a sense of rigour with an informal and elegant style that became the Oxford “house style”.

There were significant interactions between Indian mathematicians and Atiyah and his mathematics. I already touched upon Patodi, who was a student of M.S. Narasimhan at the TIFR and wrote a series of ground-breaking papers (first on his own and then jointly with Atiyah, Bott and Singer). This work in turn had as its basis a technique pioneered by S. Minakshisundaram, which had to do with the geometric information encoded in an infinite sum (“asymptotic series”) representing solutions of the partial differential equation governing heat propagation on curved spaces. In the early 1980s, Atiyah and Bott asked the innocent question: What if strong interactions were the only forces at work and space-time was two-dimensional? This led to a new approach to a proof of a celebrated theorem of Narasimhan and C.S. Seshadri and a new derivation of the famous formulae of Gunter Harder and Narasimhan—formulae that had been derived by entirely different methods. (Atiyah arranged for many students of Narasimhan to go to Oxford. While still a graduate student, I myself spent an idyllic six months at the Mathematical Institute there and carefully read the paper of Atiyah and Bott in its “preprint” form, and it played an important part in my later work.)

Atiyah was an Honorary Fellow of the TIFR. In 2005, he chaired a review committee to assess the performance of the School of Mathematics at the TIFR in the previous decade.

Atiyah’s mathematical achievements were universally recognised. He was awarded the Fields Medal in 1966. In 2004, the Abel Prize was shared by Atiyah and Singer “...for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics”. There were many other honours, among which, I suspect, he particularly cherished the knighthood (1983) and the presidentship of the Royal Society (1990-95).

T.R. Ramadas is Distinguished Professor at the Chennai Mathematical Institute. His career has seen long stints at the Tata Institute of Fundamental Research and at the International Centre for Theoretical Physics, Trieste, Italy, from where he retired as Head of Mathematics. He is a member of the executive committee of the International Mathematical Union.

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