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Celebratio Mathematica

Michael F. Atiyah

Sir Michael Atityah OM
22 April 1929–11 January 2019

by Nigel Hitchin

Mi­chael Atiyah (Elec­ted FRS 1962; Pres­id­ent 1990–1995) was the dom­in­ant fig­ure in UK math­em­at­ics in the lat­ter half of the twen­ti­eth cen­tury. He made out­stand­ing con­tri­bu­tions to geo­metry, to­po­logy, glob­al ana­lys­is and, par­tic­u­larly over the last 30 years, to the­or­et­ic­al phys­ics. Not only was he held in high es­teem at a world­wide level, win­ning a Fields Medal in 1966, the Abel Prize in 2004 and in­nu­mer­able oth­er in­ter­na­tion­al awards, but his ir­re­press­ible en­ergy and broad in­terests led him to take on many na­tion­al roles too, in­clud­ing the pres­id­ency of the Roy­al So­ci­ety, the mas­ter­ship of Trin­ity Col­lege, Cam­bridge, and the found­ing dir­ect­or­ship of the Isaac New­ton In­sti­tute for Math­em­at­ic­al Sci­ences. His most not­able math­em­at­ic­al achieve­ment, with Is­ad­ore Sing­er, is the in­dex the­or­em, which oc­cu­pied him for over 20 years, gen­er­at­ing res­ults in to­po­logy, geo­metry and num­ber the­ory us­ing the ana­lys­is of el­lipt­ic dif­fer­en­tial op­er­at­ors. Then, in mid life, he learned that the­or­et­ic­al phys­i­cists also needed the the­or­em and this opened the door to an in­ter­ac­tion between the two dis­cip­lines that he pur­sued en­er­get­ic­ally un­til the end of his life. It led him not only to math­em­at­ic­al res­ults on the Yang–Mills equa­tions that the phys­i­cists were seek­ing, but also to en­cour­aging the im­port­a­tion of con­cepts from quantum field the­ory in­to pure math­em­at­ics.

Early life

Figure 1. Sudanese childhood, aged two.
Mi­chael Atiyah was born on 22 April 1929 in Hamp­stead, Lon­don, the eld­est of four chil­dren. His fath­er, Ed­ward Atiyah, came from a fam­ily of West­ern-ori­ented Le­banese Chris­ti­ans. Pre­vi­ous gen­er­a­tions had worked for the Anglo-Egyp­tian ad­min­istered Su­dan gov­ern­ment, and Ed­ward was no ex­cep­tion. Thus Khar­toum was where Mi­chael spent many child­hood years (Fig­ure 1).

Ed­ward’s ca­reer was closely as­so­ci­ated with Brit­ish cul­ture, and, after at­tend­ing Vic­tor­ia Col­lege in Al­ex­an­dria, he gradu­ated with a de­gree in his­tory from Brasen­ose Col­lege, Ox­ford. His sub­sequent life was led as a civil ser­vant, au­thor (in­clud­ing de­tect­ive stor­ies), broad­caster and pro­moter of the Ar­ab cause; he be­came sec­ret­ary of the Lon­don of­fice of the Ar­ab League. It was as a stu­dent in Ox­ford that he met Jean Levens, Mi­chael’s in­de­pend­ent-minded Scot­tish moth­er. Ed­ward had en­countered her broth­er, Robert, at the Ox­ford Uni­on and gradu­ally be­came a fre­quent vis­it­or at the fam­ily home. In 1928 they mar­ried, spent their hon­ey­moon in Florence and named their first child after Michelan­gelo.

Mi­chael’s first school was the Clergy House in Khar­toum, con­sist­ing of a small group of 25 pu­pils of all ages. When the time came to move to sec­ond­ary school, the start of the Second World War nar­rowed his op­tions. He spent a term in a French school in Le­ban­on and then moved briefly to an Itali­an school in Khar­toum, but sur­vived only four days of be­ing taught in Ar­ab­ic (al­though this was his first spoken lan­guage) and then re­turned to his former school. Fi­nally in 1941 his par­ents de­cided, with the help of a schol­ar­ship, to send him to his fath­er’s old school, Vic­tor­ia Col­lege, which by that time had moved to Cairo as the ori­gin­al build­ings had been con­ver­ted in­to a mil­it­ary hos­pit­al. Here he was placed in classes two years ahead of his age (he was do­ing cal­cu­lus at 13) and was sur­roun­ded by a stim­u­lat­ing group of in­ter­na­tion­al pu­pils and Eng­lish teach­ers. His par­ents already re­cog­nized that math­em­at­ics was one of his strengths.

From an early age, the heat of the Su­danese sum­mer meant that the fam­ily moved away for four months, vis­it­ing re­l­at­ives in Eng­land, the Le­ban­on, Palestine and Egypt. Travel was part of fam­ily life, and even re­turn­ing to school after the sum­mer va­ca­tion meant a four-day jour­ney by train and Nile boat with his young­er broth­er, Patrick. Life in Egypt was not without its dangers and aged 13 he was struck down with men­ingit­is and only cured after a hec­tic rush around Cairo to find some peni­cil­lin.

Figure 2. Victoria College, Alexandria, 1945.

In Au­gust 1944 Vic­tor­ia Col­lege moved back to Al­ex­an­dria (Fig­ure 2) and from there, Mi­chael’s par­ents de­cided that his math­em­at­ic­al abil­ity re­quired him to aim for Cam­bridge. Thus, in 1945, at the age of 16, he was sent to Manchester Gram­mar School to pre­pare ap­pro­pri­ately, liv­ing in­de­pend­ently in a lodging house. His class­mates, know­ing only that he was from the Middle East, wel­comed him with a camel drawn on the black­board.

Hav­ing been two years ahead in Vic­tor­ia Col­lege, Mi­chael had already taken his High­er School Cer­ti­fic­ate ex­am­in­a­tions, but he was now ex­posed to a new level of math­em­at­ic­al study that re­quired a great deal of hard work and in a com­pet­it­ive en­vir­on­ment. He was taught by the chief math­em­at­ics mas­ter F. L. Hey­wood (an Ox­ford first in math­em­at­ics), who dis­liked text­books and pro­duced de­tailed hand-writ­ten notes de­scrib­ing the cor­rect ap­proach as he saw it. The high pres­sure at­mo­sphere yiel­ded res­ults, with three schol­ar­ships to Trin­ity Col­lege, Cam­bridge, that year and two to St. John’s. At the time, Mi­chael had little idea of which col­lege he should ap­ply to, but the school had a def­in­ite peck­ing or­der and Trin­ity was chosen for the top math­em­aticians. And so began his long as­so­ci­ation with the col­lege.

Be­fore go­ing to Cam­bridge, Mi­chael de­cided to spend two years do­ing his Na­tion­al Ser­vice. This could have been avoided, or post­poned as most of his con­tem­por­ar­ies did, but largely out of a sense of duty he left school in April 1947 with the aim of join­ing the Roy­al Elec­tric­al and Mech­an­ic­al En­gin­eers. However, there was a hitch — he was re­garded as hav­ing dual na­tion­al­ity and re­com­men­ded to de­fer un­til he was 21, an age at which he could choose between the two. With char­ac­ter­ist­ic lo­gic, he poin­ted out that his fath­er was in fact state­less and so really he had sole Brit­ish na­tion­al­ity. When this was ac­cep­ted, he was promptly summoned to re­gister with­in 24 hours and in Oc­to­ber of that year he be­came a clerk in the re­gi­ment­al headquar­ters.

Look­ing back on his ex­per­i­ence years later, Mi­chael felt that it was worth­while to be im­mersed in a very dif­fer­ent so­cial en­vir­on­ment, his phys­ic­al fit­ness im­proved with run­ning and ten­nis, and he had plenty of spare time to think about math­em­at­ics. This was partly through books and prob­lems sent on from Cam­bridge by his peers at school, but also through his own ex­per­i­ments; for ex­ample, with prop­er­ties of high­er di­men­sion­al poly­hedra. What it did provide was an in­ter­lude that gen­er­ated enorm­ous en­thu­si­asm for the sub­ject when he fi­nally ar­rived at Trin­ity, which was per­haps ab­sent from those who had gone dir­ectly from the Sixth Form. He failed to rise through the ranks in the army, but was com­men­ded for his loud voice on the parade ground, a fore­taste for his fu­ture stu­dents and col­lab­or­at­ors!

Cambridge

Mi­chael ar­rived in Cam­bridge in 1949, armed with an ex-ser­vice­man’s grant and a ma­jor schol­ar­ship from Trin­ity Col­lege that left him quite well provided for. His first con­tacts were his Trin­ity su­per­visors. One was A. S. Be­sicov­itch FRS, a Rus­si­an ana­lyst who had been there since 1927. He was a for­mid­able char­ac­ter who would play eight games of chess sim­ul­tan­eously with his stu­dents. By con­trast there was also the ta­cit­urn J. A. Todd FRS 1948, an al­geb­ra­ic geo­met­er whose book he had already stud­ied and en­joyed. Per­haps more im­port­ant were his fel­low Trin­ity stu­dents with whom he would en­gage in math­em­at­ic­al dis­cus­sions and com­pete in ex­ams. These in­cluded J. F. (Frank) Adams (FRS 1964), John Polk­ing­horne (FRS 1974), Ian Mac­don­ald (FRS 1979), the stat­ist­i­cian John Aitchis­on and James Mack­ay (later Bar­on Mack­ay of Clash­fern, Lord High Chan­cel­lor of Great Bri­tain 1987–1997). Mack­ay re­calls as a stu­dent sit­ting on the pil­lion of a mo­tor­bike go­ing to Ox­ford to vis­it Mi­chael’s broth­er, Patrick. They re­mained lifelong friends — in later years Mi­chael per­suaded him as Lord Chan­cel­lor to re­lease the files of Op­er­a­tion Ep­si­lon, the sur­veil­lance of de­tained Ger­man sci­ent­ists who had worked on the Nazi nuc­le­ar pro­gramme.

Col­lege life was quite spartan in that post­war peri­od, with food ra­tion­ing, in­ad­equate heat­ing and lim­ited wash­ing fa­cil­it­ies, but there were many activ­it­ies to join in, such as chess and squash and the stu­dents’ math­em­at­ic­al so­ci­ety, the Archimedeans. In fact “at a meet­ing of the Archimedeans I came across a little man sit­ting on a table with his legs not even reach­ing the ground” was how Lily Brown re­called first meet­ing Mi­chael. The strong-minded Scot, the first in her fam­ily to go to uni­versity, had read math­em­at­ics in Ed­in­burgh with Mack­ay and then moved to Gir­ton Col­lege, Cam­bridge, to com­plete the Tri­pos. That first en­counter with Mi­chael led to a long court­ship and fi­nally mar­riage in 1955.

For the first two years of the Tri­pos, Mi­chael and John Polk­ing­horne worked over past pa­pers in­tens­ively, which led them to come top of the list in the res­ults, but by the time of Part 3 Mi­chael in par­tic­u­lar was be­gin­ning to fo­cus his in­terests and be­com­ing less con­cerned with ex­am­in­a­tion res­ults. Geo­metry was cent­ral to these in­terests. This was a peri­od in Cam­bridge of trans­ition between the (to Mi­chael) at­tract­ive clas­sic­al geo­met­ric ap­proach and the more rig­or­ous one based on al­gebra. The two rel­ev­ant texts were Todd’s book, Pro­ject­ive and Ana­lyt­ic­al Geo­metry [e1], which in­cluded the al­gebra re­luct­antly but ne­ces­sar­ily, and Meth­ods of al­geb­ra­ic geo­metry by W. V. D. Hodge FRS and D. Pe­doe [e2], which ac­cep­ted it face-on, and Mi­chael had to ab­sorb both. His first pa­per [1], pro­duced in his second un­der­gradu­ate year, is writ­ten in the purely clas­sic­al vein, but this was not to last.

When the time came to en­rol for a PhD, stu­dents had to choose a su­per­visor, and Mi­chael, want­ing to pur­sue re­search in geo­metry, had the choice of Todd or Hodge. He had been the star pu­pil for Todd at Trin­ity, but, per­haps to Todd’s dis­ap­point­ment, chose Hodge as hav­ing the big­ger in­ter­na­tion­al repu­ta­tion. Hodge, un­usu­ally, had as many as four stu­dents that year, one of whom was Ro­ger Pen­rose (FRS 1972). Hodge’s re­search in har­mon­ic in­teg­rals was an early con­tri­bu­tion to what be­came a whole new as­pect of geo­metry, mix­ing to­po­lo­gic­al, dif­fer­en­tial and al­geb­ra­ic tech­niques. It was gen­er­ally thought to be hard, which per­haps put off oth­er stu­dents from work­ing with him, but Mi­chael had at­ten­ded lec­tures on al­geb­ra­ic to­po­logy by P. J. Hilton and S. Wylie, had some ba­sic Rieman­ni­an geo­metry and was reas­on­ably well pre­pared.

His first task was to un­der­stand char­ac­ter­ist­ic classes of vec­tor bundles, fol­low­ing S.-S. Chern (ForMem­RS 1985) and C. B. Al­lendo­er­fer, but quite rap­idly in his first year, Mi­chael latched on to the new tech­niques of sheaf the­ory de­veloped by Henri Cartan (ForMem­RS 1971) and Jean-Pierre Serre (ForMem­RS 1974) in Par­is. Read­ing the latest Comptes Ren­dus notes in the lib­rary, he rap­idly ab­sorbed the new tech­niques and by the end of his first year pro­duced a prize-win­ning es­say on ruled sur­faces via the the­ory of holo­morph­ic vec­tor bundles, stim­u­lated by both lec­tures of A. Weil (ForMem­RS 1966) and dis­cus­sions with the vis­it­ing Amer­ic­an math­em­atician, N. S. Haw­ley. Iron­ic­ally, the dis­cov­ery of a mis­take in one of Haw­ley’s pa­pers was a pivotal event for this work, an is­sue that his su­per­visor re­com­men­ded he tone down a little in the res­ult­ing pa­per [2], which is writ­ten in a thor­oughly mod­ern style.

Figure 3. Research fellow, Trinity College, 1954.

Watch­ing a num­ber of his con­tem­por­ar­ies leave after a year, the prize es­say gave him enough con­fid­ence to pro­ceed fur­ther, but not without self-doubts. He tempered the loneli­ness of math­em­at­ic­al re­search by go­ing to lec­tures on ar­chi­tec­ture and ar­chae­ology, which he found in­ter­est­ing enough but con­firmed his be­lief that math­em­at­ic­al re­search was harder and more worth­while. A key event in the fol­low­ing years was at­tend­ing the 1954 In­ter­na­tion­al Con­gress of Math­em­aticians in Am­s­ter­dam, where the award of Fields Medals to Serre and K. Kodaira offered fur­ther veri­fic­a­tion that his re­search field was of great im­port­ance.

By the end of his second year as a PhD stu­dent Mi­chael had his prize work and a pa­per with Hodge on in­teg­rals of the second kind, and on this basis was elec­ted a Fel­low of Trin­ity (Fig­ure 3), but, sat­is­fy­ing as this was, Cam­bridge offered few new ideas in his third year or op­por­tun­it­ies for vis­it­ors. Dur­ing 1953–1954, Hodge had been to a con­fer­ence in Prin­ceton and ex­plained what Mi­chael had been work­ing on and, as a con­sequence, D. C. Spen­cer in­vited him to the uni­versity, but then the award of a Com­mon­wealth Fel­low­ship en­abled him to travel and stay longer, at the In­sti­tute for Ad­vanced Study in­stead of Prin­ceton Uni­versity.

His do­mest­ic life, however, had changed. On his elec­tion as a Fel­low, Mi­chael and Lily be­came en­gaged and she, be­ing a year ahead of him, had by 1955 gradu­ated with a PhD un­der the su­per­vi­sion of Mary Cartwright FRS and was em­ployed as a lec­turer at Bed­ford Col­lege, Lon­don. She gave up her job, they mar­ried in Ed­in­burgh (with James Mack­ay as best man) and set off for Prin­ceton.

Princeton

The stay in Prin­ceton was a key peri­od in Mi­chael’s math­em­at­ic­al life. It was not only the pres­ence of Kodaira and Serre, the re­cent Fields medal­lists, but also a stream of vis­it­ors bring­ing new ideas and per­spect­ives. It was here that Mi­chael met his three fu­ture prime col­lab­or­at­ors and per­son­al friends, Raoul Bott (ForMem­RS 2005), Is­ad­ore Sing­er and Friedrich Hirzebruch (ForMem­RS 1994), the lat­ter he had briefly en­countered be­fore when Hodge in­vited them to talk in Cam­bridge. Bott lec­tured on Morse the­ory, Kodaira on sheaf the­ory, cur­rents and al­geb­ra­ic geo­metry, and Hirzebruch on char­ac­ter­ist­ic classes. Three days a week, a car­load of math­em­aticians in­clud­ing Mi­chael, Bott and Sing­er eagerly trav­elled to the uni­versity to hear these sem­inars.

Mi­chael spoke three times in Serre’s sem­in­ar on ana­lyt­ic vec­tor bundles, but was ini­tially taken aback by the de­mand from a Bourbaki-in­flu­enced audi­ence de­mand­ing pre­ci­sion and clar­ity, quite re­moved from the gen­tle­manly at­mo­sphere in Cam­bridge. The in­flu­ence of this sem­in­ar 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 trav­elled ex­tens­ively for three months around Amer­ica, part of the con­di­tions of a Com­mon­wealth Fel­low­ship. He met Chern in Chica­go and spent a month in Mex­ico, where, in 1956, S. Lef­schetz (ForMem­RS 1961) had or­gan­ized an in­ter­na­tion­al sym­posi­um on to­po­logy in Mex­ico City. Os­tens­ibly con­cern­ing to­po­logy, it is no­tice­able from the titles of talks that fibre bundles, char­ac­ter­ist­ic classes and the Riemann–Roch the­or­em were already chan­ging the face of that sub­ject.

It was a peri­od when vari­ous strands of math­em­at­ics, per­haps not ap­par­ent at the time, were com­ing to­geth­er to form a new dis­cip­line. Par­is was a sig­ni­fic­ant source, but, when trans­por­ted to Prin­ceton, the French al­geb­ra­ic geo­metry merged with dif­fer­en­tial to­po­logy to form a much lar­ger frame­work. And it was not the per­man­ent mem­bers at the In­sti­tute who were lead­ing this move­ment, but the young vis­it­ors. It was a real melt­ing pot of ideas and a form­at­ive ex­per­i­ence for Mi­chael.

From Cambridge to America

Figure 4. With Friedrich Hirzebruch at the Arbeitstagung, Bonn, in 1977.
Taken by Konrad Jacobs, Erlangen. Copyright: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

At the be­gin­ning of 1957 Mi­chael re­turned to Cam­bridge as a uni­versity lec­turer, and then the fol­low­ing year was also ap­poin­ted as a teach­ing fel­low at Pem­broke Col­lege. He was ex­pec­ted to bring back some of the ex­cite­ment of do­ing math­em­at­ics in the United States and, some­what sur­pris­ingly for a young don, the pro­fess­ors gave him a free hand.

Those who knew him would un­der­stand this — he was an ebul­li­ent char­ac­ter, eager to con­vey his thoughts and ideas in a strong voice, which would be­come fa­mil­i­ar across many a de­part­ment­al com­mon room. It was not just his own work, but any new piece of math­em­at­ics he found in­ter­est­ing could be the sub­ject of his en­er­get­ic ex­pos­i­tion. Stu­dents were not ex­empt — one baffled Cam­bridge un­der­gradu­ate re­called go­ing to his su­per­visor, Peter Swin­ner­ton-Dyer (FRS 1967), to be told “Dr. Atiyah treats the syl­labus with the ut­most con­tempt”. It was the scope of the lec­ture rather than his present­a­tion — throughout his whole ca­reer he was fam­ous for his in­spir­ing sem­inars and lec­tures link­ing dif­fer­ent themes to­geth­er. Re­con­struct­ing the ar­gu­ment later was usu­ally a dif­fer­ent mat­ter.

So, in Cam­bridge, he began to or­gan­ize col­loquia. There was still little op­por­tun­ity for in­vit­ing out­side speak­ers, but pro­fess­ors find­ing out what each oth­er was do­ing was it­self a nov­elty. At the same time he was re­count­ing the ideas he had ac­quired in the United States to Hodge and oth­ers, like E. C. (Chris­toph­er) Zee­man (FRS 1975) who too had re­cently re­turned from Amer­ica. Hodge him­self was be­com­ing in­creas­ingly oc­cu­pied with oth­er activ­it­ies — Mas­ter of Pem­broke and Phys­ic­al Sec­ret­ary of the Roy­al So­ci­ety — and he handed over to Mi­chael many oth­er de­part­ment­al activ­it­ies and some of his gradu­ate stu­dents too, such as Rolf Schwar­zen­ber­ger and Ian Porteous.

Figure 5. With Grothendieck, Bonn, 1958.
From the Hirzebruch family collection; copyright unknown.

At this time, in 1957, the Bonn Arbeit­sta­gung (Fig­ures 4 and 5) began and Mi­chael be­came a reg­u­lar at­tendee, for­ging closer links with Hirzebruch, who, at a young age, had been ap­poin­ted pro­fess­or there. It was a very act­ive peri­od for to­po­logy, with Mil­nor’s work on man­i­fold clas­si­fic­a­tion, Thom’s on cobor­d­ism the­ory and Hirzebruch show­ing the links with al­geb­ra­ic geo­metry. The top­ics were not chosen in ad­vance, so that the meet­ing be­came a shop win­dow for the latest res­ults of the likes of Serre, Jacques Tits, Ar­mand Borel and Al­ex­an­dre Grothen­dieck as well as Mi­chael’s former Prin­ceton as­so­ci­ates. Grothen­dieck in par­tic­u­lar took centre stage at the be­gin­ning — the oth­ers sat back and learned from it (Fig­ure 5).

Cam­bridge also saw an­oth­er de­vel­op­ment in Mi­chael’s life — the birth of sons John (1957) and Dav­id (1958).

\( K \)-theory

Mi­chael was not by nature a the­ory-build­er, but some­times it be­came a ne­ces­sity. One of his math­em­at­ic­al achieve­ments, be­gun in 1959 col­lab­or­at­ing with Hirzebruch dur­ing a brief re­turn to Prin­ceton, was the in­ven­tion of K-the­ory. This was an ad­apt­a­tion in­to a new en­vir­on­ment — al­geb­ra­ic to­po­logy — of Grothen­dieck’s for­mu­la­tion of the Riemann–Roch the­or­em in al­geb­ra­ic geo­metry, which Mi­chael had been ex­posed to in the Arbeit­sta­gung. Its in­tro­duc­tion was part of a gen­er­al drift of ideas from al­geb­ra­ic geo­metry to to­po­logy, a big step in this dir­ec­tion hav­ing been taken by Hirzebruch in his book, Neue to­po­lo­gis­che Meth­oden in der al­geb­rais­chen Geo­met­rie [e3].

Al­though not by train­ing an al­geb­ra­ic to­po­lo­gist, Mi­chael began to in­volve him­self more and more in the sub­ject. He ob­served that a num­ber of prob­lems in to­po­logy, in par­tic­u­lar some posed by Ioan James (FRS 1968), would be­come much easi­er to prove if one could pro­duce a co­homo­logy the­ory based on to­po­lo­gic­al vec­tor bundles. Moreover, the in­teg­ral­ity res­ults of char­ac­ter­ist­ic classes in Hirzebruch’s work would be­come clear­er. To con­struct a co­her­ent the­ory, new fea­tures not present in Grothen­dieck’s work, such as the odd de­gree case and the real the­ory, were needed. Its prop­er­ties re­quired the res­ults of Bott on the peri­od­icity of the ho­mo­topy groups of the unit­ary and or­tho­gon­al groups that Mi­chael had heard about in Prin­ceton. The res­ult­ing the­ory, \( K \)-the­ory, sub­sequently be­came the nat­ur­al vehicle for the in­dex the­or­em, but with­in the realm of al­geb­ra­ic to­po­logy it had nu­mer­ous achieve­ments. One of them was, proved to­geth­er with Adams, a simple proof of the clas­sic­al Hopf in­vari­ant prob­lem con­cern­ing the de­grees of maps between spheres.

For the next three years or so, Mi­chael’s pa­pers were de­voted to ap­plic­a­tions of \( K \)-the­ory, and he gave an in­vited talk, “The Grothen­dieck ring in geo­metry and to­po­logy”, at the 1962 In­ter­na­tion­al Con­gress in Stock­holm, driv­ing across Nor­way with his fam­ily and moth­er (a stop­over in a log cab­in was the im­petus for buy­ing one in Perth­shire 15 years later). However, dur­ing this peri­od there was an­oth­er change in his life.

Oxford

While in Cam­bridge, Mi­chael’s work at­trac­ted the at­ten­tion of the to­po­lo­gist J. H. C. (Henry) White­head FRS in Ox­ford, and he went there to give talks, but in 1960 White­head died sud­denly. The Waynf­lete Chair, which he had oc­cu­pied, be­came va­cant and Mi­chael, at the age of 31, ap­plied for it. In the event, it went to the more seni­or Gra­ham Hig­man FRS, a group the­or­ist who was already there as a read­er. So Mi­chael suc­cess­fully ap­plied for this read­er­ship in­stead of the chair, and took it up in 1961. He chose not to en­gage in col­lege teach­ing and could then spend more time do­ing re­search.

Math­em­at­ics at the time was not one of Ox­ford’s strengths, al­though there was at least a cent­ral build­ing — the Math­em­at­ic­al In­sti­tute. The closest in­terest for Mi­chael was to­po­logy, rep­res­en­ted by Ioan James, also a read­er, but Mi­chael, who be­came a Fel­low of the young St. Cath­er­ine’s Col­lege, began in­stead to ac­quire a group of gradu­ate stu­dents, many sent on from Cam­bridge by Hodge. Graeme Segal (FRS 1982) was one of them. He re­calls Mi­chael’s ad­vice at the time: “People are al­ways ask­ing me to give them a prob­lem, but they don’t mean that; what they want is a meth­od. I give my stu­dents a meth­od.” By that time he had ac­quired a group of six stu­dents and “a little cloud of de facto stu­dents of­fi­cially be­long­ing to oth­ers”. To com­pensate for the lack of act­ive geo­met­ers, Mi­chael spent any avail­able time go­ing to Har­vard or Prin­ceton or in­vit­ing Bott or Sing­er to Ox­ford.

In 1963 the num­ber the­or­ist E. C. Titch­marsh FRS died, and the Sa­vil­ian Pro­fess­or­ship of Geo­metry that he had held was offered to Mi­chael, with a fel­low­ship at New Col­lege. Since Titch­marsh’s pre­de­cessor was G. H. Hardy FRS, Mi­chael was the first geo­met­er to hold that chair for some time. Mi­chael and Lily bought a house in Head­ing­ton, and ex­ten­ded it in 1966. The long garden was not only a chance for Mi­chael to in­dulge in one of his hob­bies (mu­sic and hill-walk­ing be­ing oth­ers), but also a place to pace up and down with his math­em­at­ic­al thoughts. A third son, Robin, was born in 1963.

The index theorem

The Atiyah–Sing­er in­dex the­or­em is Mi­chael’s best known res­ult. It won its au­thors the Abel Prize in 2004 and pa­pers on the top­ic oc­cupy two volumes of Mi­chael’s col­lec­ted works.

Its ori­gins lie again in the os­mos­is of con­cepts from al­geb­ra­ic geo­metry in­to to­po­logy. The Riemann–Roch the­or­em in its mod­ern for­mu­la­tion con­cerns the al­tern­at­ing sum of di­men­sions of sheaf co­homo­logy groups \[ \sum^n_{i=0} (-1)^i \dim H^i (M,V) \] and is fun­da­ment­al for many pro­cesses in al­geb­ra­ic geo­metry. In Hirzebruch’s early work it is ex­pressed as a com­bin­a­tion of char­ac­ter­ist­ic classes — a to­po­lo­gic­al in­vari­ant. Col­lect­ing to­geth­er the even de­grees and the odd ones, this is a dif­fer­ence of the di­men­sions of two vec­tor spaces. Hirzebruch’s meth­ods also ap­plied to the sig­na­ture of a dif­fer­en­ti­able man­i­fold, which, through the the­ory of Hodge’s har­mon­ic forms, is also a dif­fer­ence of di­men­sions. Both cases can be ex­pressed as the dif­fer­ence in the di­men­sions of the null spaces of two as­so­ci­ated dif­fer­en­tial op­er­at­ors on a com­pact man­i­fold. This is the set­ting of the in­dex the­or­em.

Through care­ful ma­nip­u­la­tions, Hirzebruch had been show­ing at the Arbeit­sta­gun­gen how cer­tain com­bin­a­tions of char­ac­ter­ist­ic num­bers were al­ways in­tegers, and this also de­man­ded an ex­plan­a­tion in terms of di­men­sions. These ob­ser­va­tions con­sti­tuted an an­swer in search of a prob­lem, and that gen­er­al prob­lem was to de­term­ine the dif­fer­ence in di­men­sions of the null space of an el­lipt­ic op­er­at­or \( D \) and its ad­joint \( D^{\ast} \), the so-called in­dex of \( D \): \[ \operatorname{ind} D=\dim \ker D - \dim\ker D^{\ast}. \] The Riemann–Roch the­or­em was a very spe­cial case.

The fact that the in­dex was a to­po­lo­gic­al in­vari­ant was known to the Rus­si­an ana­lysts, but not a pre­cise for­mula. In the spring of 1962, Sing­er came to spend a sab­bat­ic­al term in Ox­ford and posed a spe­cif­ic ques­tion about the so-called \( \hat{A} \)-genus that was an in­teger for a spin man­i­fold, ac­cord­ing to a res­ult of Hirzebruch. Once they had real­ized, without a proof, that this must be the in­dex of the Dir­ac op­er­at­or in its Rieman­ni­an ex­pres­sion, they had enough in­form­a­tion to be­gin the pro­gramme, which took 18 months be­fore a first proof was found. Sing­er’s early back­ground in math­em­at­ics was in the area of ana­lys­is and he brought that ex­pert­ise to the table to com­ple­ment the to­po­logy and al­geb­ra­ic geo­metry that were Mi­chael’s forte.

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 the work pro­gressed, the lan­guage of \( K \)-the­ory came to be seen as the ap­pro­pri­ate frame­work for in­dex prob­lems, and in 1968 came the first of the five pa­pers in An­nals of Math­em­at­ics with Sing­er which gave proofs of the in­dex the­or­em in many dif­fer­ent con­texts with many dif­fer­ent ap­plic­a­tions [4], [3], [5], [7], [6].

By this time, Mi­chael had been elec­ted a Fel­low of the Roy­al So­ci­ety in 1962 and won a Fields Medal at the 1966 Mo­scow In­ter­na­tion­al Con­gress for his work on \( K \)-the­ory and the in­dex the­or­em. The year fol­low­ing the Fields Medal, he paid his third vis­it to the In­sti­tute in Prin­ceton, and, as with oth­er Fields medal­lists, was ap­proached to see if he would like to go there as a per­man­ent mem­ber. After some de­lib­er­a­tion back in Ox­ford, in 1969 he de­cided to move to a place that had good memor­ies and bet­ter fa­cil­it­ies for in­ter­ac­tion. He did not fully leave Ox­ford, for the In­sti­tute term fin­ished in April and the fam­ily re­turned to their house and Trin­ity term each year.

Back in Princeton

The in­dex the­or­em con­tin­ued to be a sub­ject of re­search in Prin­ceton, largely through the pres­ence of a young In­di­an math­em­atician, V. K. Pat­odi, who brought with him a new meth­od of proof.

Even after the first an­nounce­ment, the ori­gin­al proof of the the­or­em was felt not to be the last word. It was not, in Mi­chael’s words, “A proof con­sist­ent with the el­eg­ance of the prob­lem”. It con­sisted first of ana­lyt­ic­al work show­ing that the in­dex was a par­tic­u­lar type of in­vari­ant in al­geb­ra­ic to­po­logy, namely a cobor­d­ism in­vari­ant — a weak­er no­tion than to­po­lo­gic­al equi­val­ence. To prove the the­or­em it was then enough to check it on gen­er­at­ors of the cobor­d­ism ring, a trick that Hirzebruch had used earli­er. This had lim­it­a­tions for the fu­ture de­vel­op­ment of the the­or­em, and the more sat­is­fact­ory proofs that ap­peared in the An­nals in­volved \( K \)-the­ory and were es­sen­tially mod­elled on Grothen­dieck’s ap­proach, but re­pla­cing the prop­er­ties of sheaves by those of pseudodif­fer­en­tial op­er­at­ors.

Fol­low­ing on from this second proof, Pat­odi, build­ing on the work of Sing­er and H. McK­ean, came up with an ap­proach us­ing the heat ker­nel, form­ally \( \operatorname{tr} \exp(-t D D^{\ast}) \). Since the ei­gen­spaces of \( DD^{\ast} \) and \( D^{\ast} D \) for nonzero ei­gen­vectors are in­ter­changed by \( D \), the in­dex is the dif­fer­ence \[ \operatorname{tr}\exp(-tD^{\ast}D)-\operatorname{tr}\exp(-tDD^{\ast}) \] but there is an asymp­tot­ic ex­pres­sion as \( t \) tends to zero, which in­volves in­teg­rals of loc­ally-defined terms. Pat­odi had deftly ma­nip­u­lated these to prove the in­dex the­or­em in spe­cial cases. The loc­al­ity of the ap­proach offered the op­por­tun­ity to de­vel­op an in­dex the­ory for man­i­folds with bound­ary and a means of ex­plain­ing work that Hirzebruch had been car­ry­ing out on sig­na­ture de­fects, which yiel­ded num­ber-the­or­et­ic Dede­kind sums. This re­quired a more con­cep­tu­al ap­proach to Pat­odi’s “mi­ra­cu­lous can­cel­la­tions”, and fi­nally gave rise to a new type of glob­al bound­ary value prob­lem and a real-val­ued in­dex, the eta-func­tion. This later came to play a sig­ni­fic­ant role in the phys­i­cists’ the­ory of an­om­alies and de­term­in­ants.

While resolv­ing the prob­lems in­volved in un­der­stand­ing these new is­sues, Mi­chael made an­oth­er de­cision. Des­pite the heady at­mo­sphere of re­search in Prin­ceton, he, like oth­ers be­fore and after him, de­cided to leave the In­sti­tute after three years. Par­tially this was for fam­ily reas­ons — Lily was un­able to teach in Amer­ica, and they de­cided that they pre­ferred a Brit­ish edu­ca­tion for the chil­dren. Mi­chael had been offered a Roy­al So­ci­ety Re­search Pro­fess­or­ship for his re­turn to the UK. It could have been held any­where, but he chose Ox­ford.

Back to Oxford

Back in Ox­ford in 1972, do­ing re­search full time without un­der­gradu­ate teach­ing, Mi­chael took on gradu­ate stu­dents. In Prin­ceton this was hardly pos­sible, the young people around him be­ing postdocs. He was also freer to travel and col­lab­or­ate, and after each vis­it he brought back new res­ults and con­cepts, ex­pound­ing them in a weekly sem­in­ar that at­trac­ted much at­ten­tion, and not just in Ox­ford. This al­tern­at­ive chan­nel­ling of his ideas proved to be very fruit­ful.

Figure 6. With James Eells and Roger Penrose, Durham, 1982.
Photography by Dirk Ferus. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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 (Fig­ure 6). Pen­rose 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, the corner­stone of twis­tor the­ory, which they had both learned from Todd’s book all those years ago. The first fruit of the in­ter­ac­tion was Mi­chael’s re­cog­ni­tion that the sheaf the­ory he had first learned 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 that Pen­rose was work­ing on. This provided a ready-made reser­voir of soph­ist­ic­ated tech­niques to ap­ply to these lin­ear equa­tions, which was eagerly as­sim­il­ated by Pen­rose’s stu­dents. 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 in MIT of the phys­i­cists’ work on in­stan­tons — self-dual solu­tions of the Yang–Mills equa­tions on Eu­c­lidean 4-space. For the rest of the term each weekly sem­in­ar was de­voted to this top­ic. Two de­vel­op­ments oc­curred at this time.

The first was the work of Richard Ward (FRS 2005), a stu­dent of Pen­rose, who had shown us­ing twis­tor the­ory that a com­plex solu­tion of the self-du­al­ity equa­tions arose from the data of a holo­morph­ic vec­tor bundle on pro­ject­ive 3-space. 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, provid­ing a link to al­geb­ra­ic geo­metry.

The second de­vel­op­ment was that the in­dex the­or­em could be put to use to ac­tu­ally cal­cu­late the di­men­sion of the mod­uli space of in­stan­tons — all solu­tions mod­ulo gauge equi­val­ence. The di­men­sion­al­ity of the mod­uli space co­in­cided with a di­men­sion that W. Barth had cal­cu­lated for the mod­uli of cer­tain stable holo­morph­ic bundles on pro­ject­ive space, and this it­self was based on a very con­crete con­struc­tion that G. Hor­rocks had spoken about 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 the au­thor of this mem­oir had found this, we heard from Y. Man­in that he and his stu­dent, 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 (Atiyah, Drin­feld, Hitchin, Man­in) con­struc­tion.

This piece of work was in­flu­en­tial in dif­fer­ent ways. First, it provided a fur­ther move­ment of ideas from al­geb­ra­ic geo­metry in­to dif­fer­en­tial to­po­logy, this time, un­like Riemann–Roch or the in­dex the­or­em, in­volving non­lin­ear dif­fer­en­tial equa­tions. Mod­uli spaces had tra­di­tion­ally been in­tro­duced study­ing fam­il­ies of al­geb­ra­ic vari­et­ies or sheaves, but now there were im­port­ant ex­amples in dif­fer­en­tial geo­metry. In this new world the in­dex the­or­em played a sup­port­ing role in de­scrib­ing the tan­gent space, a lin­ear ap­prox­im­a­tion to the non­lin­ear mod­uli space. This had pro­found con­sequences when Mi­chael’s stu­dent, Si­mon Don­ald­son (FRS 1986), took the Yang–Mills equa­tions much fur­ther, re­placed Eu­c­lidean space by a more gen­er­al 4-di­men­sion­al man­i­fold and proved highly un­ex­pec­ted res­ults in dif­fer­en­tial to­po­logy that earned him a Fields Medal at the early age of 29.

It also had an in­flu­ence on Mi­chael’s at­ti­tude to the in­ter­ac­tion of math­em­at­ics and phys­ics. Here was a piece of con­tem­por­ary pure math­em­at­ics that provided a com­plete solu­tion to a prob­lem posed by phys­i­cists. Surely this could be taken fur­ther?

Mathematics and physics

Mi­chael’s in­ter­ac­tion with phys­i­cists had already be­gun be­fore the work on in­stan­tons — one of his vis­its to MIT to see Sing­er had in­volved a dis­cus­sion with R. Jackiw on an­om­alies, the Dir­ac op­er­at­or and the in­dex the­or­em. But after 1977 this took up a more sub­stan­tial part of his life, much of which was in­volved with fa­cil­it­at­ing the in­ter­ac­tion and edu­cat­ing phys­i­cists about con­cepts they wer­en’t aware of.

Con­versely, he also began to ap­ply ideas from Yang–Mills the­ory to math­em­at­ic­al prob­lems, per­haps the most not­able his pa­per with Bott on Riemann sur­faces and the Narasim­han–Se­shadri the­or­em [8].

The most im­port­ant go-between was the the­or­et­ic­al phys­i­cist Ed­ward Wit­ten (ForMem­RS 1999). They first met in MIT in 1977 while Wit­ten was a stu­dent, and an im­pressed Mi­chael in­vited him to Ox­ford for a few weeks for dis­cus­sions that were the be­gin­ning of a long as­so­ci­ation. Mi­chael in­tro­duced Wit­ten to some es­tab­lished pieces of math­em­at­ics that were un­known to the phys­ics com­munity — Morse the­ory, equivari­ant co­homo­logy and mo­ment maps. In turn, Wit­ten re­in­ter­preted these and fed back new in­sights to the math­em­at­ic­al com­munity. Soon he was ap­poin­ted as a per­man­ent mem­ber at the In­sti­tute for Ad­vanced Study and Mi­chael’s vis­its there were ba­sic­ally to see him.

In 1987–1988 Mi­chael made sev­er­al trips and in par­tic­u­lar set Wit­ten vari­ous chal­lenges. One was see­ing Don­ald­son the­ory in quantum-the­or­et­ic terms, which was achieved through a twis­ted ver­sion of su­per­sym­met­ric Yang–Mills the­ory. An­oth­er was un­der­stand­ing the Jones (V. F. R. Jones, FRS 1990) poly­no­mi­al of a knot via quantum field the­ory. This work brought in­to play a pro­duct­ive fu­sion of ideas from dif­fer­ent parts of math­em­at­ics and phys­ics and con­trib­uted to Wit­ten win­ning a Fields Medal in 1990. As Mi­chael wrote of Wit­ten in his cita­tion for the In­ter­na­tion­al Con­gress: “Al­though he is def­in­itely a phys­i­cist (as his list of pub­lic­a­tions clearly shows) his com­mand of math­em­at­ics is ri­valled by few math­em­aticians, and his abil­ity to in­ter­pret phys­ic­al ideas in math­em­at­ic­al form is quite unique. Time and again he has sur­prised the math­em­at­ic­al com­munity by a bril­liant ap­plic­a­tion of phys­ic­al in­sight lead­ing to new and deep math­em­at­ic­al the­or­ems.” The con­cepts that arose out of this work, not­ably (build­ing on ideas of Segal) that of a To­po­lo­gic­al Quantum Field The­ory, formed the theme of Mi­chael’s Ox­ford Sem­in­ar for one term, the out­come be­ing a pub­lished col­lec­tion of stim­u­lat­ing ideas rather than a math­em­at­ic­al pa­per [9].

Mi­chael had a re­mark­able in­tu­ition re­gard­ing the math­em­at­ics that was rel­ev­ant to quantum field the­ory, and his rap­port with Wit­ten fed a two-way ex­change of ideas — he had a repu­ta­tion as a talk­er, but he was also a good listen­er. Yet he did not enter the phys­i­cist’s ter­rit­ory to at­tempt to make quantum field the­ory math­em­at­ic­ally rig­or­ous. It was enough to see that the math­em­at­ic­al the­ory had a nat­ur­al home there with the prom­ise of new in­sights. In­stead, the phys­ics-ori­ented top­ics he wrote about con­cerned the clas­sic­al rather than quantum as­pects of gauge the­or­ies — the to­po­logy and geo­metry of mod­uli spaces of in­stan­tons and mono­poles and the Skyrme mod­el in nuc­le­ar phys­ics. He col­lab­or­ated, on the sug­ges­tion of his Trin­ity con­tem­por­ary John Polk­ing­horne (soon to be­come an Anglic­an priest), with the Cam­bridge the­or­et­ic­al phys­i­cist Nich­olas Man­ton (FRS 1996) on these themes, study­ing their geo­metry and dy­nam­ics as ex­ten­ded ob­jects and not point particles.

If his ac­tu­al work did not en­gage dir­ectly with the phys­i­cists’ main aims, Mi­chael nev­er­the­less thought ser­i­ously about the re­la­tion­ship. In a Har­vard con­fer­ence for Bott’s sev­en­ti­eth birth­day in 1993, he out­lined four re­ac­tions to the way in which math­em­aticians could in­ter­act with phys­i­cists:

  • Try to give rig­or­ous proofs of math­em­at­ic­al res­ults pre­dicted by the phys­i­cists.
  • Try and un­der­stand the phys­ics and enter in­to a dia­logue us­ing their lan­guage.
  • De­vel­op the phys­ics on a rig­or­ous basis with form­al proofs.
  • Un­der­stand the deep­er mean­ing of the con­nec­tion between math­em­at­ics and phys­ics.

While ac­cept­ing the valid­ity of each ap­proach, he ad­mit­ted that he was at­trac­ted to the last one, even if it had an es­sen­tially philo­soph­ic­al con­tent, rais­ing the ques­tion of wheth­er math­em­at­ics is cre­ated or dis­covered.

The dia­logue con­tin­ued even later. In 2002 he and Lily spent two winter months at Cal­Tech, where Wit­ten held a vis­it­ing po­s­i­tion. Mi­chael re­called: “Dur­ing my stay I felt like a gradu­ate stu­dent again, with Wit­ten as my su­per­visor. It was in­tel­lec­tu­ally de­mand­ing but very re­ward­ing. It res­ul­ted in a mam­moth joint pa­per in which I de­veloped the de­tails of the math­em­at­ics that Wit­ten had as­signed to me and he pur­sued the phys­ics bey­ond my reach.”

Trinity College

Trin­ity Col­lege, Cam­bridge, was al­ways a mag­net for Mi­chael. In 1983 he had hoped to be­come Mas­ter and when An­drew Hux­ley (PRS 1980–1985) was ap­poin­ted, he thought his chance had gone — ”his­tory only knocks once”. But in 1990 he was suc­cess­ful and left Ox­ford to move in­to the Mas­ter’s Lodge. At the same time, he be­came Pres­id­ent of the Roy­al So­ci­ety and found­ing dir­ect­or of the Isaac New­ton In­sti­tute for Math­em­at­ic­al Sci­ences, a na­tion­al fa­cil­ity based in Cam­bridge. This busy life, to and fro between Cam­bridge and Lon­don, left little time for math­em­at­ic­al re­search.

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, a pres­id­ent of both 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. The two rather dif­fer­ent audi­ences served by Trin­ity and the Roy­al So­ci­ety en­tailed an ex­ten­sion of themes on which Mi­chael was re­quired to speak and write.

As for Trin­ity, he and Lily en­joyed en­ter­tain­ing both em­in­ent vis­it­ors and stu­dents in the Lodge, al­though bal­an­cing their wishes for a re­lax­ing home en­vir­on­ment with col­lege pro­tocol took some time. One col­lege Fel­low re­calls Lily as be­ing the “last tra­di­tion­al Mas­ter’s wife”. In fact in later years, Mi­chael re­gret­ted that the con­ven­tions of the time had pre­ven­ted Lily con­tinu­ing an in­de­pend­ent aca­dem­ic ca­reer, al­though she had been a very suc­cess­ful teach­er at Head­ing­ton School in Ox­ford. He gave fre­quent ad­dresses at the vari­ous col­lege events and spe­cial oc­ca­sions and even gave a ser­mon in the chapel (on Trin­ity Sunday 1997, which he hoped would be­come a con­ven­tion).

Mi­chael was proud of Trin­ity’s fame through the achieve­ments of its alumni, and was quick to point out that some con­tem­por­ary or his­tor­ic­al per­son­age “was a Trin­ity man”, which must have been the res­ult of some con­sid­er­able re­search while Mas­ter. On his watch, a No­bel Prize in eco­nom­ics was awar­ded to James Mir­rlees, whom Mi­chael had su­per­vised when he was an un­der­gradu­ate.

He did not ar­rive with any par­tic­u­lar agenda for the col­lege, but nev­er­the­less some of his ideas were voted down. In par­tic­u­lar it seems as if the Fel­lows did not wish to cel­eb­rate 450 years of ex­ist­ence and pre­ferred to wait an­oth­er 50 for a round num­ber. In his re­tire­ment speech he offered a cyn­ic’s view of a Mas­ter’s ex­per­i­ence: “First you have to find out what the situ­ation is: this takes sev­er­al years. Next you have to for­mu­late the changes you would like to see: this takes even longer. Fi­nally you real­ize it is im­possible to make the changes: this you dis­cov­er pretty quickly!” It is not clear that he was en­tirely jok­ing, but two changes did oc­cur: the re­search fel­low­ships were opened up to non-Trin­ity ap­plic­ants, and a por­trait of one of Mi­chael’s sci­entif­ic her­oes, James Clerk Max­well FRS, was hung in the Col­lege Hall. This was some­what short of a statue next to New­ton in the chapel, which he thought Max­well de­served.

The Royal Society

With­in a few weeks of be­ing in­vited to be head of Trin­ity Col­lege, Mi­chael was ap­proached to be Pres­id­ent of the Roy­al So­ci­ety, which was some­what more than he had ex­pec­ted. His ap­point­ment con­veni­ently broke the ri­gid A-side/B-side \( + \) No­bel Prize con­ven­tion that ad­hered to bound­ar­ies which were chan­ging in the sci­entif­ic world.

Soon after tak­ing of­fice, he made some or­gan­iz­a­tion­al changes with fixed five-year terms for the of­ficers, one fall­ing va­cant each year. This lo­gic­al pro­ced­ure also served as a cre­at­ive solu­tion to some sens­it­ive in­tern­al is­sues. He also began a demo­crat­iz­a­tion pro­cess, with more open­ness in the elec­tion of mem­bers of coun­cil and of­ficers. In par­tic­u­lar, postal votes and con­tested elec­tions for coun­cil mem­bers began.

The pres­id­ency ne­ces­sit­ated speeches and meet­ings rep­res­ent­ing as­pects of UK sci­ence to many dif­fer­ent bod­ies, and al­lowed Mi­chael to de­vel­op his own views that had been sub­dued in the years de­voted to math­em­at­ics. Ex­tern­al events, too, de­man­ded re­sponses, and one of these was the plight of sci­ent­ists fol­low­ing the break-up of the So­viet Uni­on. Re­search sci­ent­ists were leav­ing the af­fected coun­tries or be­ing forced in­to un­skilled jobs be­cause of budget cuts. Mi­chael was well aware of the work his old friend Fritz Hirzebruch was do­ing in find­ing aca­dem­ic jobs for East Ger­man math­em­aticians in the same pre­dic­a­ment. A re­port was sent to the UK prime min­is­ter and fund­ing re­leased to keep Rus­si­an sci­ence con­nec­ted to the best work in the West.

One in­nov­a­tion in­tro­duced dur­ing this time was the es­tab­lish­ment of the Academy of Med­ic­al Sci­ences in 1998, fol­low­ing the re­com­mend­a­tions of a work­ing group chaired by Mi­chael. The founder pres­id­ent was Peter Lach­mann FRS, the So­ci­ety’s Bio­lo­gic­al Sec­ret­ary from 1993 to 1995, who first got to know Mi­chael play­ing sim­ul­tan­eous chess games as a stu­dent in Trin­ity. He has ac­know­ledged that Mi­chael played “a vi­tal role in over­com­ing wide­spread op­pos­i­tion among med­ic­al or­gan­iz­a­tions to the whole idea and per­suaded many in­flu­en­tial people of its mer­its, tak­ing de­cis­ive ac­tion at a crit­ic­al mo­ment”. Mi­chael’s out­ward-look­ing con­vic­tions not only helped to bring to­geth­er oth­er na­tion­al academies, such as the Roy­al Academy of En­gin­eer­ing, but also was in­stru­ment­al in found­ing the Inter-Academy Pan­el, play­ing a sim­il­ar in­ter­na­tion­al role.

As pres­id­ent, Mi­chael found him­self ad­dress­ing academies, so­ci­et­ies and in­sti­tutes of all kinds, from the Carib­bean academy to the In­di­an one, and from the is­sue of to­bacco con­trol to sci­ence in the Na­tion­al Por­trait Gal­lery. However, there was one theme in the re­la­tion­ship between sci­ence and so­ci­ety that he re­turned to again and again: the threat posed by nuc­le­ar weapons. He re­served his fi­nal An­niversary Ad­dress, in 1995, 50 years after the drop­ping of the first atom­ic bomb, for an ex­pos­i­tion of his per­son­al views without the re­straints of rep­res­ent­ing the So­ci­ety. He con­tras­ted the tech­nic­al sci­entif­ic ac­com­plish­ment of nuc­le­ar weapons with the pub­lic sus­pi­cion of the sci­ent­ist that fol­lowed. The main thrust of his ar­gu­ment was that “the in­sist­ence on a UK nuc­le­ar cap­ab­il­ity was fun­da­ment­ally mis­guided, a total waste of re­sources and a sig­ni­fic­ant factor in our re­l­at­ive eco­nom­ic de­cline over the past 50 years”. There was un­der­stand­ably sig­ni­fic­ant press cov­er­age, but in fact these were long-held views that he con­tin­ued to ex­pound when he be­came pres­id­ent of the Pug­wash Move­ment in 1997, in suc­ces­sion to its founder, Joseph (later Sir Joseph) Rot­blat (FRS 1995). Mi­chael was par­tic­u­larly pleased when Rot­blat was elec­ted to the Roy­al So­ci­ety and won the No­bel Peace Prize in the same year, 1995.

Edinburgh

On his re­tire­ment from Trin­ity Col­lege in 1997, Mi­chael moved to Lily’s home city of Ed­in­burgh. They had fre­quently been up to Scot­land to their log cab­in in Kirk­mi­chael, Perth­shire, but now they bought a pent­house flat with views of both Ar­thur’s Seat and the Pent­land Hills, yet quite close to the math­em­at­ics de­part­ment of the Uni­versity of Ed­in­burgh. The de­part­ment provided an of­fice that Mi­chael used on a daily basis un­less he was trav­el­ling. Elmer Rees, a former col­league and col­lab­or­at­or in Ox­ford, was a pro­fess­or there, and there were young­er mem­bers of staff with whom he could dis­cuss ideas.

Here, after an ab­sence of sev­er­al years from re­search, Mi­chael sought a prob­lem to work on and found one thanks to a ques­tion from the phys­i­cist Mi­chael Berry FRS arising from the spin stat­ist­ics the­or­em. This was a simply stated prob­lem about the ex­ist­ence of a con­tinu­ous map with cer­tain prop­er­ties from the con­fig­ur­a­tion space of n points in three-di­men­sion­al Eu­c­lidean space to the flag man­i­fold \( U(n)/T \). He de­voted six pa­pers to this theme. A res­ol­u­tion and gen­er­al­iz­a­tion ap­peared in a joint pa­per with Ro­ger Bielawski, but this de­pended on gauge-the­or­et­ic ideas — some­how this was not to Mi­chael “a proof con­sist­ent with the el­eg­ance of the prob­lem” and he was nev­er quite sat­is­fied with it.

Mi­chael’s sub­sequent work again had an in­ter­face with phys­ics, as the string the­or­ists had real­ized that D-brane charges were best rep­res­en­ted as \( K \)-the­ory classes, and this led to a re­turn to his old sub­ject and the new twis­ted ver­sions of it. But an­oth­er fa­cet began, the change from “in­ad­vert­ent phys­i­cist” to “in­ten­tion­al phys­i­cist” in the words of Bernd Schro­ers, his col­lab­or­at­or at Heri­ot-Watt Uni­versity.

At this stage Mi­chael wanted to at­tack phys­ic­al prob­lems head-on but with un­ortho­dox ideas, avoid­ing quantum the­ory, which he dis­liked as be­ing too lin­ear des­pite its ob­vi­ous achieve­ments. These pro­pos­als were highly spec­u­lat­ive, one in­volving ad­vanced and re­tarded terms in the equa­tions, an­oth­er based on so-called “geo­met­ric mod­els of mat­ter” — four-di­men­sion­al ob­jects where to­po­lo­gic­al in­vari­ants were sup­posed to cor­res­pond to ba­ry­on and lepton num­bers and charges. These could have been viewed as meta­phors, but Mi­chael was con­vinced that they could provide new phys­ic­al in­sights. Schro­ers [e6] de­scribes this peri­od: “He gen­er­ated new ideas at a prodi­gious rate, an­noun­cing them with in­fec­tious en­thu­si­asm, but abandon­ing them cas­u­ally if a new and more prom­ising av­en­ue presen­ted it­self. To­wards the end of his life, this en­thu­si­asm car­ried him per­haps a bridge too far as he at­temp­ted to re­solve long-stand­ing ques­tions in math­em­at­ics, but he car­ried on work­ing un­til the end.

If these ec­cent­ric ap­proaches to the­or­et­ic­al phys­ics did not at­tract fol­low­ers, there was one event in his fi­nal year that would vin­dic­ate his search for to­po­lo­gic­al ideas in real phys­ics. The 2019 Break­through Prize in Phys­ics was awar­ded to C. Kane and E. Mele for their work on to­po­lo­gic­al in­su­lat­ors, a new kind of ma­ter­i­al. The the­or­et­ic­al pre­dic­tion of their ex­ist­ence, soon ex­per­i­ment­ally veri­fied, ap­peared in 2019 [e5], where we may read: “These bundles are clas­si­fied with­in the math­em­at­ic­al frame­work of twis­ted real \( K \)-the­ory…the \( Z_2 \) in­dex is re­lated to the \( \operatorname{mod} 2 \) in­dex of the real Dir­ac op­er­at­or.”

Figure 7. With Dr Alasdair Allan MSP, the Minister for Learning, Science and Scotland’s Language, in front of the James Clerk Maxwell statue in George Street, Edinburgh in 2015.
Kindly supplied by the Royal Society of Edinburgh.

Math­em­at­ics was not Mi­chael’s only in­terest in his newly ad­op­ted home. The Roy­al So­ci­ety of Ed­in­burgh be­came a sub­sti­tute for Trin­ity Col­lege, and he would en­joy en­ter­tain­ing guests there. He be­came more and more in­volved with the So­ci­ety and ap­pre­ci­ated the fact that it covered the hu­man­it­ies as well as sci­ence. Fi­nally, he be­came its pres­id­ent from 2005 un­til 2008. His time at the Roy­al So­ci­ety of Lon­don served as “a good ap­pren­tice­ship”, as he com­men­ted when elec­ted to the po­s­i­tion.

Mi­chael be­came quite well known for his con­tri­bu­tions to pub­lic life in Scot­land, felt at home, hav­ing had a Scot­tish moth­er and wife, and in­creas­ingly pre­ferred the polit­ic­al com­plex­ion of the de­volved gov­ern­ment as op­posed to the Con­ser­vat­ives in West­min­ster. He be­came well-in­formed about the Scot­tish En­light­en­ment and be­came more likely to em­phas­ize the Scot­tish di­men­sion of fam­ous men rather than their at­tach­ment to Trin­ity Col­lege. Most im­port­antly, his mis­sion to erect a statue of Max­well, denied in Trin­ity, was suc­cess­ful in Ed­in­burgh and it stands prom­in­ently in George Street, close to Max­well’s ori­gin­al home (Fig­ure 7).

Later life

Apart from in­teg­rat­ing in­to the so­cial and aca­dem­ic life of Ed­in­burgh, there were sev­er­al oth­er activ­it­ies that Mi­chael en­gaged in after leav­ing Cam­bridge. There was the Or­der of Mer­it, tra­di­tion­ally awar­ded to re­tir­ing Pres­id­ents of the Roy­al So­ci­ety. He en­joyed the reg­u­lar lunches with the Queen and oth­er mem­bers of the Or­der. There were the Chan­cel­lor­ship of the Uni­versity of Leicester from 1995 to 2012, nu­mer­ous hon­or­ary de­grees and for­eign mem­ber­ship of academies provid­ing yet more scope for speeches. One activ­ity he par­tic­u­larly en­joyed was the Heidel­berg Laur­eates For­um, where he could talk with en­thu­si­asm and en­cour­age­ment to young sci­ent­ists, of­fer­ing ad­vice and in­spir­a­tion. He was a won­der­ful ra­con­teur at this stage, re­gal­ing young and old with stor­ies, stop­ping just short of in­dis­cre­tion. Des­pite oc­ca­sion­al health is­sues, Mi­chael con­tin­ued to travel abroad and he ad­dressed the last In­ter­na­tion­al Con­gress of Math­em­aticians in Rio de Janeiro just a few months be­fore he died.

There was also a re­turn to his roots in the Middle East. The civil war in Le­ban­on ended in 1990 and Mi­chael went back at reg­u­lar in­ter­vals. The Amer­ic­an Uni­versity of Beirut, foun­ded in 1866, was the premi­er uni­versity in the re­gion for many years. Since 1887 the lan­guage of in­struc­tion was Eng­lish and it was there that Mi­chael’s grand­fath­er stud­ied medi­cine in the 1890s, in­tro­du­cing the Eng­lish lan­guage in­to the fam­ily. Sev­er­al cous­ins also taught there in the twen­ti­eth cen­tury, and gradu­ally Mi­chael be­came more in­volved. Even­tu­ally the Cen­ter for Ad­vanced Math­em­at­ics was set up with Mi­chael as chair­man of its ad­vis­ory board. Later, the Si­mons Found­a­tion made a dona­tion of two mil­lion dol­lars to es­tab­lish a chair in his name. Con­tinu­ing this as­so­ci­ation, the Lon­don Math­em­at­ic­al So­ci­ety has re­cently set up the Atiyah UK–Le­ban­on Fel­low­ship scheme in his memory.

Lily Atiyah died on 13 March 2018 at the age of 90, Mi­chael on 11 Janu­ary 2019. He is sur­vived by two sons, Dav­id and Robin. His eld­est son, John, died in 2002 while on a walk­ing hol­i­day in the Pyren­ees.

Awards and recognition

1962 Fel­low of the Roy­al So­ci­ety
1966 Fields Medal
1968 Roy­al Medal
1980 De Mor­gan Medal, Lon­don Math­em­at­ic­al So­ci­ety
1981 Ant­o­nio Fel­trinelli Prize, Ac­ca­demia dei Lincei
1983 Knight Bach­el­or
1985 Fel­low, Roy­al So­ci­ety of Ed­in­burgh
1987 King Fais­al Prize
1988 Co­pley Medal
1992 Or­der of Mer­it
1993 Ben­jamin Frank­lin Medal, Amer­ic­an Philo­soph­ic­al So­ci­ety
1993 Nehru Cen­ten­ary Medal, In­di­an Na­tion­al Sci­ence Academy
1993 Hon­or­ary Fel­low, Roy­al Academy of En­gin­eer­ing
1993 Com­mand­er of the Or­der of the Ce­dars, Le­ban­on
1996 Free­dom of the City of Lon­don
1997 Or­der of An­dres Bello, Venezuela
1999 Hon­or­ary Fel­low, Fac­ulty of Ac­tu­ar­ies
2001 Hon­or­ary Fel­low, Academy of Med­ic­al Sci­ences
2004 Abel Prize
2005 Or­der of Mer­it (Gold) Le­ban­on
2008 Pres­id­ent’s Medal, In­sti­tute of Phys­ics
2010 Grande Médaille, French Academy of Sci­ences
2010 Na­tion­al Or­der of Sci­entif­ic Mer­it of Brazil
2011 Grand Of­fi­ci­er Légion d’Hon­neur
2012 Fel­low, Amer­ic­an Math­em­at­ic­al So­ci­ety

Foreign membership

1969 Amer­ic­an Academy of Arts and Sci­ences
1972 Roy­al Swedish Academy
1977 Akademie Leo­pold­ina
1978 US Na­tion­al Academy of Sci­ences
1978 Académie des Sci­ences
1979 Roy­al Ir­ish Academy
1983 Third World Academy of Sci­ence
1992 Aus­trali­an Academy of Sci­ence
1992 Ukrain­i­an Academy of Sci­ence
1993 In­di­an Na­tion­al Sci­ence Academy
1994 Rus­si­an Academy of Sci­ence
1996 Geor­gi­an Academy of Sci­ence
1998 Amer­ic­an Philo­soph­ic­al So­ci­ety
1997 Venezuelan Academy of Sci­ence
1999 Ac­ca­demia dei Lincei
1999 Chinese Academy of Sci­ence
2001 Czechoslov­akia Uni­on of Math­em­at­ics
2001 Nor­we­gi­an Academy of Sci­ence and Let­ters
2001 Mo­scow Math­em­at­ic­al So­ci­ety
2002 Roy­al Span­ish Academy of Sci­ence
2008 Le­banese Academy of Sci­ences

Honorary degrees

1968 Bonn Uni­versity
1969 Uni­versity of War­wick
1979 Uni­versity of Durham
1981 Uni­versity of St. An­drews
1983 Uni­versity of Chica­go
1983 Trin­ity Col­lege, Dub­lin
1984 Uni­versity of Cam­bridge
1984 Uni­versity of Ed­in­burgh
1985 Uni­versity of Lon­don
1985 Uni­versity of Es­sex
1987 Uni­versity of Ghent
1990 Uni­versity of Read­ing
1990 Uni­versity of Hel­sinki
1991 Uni­versity of Leicester
1992 Rut­gers Uni­versity
1992 Uni­versity of Sala­manca
1993 Uni­versity of Montreal
1993 Uni­versity of Wales
1993 Uni­versity of Wa­ter­loo
1994 Le­banese Uni­versity
1994 Queen’s Uni­versity, King­ston, Canada
1994 Uni­versity of Keele
1994 Uni­versity of Birm­ing­ham
1995 Open Uni­versity
1996 UM­IST
1996 Chinese Uni­versity of Hong Kong
1997 Brown Uni­versity
1998 Uni­versity of Ox­ford
1998 Charles Uni­versity, Prague
1999 Heri­ot-Watt Uni­versity
2001 Na­tion­al Uni­versity of Mex­ico
2004 Amer­ic­an Uni­versity of Beirut
2005 Uni­versity of York
2006 Har­vard Uni­versity
2007 Scuola Nor­male, Pisa
2008 Poly­tech­nic Uni­versity of Cata­lonia

Acknowledgements

A prime source for this mem­oir was an ex­ten­ded series of in­ter­views, with my­self as fa­cil­it­at­or, which took place while Mi­chael was Mas­ter of Trin­ity; these are avail­able on­line [10]. They also formed the basis of a bio­graphy in [e4], wherein oth­er re­col­lec­tions can be found. Memor­ies from a more re­cent peri­od are in [e6]. The col­lec­ted works con­tain Mi­chael’s com­ments fo­cused on the back­ground to the math­em­at­ic­al pa­pers. I also wish to thank Dav­id Atiyah and par­ti­cipants at the two me­mori­al events in Ed­in­burgh and Cam­bridge for sev­er­al en­light­en­ing con­tri­bu­tions.

Nigel Hitchin stud­ied math­em­at­ics in Ox­ford, gain­ing a D.Phil. in 1972. He worked with Mi­chael Atiyah in Prin­ceton and Ox­ford, in par­tic­u­lar on vari­ous as­pects of gauge the­ory in pure math­em­at­ics. After 1990 he held chairs at War­wick and Cam­bridge be­fore re­turn­ing to Ox­ford as Sa­vil­ian Pro­fess­or of Geo­metry in 1997. His work links to­geth­er dif­fer­en­tial geo­metry, al­geb­ra­ic geo­metry and the equa­tions of the­or­et­ic­al phys­ics. Since 2016 he has been an emer­it­us pro­fess­or.

Works

[1]M. F. Atiyah: “A note on the tan­gents of a twis­ted cu­bic,” Proc. Cam­bridge Philos. Soc. 48 (1952), pp. 204–​205. MR 0048079 Zbl 0046.​14604 article

[2]M. F. Atiyah: “Com­plex fibre bundles and ruled sur­faces,” Proc. Lon­don Math. Soc. (3) 5 (1955), pp. 407–​434. MR 0076409 Zbl 0174.​52804 article

[3]M. F. Atiyah and G. B. Segal: “The in­dex of el­lipt­ic op­er­at­ors, II,” Ann. Math. (2) 87 : 3 (1968), pp. 531–​545. Rus­si­an trans­la­tion pub­lished in Us­pehi Mat. Nauk 23:6(144) (1968). MR 0236951 Zbl 0164.​24201 article

[4] M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, I,” Ann. Math. (2) 87 : 3 (May 1968), pp. 484–​530. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 23:5(143). MR 236950 Zbl 0164.​24001 article

[5] M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, III,” Ann. Math. (2) 87 : 3 (May 1968), pp. 546–​604. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 24:1(145). MR 236952 Zbl 0164.​24301 article

[6] M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, V,” Ann. Math. (2) 93 : 1 (January 1971), pp. 139–​149. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 27:4(166) (1972). MR 279834 article

[7] M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, IV,” Ann. Math. (2) 93 : 1 (January 1971), pp. 119–​138. A Rus­si­an trans­la­tion was pub­lished in Us­pehi Mat. Nauk 27:4(166) (1972). MR 279833 Zbl 0212.​28603 article

[8]M. F. Atiyah and R. Bott: “The Yang–Mills equa­tions over Riemann sur­faces,” Philos. Trans. R. Soc. Lond., A 308 : 1505 (1983), pp. 523–​615. MR 702806 Zbl 0509.​14014 article

[9]M. Atiyah: The geo­metry and phys­ics of knots. Lezioni Lincee. Cam­bridge Uni­versity Press, 1990. These notes arise from lec­tures presen­ted in Florence un­der the aus­pices of the Ac­ca­demia dei Lincei. Rus­si­an trans­la­tion pub­lished as Geo­met­riya i fiz­ika uzlov (1995). See also Minicon­fer­ence on geo­metry and phys­ics (1989). MR 1078014 Zbl 0729.​57002 book

[10] N. Hitchin and M. Atiyah: Web of stor­ies, 1990–1997. Col­lec­tion of au­dio re­cord­ings: in­ter­views of Sir Mi­chael Atiyah by Nigel Hitchins dur­ing the lat­ter’s ten­ure as Mas­ter of Trin­ity Col­lege. misc