Michael Francis Atiyah

Born: 22 April 1929

Michael Atiyah's father, Edward Selim Atiyah (1903-1964), was Lebanese and his mother, Jean Levens, was Scottish. Edward, whose father was a medical doctor in Khartoum, had been educated at Brasenose College, Oxford, and became a civil servant in Khartoum. He was also an author and set up a radio broadcasting service during World War II. He was a strong supporter of the Palestinian cause. Michael Atiyah, when interviewed in [32], spoke about his father:-

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.

Michael's mother Jean, although of Scottish descent, was the daughter of a minister of a church in Yorkshire. She lived in Oxford and had studied at the university there. It was in Oxford that Edward and Jean met. They had four children, three sons Michael (the eldest and subject of this biography), Patrick Selim (born 5 March 1931, who went on to become an English lawyer and academic) and Joseph (known as Joe, the youngest of the four children who after a mathematics degree from Cambridge University, became a computer scientist working in computer software and telecommunications), and a daughter Selma (who studied English at an American University and lives in America). Although he was born in London, Michael grew up in Khartoum. However, to avoid the summer heat there the family usually returned to England at that time. Michael's primary school education was at the Diocesan school in Khartoum which he entered in 1934 at the age of five. He completed his primary education in 1941 and the family, as usual, returned to England.

Lebanon had been controlled by the French and, after the fall of France in 1940, it came under the control of the Vichy government. After their trip to England, the Atiyah family returned to Lebanon via France in 1941 and Michael returned to a French school. However, just after this began, the British and Free French began fighting to gain control of the Lebanon. Michael was sent to Victoria College in Cairo. This was a boarding school modelled on the English boarding school system and it was a school that Edward Atiyah had attended. Atiyah writes in the autobiography [3]:-

At Victoria College I got a good basic education but had to adapt to being two years younger than most others in my class. I survived by helping bigger boys with their homework and so was protected by them from the inevitable bullying of a boarding school.

Atiyah talked in [35] about how he came to chose mathematics:-

I was always interested in mathematics from a very young age. ... My parents always thought that I was cut out to be a mathematician from a very young age, all the way through. ... But there was a stage [at Victoria College in Cairo] when I got very interested in chemistry, and I thought that would be a great thing; after about a year of advanced chemistry I decided that it wasn't what I wanted to do and I went back to mathematics. I never seriously considered doing anything else.

He gave a somewhat fuller description of his decision between chemistry and mathematics in the interview. He said that it was inorganic chemistry that put him off the subject [15]:-

It was how to make sulphuric acid and all that sort of stuff. Lists of facts, just facts, you had to memorize a vast amount of material. Organic chemistry was more interesting, there was a bit of structure to it. But inorganic chemistry was just a mountain of facts in books like this. It's true that in mathematics you don't really need an enormous memory. You can work most things out for yourself, remember a few principles. If you're good at that, then it comes easily. If you want to do other things, you've got to work hard to learn a lot of facts. There was one reason, I think. But I enjoyed thinking, I'm good at it, and will continue with it.

After the war ended in 1945, Edward Atiyah returned to live permanently in England. Michael Atiyah attended Manchester Grammar School, one of the best schools for mathematics in the country. Although he was only sixteen years old, he had already taken his A-level examinations having been two years ahead of his age groups in Victoria College, Cairo. His two years at Manchester Grammar School were spent training to take the Cambridge scholarship examinations. However, it was at this school that he came to love geometry [3]:-

I found that I had to work very hard to keep up with the class and the competition was stiff. We had an old-fashioned but inspiring teacher who had graduated from Oxford in 1912 and from him I acquired a love of projective geometry, with its elegant synthetic proofs, which has never left me. I became, and remained, primarily a geometer though that word has been reinterpreted in different ways at different levels. I was also introduced to Hamilton's work on quarternions, whose beauty fascinated me, and still does.

He won a scholarship to Trinity College, Cambridge in 1947. However, rather than go straight to university, which was an option, he decided to do his two-years National Service, which was compulsory at the time. He served as a clerical officer and took the opportunity to read mathematics books and articles. He read Hardy and Wright's Number Theory at this time and also read articles on group theory. He was granted special permission to cut short the final year of his military service and spend it at Cambridge. There he played a lot of tennis and avidly studied mathematics on his own in the library. He matriculated at Trinity College in the autumn of 1949. Many of his fellow students had decided to postpone their National Service, so Atiyah was one of the older of the students in his year. With his exceptional talent, his extra maturity, and the studying he had done before starting his course, it is not at all surprising that he came out ranked first despite having many very talented fellow students. While still an undergraduate, he wrote his first paper A note on the tangents of a twisted cubic (1952).

After graduating with his BA in 1952, Atiyah continued to undertake research at Trinity College, Cambridge obtaining his doctorate in 1955 with his thesis Some Applications of Topological Methods in Algebraic Geometry. His thesis advisor was William V D Hodge. Speaking of the work for his thesis, Atiyah said [35]:-

I'd come up to Cambridge at a time when the emphasis in geometry was on classical projective algebraic geometry of the old-fashioned type, which I thoroughly enjoyed. I would have gone on working in that area except that Hodge represented a more modern point of view - differential geometry in relation to topology; I recognized that. It was a very important decision for me. I could have worked in more traditional things, but I think that it was a wise choice, and by working with him I got much more involved with modern ideas. He gave me good advice and at one stage we collaborated together. There was some recent work in France at the time on sheaf theory. I got interested in it, he got interested in it, and we worked together and wrote a joint paper which was part of my thesis. That was very beneficial for me.

Atiyah published two joint papers with his thesis advisor William Hodge, Formes de seconde espèce sur une variété algébrique (1954) and Integrals of the second kind on an algebraic variety (1955). He also published the single author papers Complex fibre bundles and ruled surfaces (1955). He was made a fellow of Trinity College, Cambridge in 1954. He married Lily Brown on 30 July 1955; they had three sons John, David and Robin. Lily, born in Edinburgh in 1928, was the daughter of a dock worker at the Rosyth naval yard. She had studied mathematics first at the University of Edinburgh and then took the Cambridge Tripos. She went on to obtain a doctorate, working under Mary Cartwright. Lily had met Michael Atiyah at Cambridge but, by the time they married, she was a lecturer at Bedford College, London. Atiyah was awarded a Commonwealth Fellow to study at the Institute for Advanced Study in Princeton during session 1955-56. Lily had to decide whether to keep her job at Bedford College or go to Princeton with her husband. She chose to go to Princeton with her husband and resigned her position at Bedford College. This was an important year for Atiyah who met, among others, Jean-Pierre Serre, Friedrich Hirzebruch, Kunihiko Kodaira, Donald Spencer, Raoul Bott and Isadore Singer. Returning to Cambridge, he was a college lecturer from 1957 and a Fellow of Pembroke College from 1958. He remained at Cambridge until 1961 when he moved to a readership at the University of Oxford where he became a Fellow of St Catherine's College.

Atiyah was soon to fill the highly prestigious Savilian Chair of Geometry at Oxford from 1963, holding this chair until 1969 when he was appointed professor of mathematics at the Institute for Advanced Study in Princeton. After three years in Princeton, Atiyah returned to England, becoming a Royal Society Research Professor at Oxford. He was also elected a Fellow of St Catherine's College, Oxford. Oxford was to remain Atiyah's base until 1990 when he became Master of Trinity College, Cambridge and Director of the newly opened Isaac Newton Institute for Mathematical Sciences in Cambridge.

Atiyah showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory, called K-theory. Grothendieck also contributed substantially to the development of K-theory. In [13] Atiyah's early mathematical work is described as follows:-

Michael Atiyah has contributed to a wide range of topics in mathematics centring around the interaction between geometry and analysis. His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding difficult problems. Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations. This 'index theorem' had antecedents in algebraic geometry and led to important new links between differential geometry, topology and analysis. Combined with considerations of symmetry it led (jointly with Raoul Bott) to a new and refined 'fixed point theorem' with wide applicability.

For these early achievements Atiyah was awarded a Fields Medal at the International Congress at Moscow in 1966. An address concerning Atiyah's contributions was given at the Congress by Henri Cartan, see [18]. The K-theory and the index theorem are studied in Atiyah's book K-theory (1967, reprinted 1989) and his joint work with G B Segal, The Index of Elliptic Operators I-V, in the Annals of Mathematics, volumes 88 and 93 (1968, 1971). Atiyah also described his work on the index theorem in The index of elliptic operators given as an American Mathematical Society Colloquium Lecture in 1973.

The ideas which led to Atiyah being awarded a Fields Medal were later seen to be relevant to gauge theories of elementary particles. Again we quote [13]:-

The index theorem could be interpreted in terms of quantum theory and has proved a useful tool for theoretical physicists. Beyond these linear problems, gauge theories involved deep and interesting nonlinear differential equations. In particular, the Yang-Mills equations have turned out to be particularly fruitful for mathematicians. Atiyah initiated much of the early work in this field and his student Simon Donaldson went on to make spectacular use of these ideas in 4-dimensional geometry. More recently Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists, notably E Witten, to the attention of the mathematical community.

The theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways, were all areas of theoretical physics which developed using the ideas which Atiyah was introducing.

Atiyah has published a number of highly influential books: K-theory (1967); (with I G Macdonald) Introduction to commutative algebra (1969); Vector fields on manifolds (1970); Elliptic operators and compact groups (1974); Geometry on Yang-Mills fields (1979); (with N J Hitchin) The geometry and dynamics of magnetic monopoles (1988); The geometry and physics of knots (1990); (Video) The mysteries of space (1992); Siamo tutti Matematici (2007); and Edinburgh Lectures on Geometry, Analysis and Physics (2010).

Atiyah and John Tate described the Clay Mathematics Institute Millennium Prize Problems in a lecture in Paris on 24 May 2000. Atiyah's lecture covered the Poincaré conjecture, the Hodge conjecture, quantum Yang-Mills theory and the Navier-Stokes equation. He explained the problems and placed them in their historical context. He also discussed the implications for various fields of mathematics and physics if solutions to these problems were found. A 60-minute video of the lecture is available entitled The millennium prize problems.

Six volumes of Atiyah's Collected Works have been published. These contain a commentary by Atiyah and in the Preface he comments on the practice of publishing 'collected works' during the lifetime of their author:-

It appears to be increasingly fashionable to publish 'collected works' long before the author's demise. There are several clear advantages to all parties: posterity is saved the trouble of undertaking the collection, while the author can add some personal touches by way of a commentary. There are also disadvantages: the commentary will be biased, and the author may feel that he is being pensioned off.

Another important aspect of Atiyah's contribution is the remarkable collection of doctoral students he supervised.

Atiyah has received many honours during his career, in addition to the Fields Medal referred to above, and although we cannot list them all we will give a fairly full account. He was elected a Fellow of the Royal Society of London in 1962 at the age of 32. He received the Royal Medal of the Society in 1968 and its Copley Medal in 1988. He gave the Royal Society's Bakerian Lecture on Global geometry in 1975 and was President of the Royal Society from 1990 to 1995.

Among the prizes that he has received are the Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Gunning Victoria Jubilee Prize from the Royal Society of Edinburgh in 1990, the Benjamin Franklin Medal in 1993, the Jawaharlal Nehru Memorial Medal in 1993, the Order of Andres Bello (1st Class) from the Republic of Venezuela in 1997, the Royal Medal from the Royal Society of Edinburgh in 2003, the Order of Merit (Gold) from the Lebanon in 2005, and the President's Medal from the Institute of Physics in 2008. In 2004 Atiyah and Isadore Singer were awarded the Neils Abel prize of £480 000 by the Norwegian Academy of Science and Letters:-

... 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.

They were presented with the prize by King Harald V of Norway at a ceremony in Oslo.

Atiyah was the American Mathematical Society Colloquium Lecturer in 1973. He was President of the London Mathematical Society in 1974-76 receiving its De Morgan Medal in 1980. Atiyah was knighted in 1983 and made a member of the Order of Merit in 1992.

He has been elected a foreign member of many national academies including: the American Academy of Arts and Sciences (1969), Royal Swedish Academy of Sciences (1972), German Academy of Scientist Leopoldina (1977), Académie des Sciences, Paris (1978), United States National Academy of Sciences (1978), Royal Irish Academy (1979), Third World Academy of Science (1983), Australian Academy of Sciences (1992), Ukrainian Academy of Sciences (1992), Indian National Science Academy (1993), Russian Academy of Sciences (1994), Georgian Academy of Sciences (1996), Academy of Physical, Mathematical and Natural Sciences of Venezuela (1997), American Philosophical Society (1998), Accademia Nazionale dei Lincei, Rome (1999), Royal Norwegian Society of Sciences and Letters (2001), Czechoslovakia Union of Mathematics (2001), Moscow Mathematical Society (2001), Spanish Royal Academy of Sciences (2002), Lebanese Academy of Sciences (2008), Norwegian Academy of Science and Letters (2009). He has been made an Honorary Fellow or Member of: Trinity College, University of Cambridge (1976), Pembroke College, University of Cambridge (1983), Royal Institution (1991), St Catherine's College, University of Oxford, (1991), Darwin College, University of Cambridge (1992), Royal Academy of Engineering (1993), New College, University of Oxford (1999), Faculty of Actuaries (1999), Academy of Medical Sciences (2000). Many universities have awarded him an honorary degree including: Bonn (1968), Warwick (1969), Durham (1979), St Andrews (1981), Trinity College Dublin (1983), Chicago (1983), Edinburgh (1984), Cambridge (1984), Essex (1985), London (1985), Sussex (1986), Ghent (1987), Reading (1990), Helsinki (1990), Leicester (1991), Rutgers (1992), Salamanca (1992), Montreal (1993), Waterloo (1993), Wales (1993), Queen's-Kingston (1994), Keele (1994), Birmingham (1994), Open University (1995), Manchester (1996), Chinese University of Hong Kong (1996), Brown University (1997), Oxford (1998), University of Wales Swansea (1998), Charles University Prague (1998), Heriot-Watt University (1999), University of Mexico (2001), American University of Beirut (2004), York (2005), Harvard University (2006), Scuola Normale Pisa (2007), Universitat Politècnica de Catalunya (2008).

Let us end this biography by recording the sad facts that Atiyah's eldest son John died on 24 June 2002 while on a walking holiday in the Pyrenees with his wife, while Jeremy, the youngest son of Atiyah's brother Patrick, died on 12 April 2006 while walking in Italy.

Article by: J J O'Connor and E F Robertson

November 2014

MacTutor History of Mathematics