atiyah

Michael Francis AtiyahBorn: 22 April 1929

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

My father's main dream was to go to Oxford. He wanted to converthimself into an Englishman. It didn't quite work out. When he cameback 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 himover a bit. He became an Arab nationalist to some extent. All his lifewas divided between wanting passionately to be English and yetsympathising with the Arab political position within the Britishempire.

Michael's mother Jean, although of Scottish descent, was the daughter of aminister of a church in Yorkshire. She lived in Oxford and had studied at theuniversity there. It was in Oxford that Edward and Jean met. They had fourchildren, three sons Michael (the eldest and subject of this biography), PatrickSelim (born 5 March 1931, who went on to become an English lawyer andacademic) and Joseph (known as Joe, the youngest of the four children whoafter a mathematics degree from Cambridge University, became a computerscientist working in computer software and telecommunications), and adaughter Selma (who studied English at an American University and lives inAmerica). Although he was born in London, Michael grew up in Khartoum.However, to avoid the summer heat there the family usually returned toEngland at that time. Michael's primary school education was at the Diocesanschool in Khartoum which he entered in 1934 at the age of ve. He completedhis 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 in1940, it came under the control of the Vichy government. After their trip toEngland, the Atiyah family returned to Lebanon via France in 1941 and Michaelreturned to a French school. However, just after this began, the British andFree French began ghting to gain control of the Lebanon. Michael was sent to

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Victoria College in Cairo. This was a boarding school modelled on the Englishboarding 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 tobeing two years younger than most others in my class. I survived byhelping bigger boys with their homework and so was protected bythem 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. ... Myparents always thought that I was cut out to be a mathematician froma very young age, all the way through. ... But there was a stage [atVictoria College in Cairo] when I got very interested in chemistry, andI thought that would be a great thing; after about a year of advancedchemistry I decided that it wasn't what I wanted to do and I went backto mathematics. I never seriously considered doing anything else.

He gave a somewhat fuller description of his decision between chemistry andmathematics in the interview. He said that it was inorganic chemistry that puthim o the subject [15]:-

It was how to make sulphuric acid and all that sort of stu. Lists offacts, just facts, you had to memorize a vast amount of material.Organic chemistry was more interesting, there was a bit of structureto it. But inorganic chemistry was just a mountain of facts in bookslike this. It's true that in mathematics you don't really need anenormous 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 offacts. There was one reason, I think. But I enjoyed thinking, I'm goodat it, and will continue with it.

After the war ended in 1945, Edward Atiyah returned to live permanently inEngland. Michael Atiyah attended Manchester Grammar School, one of the bestschools 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 ofhis age groups in Victoria College, Cairo. His two years at Manchester GrammarSchool 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 thecompetition was sti. We had an old-fashioned but inspiring teacherwho had graduated from Oxford in 1912 and from him I acquired alove of projective geometry, with its elegant synthetic proofs, whichhas never left me. I became, and remained, primarily a geometerthough that word has been reinterpreted in dierent ways at dierent


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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, ratherthan go straight to university, which was an option, he decided to do histwo-years National Service, which was compulsory at the time. He served as aclerical ocer and took the opportunity to read mathematics books and articles.He read Hardy and Wright's Number Theory at this time and also read articleson group theory. He was granted special permission to cut short the nal yearof his military service and spend it at Cambridge. There he played a lot of tennisand avidly studied mathematics on his own in the library. He matriculated atTrinity College in the autumn of 1949. Many of his fellow students had decidedto postpone their National Service, so Atiyah was one of the older of thestudents in his year. With his exceptional talent, his extra maturity, and thestudying he had done before starting his course, it is not at all surprising thathe came out ranked rst despite having many very talented fellow students.While still an undergraduate, he wrote his rst paper A note on the tangents ofa twisted cubic (1952).After graduating with his BA in 1952, Atiyah continued to undertake research atTrinity College, Cambridge obtaining his doctorate in 1955 with his thesis SomeApplications of Topological Methods in Algebraic Geometry. His thesis advisorwas 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 geometrywas on classical projective algebraic geometry of the old-fashionedtype, which I thoroughly enjoyed. I would have gone on working inthat area except that Hodge represented a more modern point of view- dierential geometry in relation to topology; I recognized that. Itwas a very important decision for me. I could have worked in moretraditional things, but I think that it was a wise choice, and byworking with him I got much more involved with modern ideas. Hegave me good advice and at one stage we collaborated together. Therewas some recent work in France at the time on sheaf theory. I gotinterested in it, he got interested in it, and we worked together andwrote a joint paper which was part of my thesis. That was verybenecial for me.

Atiyah published two joint papers with his thesis advisor William Hodge, Formesde seconde espce sur une varit algbrique (1954) and Integrals of thesecond kind on an algebraic variety (1955). He also published the single authorpapers Complex bre bundles and ruled surfaces (1955). He was made a fellowof 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, wasthe daughter of a dock worker at the Rosyth naval yard. She had studiedmathematics rst at the University of Edinburgh and then took the CambridgeTripos. She went on to obtain a doctorate, working under Mary Cartwright. Lily


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had met Michael Atiyah at Cambridge but, by the time they married, she was alecturer at Bedford College, London. Atiyah was awarded a CommonwealthFellow to study at the Institute for Advanced Study in Princeton during session1955-56. Lily had to decide whether to keep her job at Bedford College or go toPrinceton with her husband. She chose to go to Princeton with her husband andresigned her position at Bedford College. This was an important year for Atiyahwho met, among others, Jean-Pierre Serre, Friedrich Hirzebruch, KunihikoKodaira, Donald Spencer, Raoul Bott and Isadore Singer. Returning toCambridge, he was a college lecturer from 1957 and a Fellow of PembrokeCollege from 1958. He remained at Cambridge until 1961 when he moved to areadership at the University of Oxford where he became a Fellow of StCatherine's College.Atiyah was soon to ll the highly prestigious Savilian Chair of Geometry atOxford from 1963, holding this chair until 1969 when he was appointedprofessor of mathematics at the Institute for Advanced Study in Princeton. Afterthree years in Princeton, Atiyah returned to England, becoming a Royal SocietyResearch Professor at Oxford. He was also elected a Fellow of St Catherine'sCollege, Oxford. Oxford was to remain Atiyah's base until 1990 when he becameMaster of Trinity College, Cambridge and Director of the newly opened IsaacNewton Institute for Mathematical Sciences in Cambridge.Atiyah showed how the study of vector bundles on spaces could be regarded asthe study of cohomology theory, called K-theory. Grothendieck also contributedsubstantially to the development of K-theory. In [13] Atiyah's early mathematicalwork is described as follows:-

Michael Atiyah has contributed to a wide range of topics inmathematics centring around the interaction between geometry andanalysis. His rst major contribution (in collaboration with FHirzebruch) was the development of a new and powerful technique intopology (K-theory) which led to the solution of many outstandingdicult problems. Subsequently (in collaboration with I M Singer) heestablished an important theorem dealing with the number ofsolutions of elliptic dierential equations. This 'index theorem' hadantecedents in algebraic geometry and led to important new linksbetween dierential geometry, topology and analysis. Combined withconsiderations of symmetry it led (jointly with Raoul Bott) to a newand rened 'xed point theorem' with wide applicability.

For these early achievements Atiyah was awarded a Fields Medal at theInternational Congress at Moscow in 1966. An address concerning Atiyah'scontributions was given at the Congress by Henri Cartan, see [18]. The K-theoryand the index theorem are studied in Atiyah's book K-theory (1967, reprinted1989) and his joint work with G B Segal, The Index of Elliptic Operators I-V, inthe Annals of Mathematics, volumes 88 and 93 (1968, 1971). Atiyah alsodescribed his work on the index theorem in The index of elliptic operators given


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as an American Mathematical Society Colloquium Lecture in 1973.The ideas which led to Atiyah being awarded a Fields Medal were later seen tobe relevant to gauge theories of elementary particles. Again we quote [13]:-

The index theorem could be interpreted in terms of quantum theoryand has proved a useful tool for theoretical physicists. Beyond theselinear problems, gauge theories involved deep and interestingnonlinear dierential equations. In particular, the Yang-Millsequations have turned out to be particularly fruitful formathematicians. Atiyah initiated much of the early work in this eldand his student Simon Donaldson went on to make spectacular use ofthese ideas in 4-dimensional geometry. More recently Atiyah has beeninuential in stressing the role of topology in quantum eld theoryand in bringing the work of theoretical physicists, notably E Witten, tothe attention of the mathematical community.

The theories of superspace and supergravity and the string theory offundamental particles, which involves the theory of Riemann surfaces in noveland unexpected ways, were all areas of theoretical physics which developedusing the ideas which Atiyah was introducing.Atiyah has published a number of highly inuential books: K-theory (1967);(with I G Macdonald) Introduction to commutative algebra (1969); Vector eldson manifolds (1970); Elliptic operators and compact groups (1974); Geometryon Yang-Mills elds (1979); (with N J Hitchin) The geometry and dynamics ofmagnetic monopoles (1988); The geometry and physics of knots (1990); (Video)The mysteries of space (1992); Siamo tutti Matematici (2007); and EdinburghLectures on Geometry, Analysis and Physics (2010).Atiyah and John Tate described the Clay Mathematics Institute Millennium PrizeProblems in a lecture in Paris on 24 May 2000. Atiyah's lecture covered thePoincar conjecture, the Hodge conjecture, quantum Yang-Mills theory and theNavier-Stokes equation. He explained the problems and placed them in theirhistorical context. He also discussed the implications for various elds ofmathematics and physics if solutions to these problems were found. A 60-minutevideo of the lecture is available entitled The millennium prize problems.Six volumes of Atiyah's Collected Works have been published. These contain acommentary by Atiyah and in the Preface he comments on the practice ofpublishing '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 toall parties: posterity is saved the trouble of undertaking the collection,while the author can add some personal touches by way of acommentary. There are also disadvantages: the commentary will bebiased, and the author may feel that he is being pensioned o.


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Another important aspect of Atiyah's contribution is the remarkable collectionof doctoral students he supervised.Atiyah has received many honours during his career, in addition to the FieldsMedal referred to above, and although we cannot list them all we will give afairly full account. He was elected a Fellow of the Royal Society of London in1962 at the age of 32. He received the Royal Medal of the Society in 1968 andits Copley Medal in 1988. He gave the Royal Society's Bakerian Lecture onGlobal geometry in 1975 and was President of the Royal Society from 1990 to1995.Among the prizes that he has received are the Feltrinelli Prize from theAccademia Nazionale dei Lincei in 1981, the King Faisal International Prize forScience in 1987, the Gunning Victoria Jubilee Prize from the Royal Society ofEdinburgh in 1990, the Benjamin Franklin Medal in 1993, the Jawaharlal NehruMemorial Medal in 1993, the Order of Andres Bello (1st Class) from theRepublic of Venezuela in 1997, the Royal Medal from the Royal Society ofEdinburgh in 2003, the Order of Merit (Gold) from the Lebanon in 2005, and thePresident's Medal from the Institute of Physics in 2008. In 2004 Atiyah andIsadore Singer were awarded the Neils Abel prize of 480 000 by theNorwegian Academy of Science and Letters:-

... for their discovery and proof of the index theorem, bringingtogether topology, geometry and analysis, and their outstanding rolein building new bridges between mathematics and theoretical physics.

They were presented with the prize by King Harald V of Norway at a ceremonyin Oslo.Atiyah was the American Mathematical Society Colloquium Lecturer in 1973.He was President of the London Mathematical Society in 1974-76 receiving itsDe Morgan Medal in 1980. Atiyah was knighted in 1983 and made a member ofthe 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 ofSciences (1972), German Academy of Scientist Leopoldina (1977), Acadmiedes Sciences, Paris (1978), United States National Academy of Sciences (1978),Royal Irish Academy (1979), Third World Academy of Science (1983), AustralianAcademy of Sciences (1992), Ukrainian Academy of Sciences (1992), IndianNational Science Academy (1993), Russian Academy of Sciences (1994),Georgian Academy of Sciences (1996), Academy of Physical, Mathematical andNatural Sciences of Venezuela (1997), American Philosophical Society (1998),Accademia Nazionale dei Lincei, Rome (1999), Royal Norwegian Society ofSciences 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 Scienceand Letters (2009). He has been made an Honorary Fellow or Member of:


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Trinity College, University of Cambridge (1976), Pembroke College, Universityof 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). Manyuniversities 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 HongKong (1996), Brown University (1997), Oxford (1998), University of WalesSwansea (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), UniversitatPolitcnica de Catalunya (2008).Let us end this biography by recording the sad facts that Atiyah's eldest sonJohn died on 24 June 2002 while on a walking holiday in the Pyrenees with hiswife, while Jeremy, the youngest son of Atiyah's brother Patrick, died on 12April 2006 while walking in Italy.Article by: J J O'Connor and E F RobertsonNovember 2014MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Atiyah.html]


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