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Celebratio Mathematica

Michael F. Atiyah

Geometry

Sir Michael Atiyah: A brief biography

by Nigel J. Hitchin

The early years

Mi­chael Atiyah was born in Lon­don on 22 April 1929. His fath­er Ed­ward, from a Le­banese fam­ily, was work­ing in the polit­ic­al ser­vice in the Su­dan at the time, though he was also an au­thor and broad­caster. His moth­er Jean (née Levens), of Scot­tish des­cent, grew up in Ox­ford where she stud­ied art and where she also met her hus­band. As a child, Mi­chael lived in Khar­toum, at­tend­ing the Dio­ces­an School there, but re­turned to Eng­land for long peri­ods dur­ing the spring and sum­mer to avoid the heat.

His sec­ond­ary edu­ca­tion began in 1941, trav­el­ling to Le­ban­on via France just be­fore its col­lapse. Very soon af­ter­wards he left the French school in Le­ban­on and en­rolled in Vic­tor­ia Col­lege, an Eng­lish-style board­ing school in Cairo where his fath­er had been. In this cos­mo­pol­it­an com­munity, he fared well in math­em­at­ics, stay­ing a couple of years ahead of his age group, but be­came per­haps more in­ter­ested in chem­istry as he pro­gressed through the school. Ul­ti­mately, he was van­quished by the feats of memory which chem­istry de­man­ded, and re­ver­ted to the math­em­at­ics which his par­ents al­ways felt was his real tal­ent.

Hav­ing at the age of 16 de­cided on math­em­at­ics as a sub­ject to take fur­ther, the Atiyahs’ thoughts nat­ur­ally turned to­wards Cam­bridge; to pre­pare for this, their son was sent to Manchester Gram­mar School in 1945. He had already taken his A-level ex­am­in­a­tions, and was quite con­fid­ent and not pre­pared for the hot­house at­mo­sphere of math­em­at­ics at the school, which showed that the sub­ject had more depths than he had been aware of. However, un­der the tu­tel­age of a ded­ic­ated math­em­at­ics mas­ter, who only used 19th cen­tury books and pro­duced his own notes for everything more re­cent, he won a schol­ar­ship to Trin­ity Col­lege, ac­cep­ted in Cam­bridge along with a siz­able co­hort of math­em­aticians from the same school.

At that point, however, the out­side world in­ter­vened in the form of mil­it­ary ser­vice. This could have been post­poned, but Mi­chael de­lib­er­ately chose to do it then and get it out of the way; so, after leav­ing school at East­er 1947, he joined the Roy­al Elec­tric­al and Mech­an­ic­al En­gin­eers. As a Cold War­ri­or he was con­fined to cler­ic­al du­ties at the re­gi­ment­al Headquar­ters, but it was an oc­cu­pa­tion that gave him spare time, which he spent read­ing math­em­at­ics books like Hardy and Wright, and do­ing the prob­lems sent on from Cam­bridge by his school con­tem­por­ar­ies who had gone there dir­ectly. For­tu­nately, a well-timed let­ter from his tu­tor en­abled him to leave the army early in 1949 and fi­nally reach his goal of Cam­bridge for the long va­ca­tion term.

Cambridge

After eight­een months of Na­tion­al Ser­vice, Cam­bridge was a lib­er­a­tion. His friends who went dir­ectly from school were already get­ting a little ex­hausted, where­as he felt full of en­thu­si­asm, and im­mersed him­self in read­ing in Trin­ity Col­lege lib­rary. When term ac­tu­ally began in Oc­to­ber, he found him­self be­ing su­per­vised in ana­lys­is by the idio­syn­crat­ic A. S. Be­sicov­itch and in geo­metry by the more staid J. A. Todd. It was the geo­metry — clas­sic­al geo­metry — that he en­joyed most and, in­deed, was the source of his first math­em­at­ic­al pa­per, pub­lished in the Pro­ceed­ings of the Cam­bridge Philo­soph­ic­al So­ci­ety in his second year as an un­der­gradu­ate. Mi­chael’s un­der­gradu­ate years were spent in go­ing to as many lec­tures as he could phys­ic­ally man­age and in dis­cuss­ing and ar­guing with his math­em­at­ic­al friends, who in­cluded J. F. Adams, I. G. Mac­don­ald and J. Polk­ing­horne, who be­came a well-known phys­i­cist, and also J. P. H. Mack­ay, the fu­ture Lord High Chan­cel­lor of Great Bri­tain. Mi­chael and John Polk­ing­horne pre­pared them­selves in­tens­ively for the ex­am­in­a­tions, go­ing through old pa­pers in the lib­rary on Sat­urday morn­ings, and ended up tak­ing the top hon­ours in the end-of-year ex­am­in­a­tion.

By this time, however, he was already think­ing about go­ing fur­ther in math­em­at­ics. Hav­ing writ­ten a pa­per, at­ten­ded many Part III courses and per­suaded W. V. D. Hodge to lec­ture on top­ics he wanted to hear, it was clear that math­em­at­ic­al re­search was his goal and, for­sak­ing Todd, he be­came the re­search stu­dent of Hodge. Un­usu­ally, Hodge had four re­search stu­dents that year (1952). Be­sides Mi­chael, these in­cluded R. Pen­rose, who had ar­rived from Uni­versity Col­lege Lon­don and later moved to Todd as a su­per­visor, and M. Hoskin, now a his­tor­i­an of sci­ence. The fourth stu­dent left to be­come a school­teach­er.

Hodge put his new stu­dent to work on char­ac­ter­ist­ic classes, read­ing the pa­pers of S.-S. Chern, A. Weil and C. B. Al­lendo­er­fer. This sort of glob­al dif­fer­en­tial geo­metry was nov­el at that time, and al­though courses in to­po­logy by P. Hilton and S. Wylie helped it was not, des­pite Hodge’s book on har­mon­ic in­teg­rals a dec­ade earli­er, a sub­ject easy to find out about in Cam­bridge. It was through his weekly read­ing of the Comptes Ren­dus that Mi­chael began to ab­sorb the new res­ults in sheaf the­ory that were be­ing pro­duced by the French school at that time, and in­flu­enced his re­search at this stage. A sab­bat­ic­al vis­it to Cam­bridge by N. S. Haw­ley fo­cused his at­ten­tion on prob­lems con­cern­ing holo­morph­ic vec­tor bundles, and in be­gin­ning to work in this area he dis­covered a fun­da­ment­al mis­con­cep­tion in one of Haw­ley’s pa­pers. Resolv­ing this gave rise to the re­search pa­per on ruled sur­faces and holo­morph­ic pro­ject­ive bundles, which won Mi­chael the Smith’s prize in 1954. Shortly af­ter­wards, he col­lab­or­ated with Hodge us­ing sheaf the­ory to study in­teg­rals of the second kind. At the end of that year he won a Fel­low­ship to Trin­ity Col­lege.

The award of the prize gave him the boost to stay in math­em­at­ics and not leave the aca­dem­ic en­vir­on­ment, as many of his con­tem­por­ar­ies did, some to na­tion­al se­cur­ity work at GCHQ, oth­ers like Ian Mac­don­ald to the Civil Ser­vice. The In­ter­na­tion­al Con­gress at Am­s­ter­dam in 1954, with K. Kodaira and J-P. Serre re­ceiv­ing Fields Medals, fired his ima­gin­a­tion even more as he saw the vista of pos­sib­il­it­ies in just the sort of math­em­at­ics in which he was work­ing.

Princeton

The year 1955 was an un­event­ful, un­pro­duct­ive one. Hav­ing seen the math­em­at­ic­al stars at the ICM, Cam­bridge was far too quiet for Mi­chael, and fol­low­ing Hodge’s vis­it to Prin­ceton the pre­vi­ous year, where he talked about their joint work, he jumped at D. C. Spen­cer’s in­vit­a­tion to go there too. On re­ceiv­ing a Com­mon­wealth Fel­low­ship he had the op­por­tun­ity to go to the In­sti­tute for Ad­vanced Study rather than the Uni­versity, which he read­ily took with Spen­cer’s en­cour­age­ment. Be­fore do­ing so, however, he mar­ried Lily Brown, who had been study­ing math­em­at­ics a year ahead of him in Cam­bridge. She gave up her job at Bed­ford Col­lege Lon­don to join him on this, his first, vis­it to the United States.

If Cam­bridge was an eye-open­er after Na­tion­al Ser­vice, Prin­ceton was even more so after Cam­bridge. The In­sti­tute un­der R. Op­pen­heimer had a big con­cen­tra­tion of vis­it­ors, with sev­er­al over­lap­ping gen­er­a­tions. R. Bott and Serre were there, as was I. M. Sing­er. At the Uni­versity, F. Hirzebruch was a pro­fess­or. Bott lec­tured on Morse the­ory, Kodaira on sheaf the­ory, cur­rents and al­geb­ra­ic geo­metry, Hirzebruch on char­ac­ter­ist­ic classes. Three days a week a car­load of math­em­aticians, in­clud­ing Mi­chael, Bott and Sing­er, eagerly traveled to the Uni­versity to hear these sem­inars. These were, of course, all top­ics and con­tacts which would be in­flu­en­tial in Mi­chael’s sub­sequent work. There was an­oth­er dif­fer­ence from the gen­tle­manly at­mo­sphere at Cam­bridge: sem­inars would abound with com­ments and cri­ti­cisms from the audi­ence, prob­ing for more pre­ci­sion and the cor­rect style.

The in­flu­ence of Serre’s Prin­ceton sem­in­ar on vec­tor bundles mani­fes­ted it­self in Mi­chael’s pa­pers pub­lished at the time on the sheaf-the­or­et­ic­al de­scrip­tion of char­ac­ter­ist­ic classes and on holo­morph­ic vec­tor bundles on el­lipt­ic curves. These were top­ics he spoke on as he traveled ex­tens­ively around Amer­ica, part of the con­di­tions of a Com­mon­wealth Fel­low­ship. He met Chern in Chica­go and par­ti­cip­ated in S. Lef­schetz’s con­fer­ence in Mex­ico.

Back to Cambridge

At the be­gin­ning of 1957, Mi­chael re­turned to Cam­bridge as a Lec­turer, and then, the fol­low­ing year, was also ap­poin­ted as a Tu­tori­al Fel­low at Pem­broke Col­lege. There, he began to or­gan­ize col­loquia, with pro­fess­ors find­ing out for the first time what each oth­er was do­ing. At the same time, he was re­count­ing the ideas he had ac­quired in the United States to Hodge and oth­ers in Cam­bridge, such as E. C. Zee­man who was act­ively lead­ing the to­po­logy group there. Hodge him­self was be­com­ing in­creas­ingly oc­cu­pied with oth­er activ­it­ies — the Mas­ter­ship of Pem­broke and Sec­ret­ary­ship of the Roy­al So­ci­ety — and he handed over to Mi­chael many of the De­part­ment­al activ­it­ies and some of his gradu­ate stu­dents too, such as R. Schwar­zen­ber­ger and I. Porteous.

At this time, in 1957, the Bonn Arbeit­sta­gun­gen began and Mi­chael be­came a reg­u­lar at­tendee, for­ging closer links with Hirzebruch. It was a very act­ive peri­od for to­po­logy, with J. Mil­nor’s work on man­i­fold clas­si­fic­a­tion, Thom’s on cobor­d­ism the­ory and Hirzebruch show­ing the links with al­geb­ra­ic geo­metry. Al­though not by train­ing a to­po­lo­gist, Mi­chael began to in­volve him­self more and more in the sub­ject. From a com­bin­a­tion of sources — Hirzebruch’s in­teg­ral­ity the­or­ems, A. Grothen­dieck’s Bonn lec­tures on the gen­er­al Riemann–Roch the­or­em, Bott’s peri­od­icity the­or­em and prob­lems posed by I. M. James, then a col­league in Cam­bridge — the rudi­ments of what be­came \( K \)-the­ory emerged. It be­came rap­idly clear that the \( K \)-groups were the cor­rect form­al ap­par­at­us to solve some very dif­fi­cult prob­lems in to­po­logy.

Mi­chael’s work at­trac­ted the at­ten­tion of J. H. C. White­head in Ox­ford, and he went there to give talks, but in 1960 White­head died sud­denly. The Waynf­lete Chair which he had oc­cu­pied was va­cant and Mi­chael, at the age of 31, ap­plied for it. In the event, it went to the more seni­or G. Hig­man, a group the­or­ist, but a Read­er­ship was soon after offered to Mi­chael, and since it provided a re­lief from col­lege teach­ing and more time to do re­search he took it up in 1961.

Oxford

At the time, Ox­ford was more renowned for its strength in philo­sophy than in math­em­at­ics, but Ioan James had moved there as a Read­er, so that to­po­logy was cer­tainly rep­res­en­ted. Sub­sequently, Cam­bridge was to lose Zee­man and oth­ers to found the new Math­em­at­ics De­part­ment in War­wick, and Hodge, still burdened by ad­min­is­tra­tion, was happy to send re­search stu­dents over to Ox­ford to be su­per­vised by Mi­chael, so that the math­em­at­ic­al en­vir­on­ment be­came much more act­ive in the area. The Cam­bridge stu­dents ex­por­ted to Mi­chael in­cluded G. B. Segal and P. E. News­tead; oth­er stu­dents at this time were K. D. El­worthy and B. J. Sander­son. Re­sources in Ox­ford for in­vit­ing speak­ers were still slender, so that Mi­chael would spend every sab­bat­ic­al op­por­tun­ity vis­it­ing Har­vard or Prin­ceton, or per­suad­ing Bott or Sing­er to spend their sab­bat­ic­als in Ox­ford. In 1963, the pre­ma­ture death of E. C. Titch­marsh left the Sa­vil­ian Pro­fess­or­ship of Geo­metry un­filled, and Mi­chael was duly ap­poin­ted to it.

In the spring of 1962, Sing­er spent part of his sab­bat­ic­al in Ox­ford, and there began the col­lab­or­a­tion which cul­min­ated in the in­dex the­or­em in its many forms. The prob­lem ori­gin­ated with the at­tempt to de­scribe the in­teg­ral­ity the­or­ems for char­ac­ter­ist­ic num­bers in terms of di­men­sions of vec­tor spaces. \( K \)-the­ory was a suit­able tool to de­scribe the in­teg­ral­ity, and Grothen­dieck’s or Hirzebruch’s Riemann–Roch the­or­em gave an an­swer for the Todd poly­no­mi­al. Dur­ing that vis­it, they re­dis­covered the Dir­ac op­er­at­or, which ex­plained the in­teg­ral­ity of the \( \hat{A} \)-poly­no­mi­al for a spin man­i­fold. However, it was S. Smale’s vis­it to Ox­ford on his way back from Mo­scow which put be­fore them a far more gen­er­al ques­tion — that of find­ing a for­mula for the in­dex of a gen­er­al el­lipt­ic op­er­at­or, a prob­lem of con­sid­er­able in­terest to the Rus­si­an school and on which Gel­fand and his cowork­ers had made some pro­gress. This wider con­text stim­u­lated the 18-month pur­suit of a proof. In Mi­chael’s lec­ture at the ICM in Stock­holm in 1962, the prob­lem and con­jec­tured for­mu­las ap­peared, but an an­nounce­ment of the proof was ad­ded as a foot­note to the pub­lished talk.

The peri­od that fol­lowed this first suc­cess­ful at­tack on the In­dex The­or­em led on to a large num­ber of pa­pers ex­plor­ing gen­er­al­iz­a­tions and rami­fic­a­tions of the ori­gin­al idea. Many of these in­volved group ac­tions, the ori­gins of which lay in walks along dusty paths at Woods Hole near Cape Cod with Raoul Bott. It was a con­fer­ence in al­geb­ra­ic geo­metry and num­ber the­ory, and what star­ted out as a con­jec­ture of G. Shimura on auto­morph­isms of al­geb­ra­ic curves blos­somed in­to a gen­er­al the­ory of Lef­schetz fixed-point for­mu­las for el­lipt­ic com­plexes, which in­cluded H. Weyl’s fam­ous char­ac­ter for­mula as a spe­cial case. As well as equivari­ance, the lan­guage of \( K \)-the­ory came to be seen as the ap­pro­pri­ate frame­work for in­dex prob­lems, and in 1968 the first of the five An­nals pa­pers with Sing­er gave new proofs of the in­dex the­or­em in many dif­fer­ent con­texts, with many dif­fer­ent ap­plic­a­tions. By this time, Mi­chael had been elec­ted a Fel­low of the Roy­al So­ci­ety in 1962 and won a Fields Medal at the 1966 Mo­scow ICM for his work on \( K \)-the­ory and the in­dex the­or­em. He con­tin­ued to work on in­dices for a total of over 20 years, as more and more ques­tions needed to be re­solved.

The year fol­low­ing the Fields Medal, Mi­chael paid his third vis­it to the In­sti­tute in Prin­ceton, and was ap­proached to see if he would go there as a Per­man­ent Mem­ber. L. Hör­mander, an­oth­er Fields Medal­ist, was just leav­ing, and after some de­lib­er­a­tion back in Ox­ford he de­cided to move in 1969.

Princeton again

The In­sti­tute held fond memor­ies and, now, new op­por­tun­it­ies. There was money to in­vite people to come and col­lab­or­ate and no ob­lig­a­tion to teach courses — a pure re­search po­s­i­tion for the first time. Moreover, since the Prin­ceton term ended in April, the Atiyahs could re­turn to Ox­ford where they kept their house, and Mi­chael would par­ti­cip­ate fully in the life of the Math­em­at­ic­al In­sti­tute in Trin­ity Term (in­deed, as I was a gradu­ate stu­dent there my­self then, I be­nefited greatly from this ar­range­ment).

In Prin­ceton, the work on the in­dex the­or­em con­tin­ued, for fam­il­ies and the mod-2 situ­ation, but Mi­chael soon be­came aware of a new ap­proach us­ing the heat ker­nel. Build­ing on the work of Sing­er and H. McK­ean, the young V. K. Pat­odi had found some mi­ra­cu­lous can­cela­tions in the asymp­tot­ic ex­pan­sions of the dif­fer­ence of two heat ker­nels, and pro­duced a proof of the Gauss–Bon­net the­or­em — the most prim­it­ive ex­ample of the in­dex the­or­em. In 1971, Bott and Pat­odi were in­vited to the In­sti­tute and work began in earn­est on this new, more ana­lyt­ic­al ap­proach to the in­dex the­or­em; \( K \)-the­ory for the mo­ment lay in the back­ground. With P. Gilkey’s use of form­al al­gebra to cir­cum­vent Pat­odi’s clev­er dir­ect ma­nip­u­la­tions, a new in­vari­ant-the­ory proof of the In­dex The­or­em emerged, one which had an es­sen­tially loc­al dif­fer­en­tial-geo­met­ric char­ac­ter. In col­lab­or­a­tion with Sing­er, it was ap­plied to study in­dices for op­er­at­ors on man­i­folds with bound­ary, in par­tic­u­lar the sig­na­ture and Dir­ac op­er­at­ors, which led to the non­loc­al bound­ary con­tri­bu­tion called the \( \eta \)-in­vari­ant. One of the guid­ing mo­tiv­a­tions in this work was the earli­er num­ber-the­or­et­ic cor­rec­tion to the sig­na­ture for­mula for sin­gu­lar­it­ies in Hirzebruch’s ana­lys­is of Hil­bert mod­u­lar sur­faces.

While resolv­ing the prob­lems in­volved in un­der­stand­ing these new glob­al bound­ary value prob­lems, Mi­chael made an­oth­er de­cision. Des­pite the heady at­mo­sphere of re­search in Prin­ceton, like oth­ers be­fore and after him, he de­cided to leave the In­sti­tute after three years. He had been offered a Roy­al So­ci­ety Re­search Pro­fess­or­ship to re­turn to the UK. It could have been taken up any­where, but he chose Ox­ford.

Oxford again

Back in Ox­ford do­ing re­search full-time, and without teach­ing and ad­min­is­trat­ive du­ties this time, Mi­chael took on gradu­ate stu­dents. In Prin­ceton, apart from G. Lusztig, this did not hap­pen. Moreover, the lack of teach­ing du­ties in Ox­ford meant that he was freer to travel and col­lab­or­ate. His re­search began by con­tinu­ing with to­po­lo­gic­al and geo­met­ric­al ap­plic­a­tions of the new In­dex The­or­em, and then he and Sing­er found an ap­plic­a­tion for a dif­fer­ent form of the In­dex The­or­em re­lated to von Neu­mann al­geb­ras. This \( L^2 \) in­dex the­or­em was used for study­ing in­fin­ite cov­er­ings of man­i­folds and unit­ary rep­res­ent­a­tions of Lie groups.

There was an­oth­er in­flu­ence in his math­em­at­ic­al work at Ox­ford dur­ing this peri­od, however. At the same time that Mi­chael re­turned to Ox­ford, his former col­league as a gradu­ate stu­dent in Cam­bridge, Ro­ger Pen­rose, came to take up the Rouse Ball Chair in Math­em­at­ics. Ro­ger and his stu­dents were work­ing out the con­sequences of his twis­tor-the­or­et­ic ap­proach to the equa­tions of math­em­at­ic­al phys­ics, and while Mi­chael may have been less at home with the phys­ic­al mo­tiv­a­tion, they were nev­er­the­less on the same wavelength when it came to the geo­metry of the Klein quad­ric, which they had both learned from Todd’s book. The first fruit of the in­ter­ac­tion was Mi­chael’s re­cog­ni­tion that the sheaf the­ory which he had first learnt as a brand-new sub­ject while they were re­search stu­dents to­geth­er was the ap­pro­pri­ate lan­guage in which to de­scribe the con­tour-in­teg­ral solu­tion of zero rest-mass field equa­tions which Pen­rose was work­ing on then. This provided a ready-made reser­voir of soph­ist­ic­ated tech­niques to ap­ply to these lin­ear equa­tions. Some­what later, sim­il­ar ideas would have even more re­mark­able re­per­cus­sions.

In early 1977, Sing­er paid an ex­ten­ded vis­it to Ox­ford and spoke in a series of sem­inars about what he had learned of the phys­i­cists’ work on “in­stan­tons” — self-dual solu­tions of the Yang–Mills equa­tions on \( \mathbb S^4 \). His audi­ence was well equipped to un­der­stand con­cepts like prin­cip­al bundles and curvature, but less about the equa­tions and the role of gauge trans­form­a­tions. Two de­vel­op­ments oc­curred at this time. The first was the work of R. S. Ward, a stu­dent of Pen­rose who had showed us­ing twis­tor the­ory that a com­plex solu­tion of the self-du­al­ity equa­tions arose from the data of a com­plex vec­tor bundle on the pro­ject­ive space \( \mathbb CP^3 \). By chance, Mi­chael had at­ten­ded the math­em­at­ic­al-phys­ics sem­in­ar in which Ward had spoken, and rap­idly saw how the Eu­c­lidean ver­sion of the cor­res­pond­ence worked. The second de­vel­op­ment was that the In­dex The­or­em could be put to use to ac­tu­ally cal­cu­late the di­men­sions of the mod­uli space of in­stan­tons. The \( (8k-3) \)-di­men­sion­al­ity of the mod­uli space co­in­cided with a di­men­sion that W. Barth had cal­cu­lated for the mod­uli of cer­tain stable holo­morph­ic bundles on \( \mathbb CP^3 \). This it­self was based on a very con­crete con­struc­tion which G. Hor­rocks had spoken about in Ober­wolfach the pre­vi­ous sum­mer. Ty­ing all the threads to­geth­er, with some dif­fer­en­tial-geo­met­ric van­ish­ing the­or­ems, gave the fi­nal out­come in Novem­ber 1977: a con­struc­tion of all in­stan­tons us­ing just fi­nite-di­men­sion­al matrices. At the same time that Mi­chael and I had found this, we heard from Yu. Man­in that he and V. G. Drin­feld, who had been fol­low­ing the Ox­ford de­vel­op­ments at a dis­tance in Mo­scow, had in­de­pend­ently de­rived the same res­ult, which was sub­sequently known as the ADHM con­struc­tion of in­stan­tons.

This in­ter­ac­tion with phys­ics was very in­flu­en­tial on the sub­sequent dir­ec­tion for Mi­chael’s math­em­at­ics. As it turned out, at the same time that he had been work­ing on the heat-ker­nel ap­proach to the in­dex the­or­em, phys­i­cists were in­de­pend­ently ar­riv­ing at a form of the same the­or­em through their study of an­om­alies. In fact, dur­ing an earli­er vis­it to MIT where Mi­chael spoke on the Dir­ac op­er­at­or, he had be­gun to see the role that the in­dex the­or­em played for phys­i­cists. It was dur­ing that vis­it that he first met a young postdoc­tor­al fel­low by the name of Ed­ward Wit­ten. By the time of the work on in­stan­tons, it was clear to Mi­chael that much more could be gained from both sides by closer ties.

The sub­sequent re­search in which Mi­chael en­gaged for a dec­ade or more was heav­ily in­flu­enced by ideas from phys­ics. This in­cluded the ap­plic­a­tion of the Yang–Mills equa­tions to the mod­uli of stable bundles on Riemann sur­faces, on which he worked with Bott. The sym­plect­ic as­pects of this provided a dif­fer­ent view­point on mod­uli spaces in gen­er­al, an ap­proach taken much fur­ther by his stu­dent, F. C. Kir­wan. It also provided a mod­el for ap­plic­a­tions of Yang–Mills the­ory in areas far re­moved from the ori­gin­al prob­lem of in­stan­tons in \( \mathbb S^4 \), ideas taken up in the most spec­tac­u­lar way by his stu­dent S. K. Don­ald­son. The three-di­men­sion­al ana­logue of in­stan­tons, the “mag­net­ic mono­poles”, also oc­cu­pied his at­ten­tion.

Per­haps more im­port­ant than these con­crete “clas­sic­al” in­ter­ac­tions of math­em­at­ics and phys­ics was Mi­chael’s role as both par­ti­cipant and fa­cil­it­at­or for the ex­change of ideas between the two sides, in par­tic­u­lar in dis­cus­sions with Wit­ten. These in­cluded press­ing for a quantum-field-the­ory in­ter­pret­a­tion of both the Don­ald­son in­vari­ants for 4-man­i­folds and the Jones poly­no­mi­als for knots. This was a cru­cial activ­ity in break­ing down bar­ri­ers of in­tu­ition on both sides. By the late 1980s this was a rap­idly de­vel­op­ing area, so much so that it was said that the award of the 1990 Fields Medals to S. Mori, Drin­feld, V. F. R. Jones and Wit­ten was made to “one clas­sic­al math­em­atician and three quantum ones”. In 1990, however, Mi­chael’s ca­reer took an­oth­er turn.

Cambridge and Edinburgh

The head­ship of Mi­chael’s old col­lege in Cam­bridge had be­come va­cant, and in 1990 Mi­chael be­came the Mas­ter of Trin­ity, 40 years after he had entered there as an un­der­gradu­ate. At the same time, he was per­suaded to be­come Pres­id­ent of the Roy­al So­ci­ety and, con­cur­rently, the Dir­ect­or of the newly formed Isaac New­ton In­sti­tute for Math­em­at­ic­al Sci­ences in Cam­bridge, whose cause he had pro­moted strongly. Clearly, with such a volume of ad­min­is­trat­ive activ­it­ies, math­em­at­ic­al re­search had to take a back seat.

Mi­chael had already had some ex­per­i­ence of pub­lic ser­vice and learned so­ci­et­ies at vari­ous stages of his ca­reer, hav­ing been in­volved in the Cock­croft Com­mit­tee for the re­form of math­em­at­ics teach­ing in schools, been Pres­id­ent of both of the Lon­don Math­em­at­ic­al So­ci­ety and the Math­em­at­ic­al As­so­ci­ation, a Vice Pres­id­ent of the Roy­al So­ci­ety, and act­ive in the In­ter­na­tion­al Math­em­at­ic­al Uni­on and in the form­a­tion of the European Math­em­at­ic­al So­ci­ety. Per­haps these activ­it­ies, as well as his math­em­at­ic­al prowess, con­trib­uted to his knight­hood, awar­ded in 1983. In any case, his activ­it­ies after 1990 won him the Or­der of Mer­it in 1992. The Pres­id­ency of the Roy­al So­ci­ety put him in a po­s­i­tion of hav­ing an in­flu­ence on broad­er sci­ence-policy is­sues, in­clud­ing eth­ic­al ques­tions, and his 1995 Farewell Ad­dress to the Roy­al So­ci­ety provided a much-pub­li­cized op­por­tun­ity to cri­ti­cise the nuc­le­ar-weapons policy of the UK and to de­plore the dis­tor­tions in re­search which have en­sued from the al­loc­a­tion of huge re­sources to it. He has since car­ried on this theme through his Pres­id­ency of the Pug­wash Move­ment.

In 1997 Mi­chael re­tired from the Mas­ter­ship of Trin­ity, and the Atiyahs went to live in Ed­in­burgh. They had for some­time reg­u­larly spent va­ca­tion peri­ods in their house in the Cairngorms. Mi­chael is cur­rently act­ive in a vari­ety of in­ter­na­tion­al com­mit­tees, and as an Hon­or­ary Pro­fess­or is also an act­ive pres­ence in the Math­em­at­ics De­part­ment in Ed­in­burgh.