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Celebratio Mathematica

Michael F. Atiyah

Geometry

Mathematics and culture: Geometry in Oxford 1960–1990

by Nigel J. Hitchin

Background

When J. J. Sylvester, the Sa­vil­ian Pro­fess­or of Geo­metry at Ox­ford Uni­versity, died in 1897, re­search in geo­metry vir­tu­ally stopped. His suc­cessor, W. Esson, was more con­cerned with Uni­versity busi­ness and the re­form of teach­ing meth­ods, and when G. H. Hardy took the po­s­i­tion in 1920 all was ana­lys­is, con­tin­ued by E. C. Titch­marsh who held the post from 1932 to 1963. Titch­marsh’s let­ter of ap­plic­a­tion stip­u­lated that he would only ac­cept if he did not have to teach geo­metry, and, though Hardy did in­deed en­gage in his un­der­gradu­ate geo­met­ric­al du­ties, this was at least con­sist­ent with his view: “I do not claim to know any geo­metry, but I do claim to un­der­stand quite clearly what geo­metry is” [e10].

It fell to J. H. C. White­head, who be­came the Waynf­lete Pro­fess­or in 1947, to wear the mantle, though it was geo­metry in the guise of to­po­logy which changed the face of re­search in the Uni­versity. White­head, born in Madras on Novem­ber 11, 1904, had stud­ied with O. Veblen in Prin­ceton from 1929 un­til 1932 and ob­tained his Ph.D. there in 1930. To­geth­er, they wrote a found­a­tion­al book on dif­fer­en­tial geo­metry [e1], where they not­ably gave a prop­er defin­i­tion of a man­i­fold, in­clud­ing the Haus­dorff prop­erty. It is worth re­mem­ber­ing that Cartan, even in 1946, con­sidered that “the gen­er­al no­tion of man­i­fold is quite dif­fi­cult to define with pre­ci­sion,” and that vari­ous au­thors, Hodge and Weyl in­cluded, la­boured over it at this time. Un­der the in­flu­ence of Mar­ston Morse in Prin­ceton, White­head tackled glob­al ques­tions con­cern­ing the dif­fer­en­tial geo­metry of man­i­folds, in­clud­ing the now-stand­ard proof of the ex­ist­ence of con­vex neigh­bor­hoods. He took up a Fel­low­ship at Bal­liol Col­lege, Ox­ford, in 1932, but his in­terests by then had moved on to al­geb­ra­ic to­po­logy. Dif­fer­en­tial geo­metry in Ox­ford at the time was driv­en by the study of cos­mo­lo­gic­al mod­els, with E. A. Mil­ne and A. G. Walk­er, and this work was es­sen­tially loc­al.

White­head’s in­terest in to­po­logy began also in Prin­ceton, un­der the in­flu­ence of S. Lef­schetz. They wrote a joint pa­per to­geth­er, and this set him off in a dir­ec­tion which was to oc­cupy his math­em­at­ic­al life un­til his death in 1960. It was a failed at­tempt on the Poin­caré con­jec­ture in 1934 that com­mit­ted him to the sub­ject, and over the years he de­veloped rig­or­ous com­bin­at­or­i­al and al­geb­ra­ic meth­ods for at­tack­ing prob­lems of ho­mo­topy equi­val­ence; his name is now at­tached to many fun­da­ment­al con­struc­tions and con­cepts. He gathered many stu­dents to work with him on these pro­jects. Per­haps his ebul­li­ent char­ac­ter helped; P. J. Hilton asked him “What is al­geb­ra­ic to­po­logy, Henry?” and he answered “Don’t worry, Peter. You'll love it!” [e12]. A fur­ther at­trac­tion was the widely known fact that his in­terests were equally di­vided between beer, crick­et and math­em­at­ics [e10].

Dur­ing the Second World War, like many of his con­tem­por­ar­ies, White­head worked as a codebreak­er at Bletch­ley Park, but, after the war and par­tic­u­larly after be­ing ap­poin­ted to the Chair, he gathered around him a group of al­geb­ra­ic to­po­lo­gists, so that Ox­ford was an im­port­ant centre for the sub­ject. Hilton and I. M. James were two of his early stu­dents. In 1957 James was ap­poin­ted as Read­er, and a few years later M. Bar­ratt and C. T. C. Wall had short-term po­s­i­tions there. A re­flec­tion of the strength of the sub­ject is White­head’s ap­proach in the late 1950s to Robert Max­well, chair­man of Per­ga­mon Press, to start a new Ox­ford-based journ­al to be called To­po­logy [e7], still the lead­er in the field.

Michael Atiyah

When White­head died sud­denly in 1960 at the age of 55, there were 12 gradu­ate stu­dents in al­geb­ra­ic to­po­logy in Ox­ford, both his and those of Ioan James. The va­cant Waynf­lete Chair was soon filled by one of White­head’s former stu­dents, but one whose in­terests were now group-the­or­et­ic­al rather than to­po­lo­gic­al — Gra­ham Hig­man. There was one un­suc­cess­ful ap­plic­ant, the 32-year-old Cam­bridge math­em­atician M. F. Atiyah, who was nev­er­the­less offered a Read­er­ship as a con­sol­a­tion, which he ac­cep­ted par­tially to be re­lieved of his col­lege teach­ing in Cam­bridge. With­in two years, Titch­marsh had died and Mi­chael Atiyah be­came the Sa­vil­ian Pro­fess­or of Geo­metry, restor­ing the Chair to its ori­gin­al pur­pose.

Atiyah’s back­ground at the time was in al­geb­ra­ic geo­metry. He had been a stu­dent of W. V. D. Hodge in Cam­bridge, who pushed him to learn about vec­tor bundles and char­ac­ter­ist­ic classes, to­geth­er with all the new points of view and tech­niques that were emer­ging in those post­war years. There was the work of Serre, and Kodaira and Spen­cer, on sheaf co­homo­logy; Chern, Al­lendo­er­fer and Weil, on char­ac­ter­ist­ic classes and curvature; and Hodge’s older work on har­mon­ic in­teg­rals, giv­en a new rigour by the ana­lyt­ic­al meth­ods of Weyl and Kodaira. He had then spent time at the In­sti­tute for Ad­vanced Study in Prin­ceton, where he met R. Bott, F. Hirzebruch and I. M. Sing­er, J.-P. Serre and A. Borel. Un­til 1959, most of his pa­pers were in al­geb­ra­ic geo­metry, but then be­came more to­po­lo­gic­al. To­po­logy was all around, go­ing through a big peri­od, and, al­though he didn’t re­gard him­self as a to­po­lo­gist, he “just picked it up” [e19].

It was the work of Hirzebruch that shif­ted his at­ten­tion from al­geb­ra­ic geo­metry to to­po­logy, for he had linked the Riemann–Roch the­or­em to the to­po­lo­gic­al Chern num­bers, and was pro­du­cing, via the cobor­d­ism the­ory of R. Thom, com­bin­a­tions of char­ac­ter­ist­ic num­bers for dif­fer­en­ti­able man­i­folds, not just pro­ject­ive vari­et­ies, which were in­tegers. Bott’s peri­od­icity the­or­em, to­geth­er with ques­tions of James in to­po­logy and A. Grothen­dieck’s work on the gen­er­al Riemann–Roch the­or­em, all con­spired to give birth to \( K \)-the­ory [e19]:

I saw that by mix­ing all these things to­geth­er you ended up with some in­ter­est­ing to­po­lo­gic­al con­sequences, and be­cause of that we then thought it would be use­ful to in­tro­duce the to­po­lo­gic­al \( K \)-group as a form­al ap­par­at­us in which to carry this out.

It was a bold ap­proach, draw­ing in­spir­a­tion from the gen­er­al­it­ies of Grothen­dieck, but it in­tro­duced many new fea­tures, in par­tic­u­lar, the odd groups which do not ap­pear in al­geb­ra­ic geo­metry.

Be­fore mov­ing to Ox­ford, Atiyah had already, with his to­po­lo­gic­al hat on, giv­en sem­inars which had gen­er­ated great in­terest from White­head and James, and he ar­rived there es­sen­tially as a to­po­lo­gist. By 1963 he had nine re­search stu­dents, most of whom were con­sidered as al­geb­ra­ic to­po­lo­gists, though one was de­scribed as a dif­fer­en­tial geo­met­er — the Aus­trali­an math­em­atician G. B. Segal, sent from Cam­bridge by Hodge. He has de­scribed the at­mo­sphere around Atiyah at the time [e16]:

He would dir­ect his abound­ing en­ergy at each of us in turn. I re­mem­ber how in­spired I felt after each meet­ing, but on the whole we stu­dents used to hide from him, for if he ran in­to us in the cor­ridor and found that we hadn’t made much pro­gress with yes­ter­day’s sug­ges­tions he would pour forth a tor­rent of new lines for us to try. At the same time he al­ways left us feel­ing there was something worth­while we could do; however wrong were the ideas we came up with, he nev­er crushed us, but made our muddle seem like steps in the right dir­ec­tion.

The index theorem

The most im­port­ant work that Atiyah did, when he ar­rived in Ox­ford, was the for­mu­la­tion and proof of the in­dex the­or­em jointly with Sing­er, then at MIT. This con­trib­uted to his Fields Medal in 1966, and was re­cog­nized by the award of the 2004 Abel Prize to Atiyah and Sing­er. In one sense, it grew nat­ur­ally out of his work on \( K \)-the­ory [e19]:

There was a kind of zig­zag pat­tern back­wards and for­wards where dif­fer­ent ver­sions of the proof of the in­dex the­or­em and bits of \( K \)-the­ory helped each oth­er along, un­til in the end they were so mixed up I didn’t know wheth­er I was go­ing from left to right or right to left.

It was not only to­po­logy and ana­lys­is that were linked by the the­or­em, but ul­ti­mately al­geb­ra­ic geo­metry, num­ber the­ory, dif­fer­en­tial geo­metry and the­or­et­ic­al phys­ics!

A lin­ear op­er­at­or on a Hil­bert space is called Fred­holm if it has a fi­nite-di­men­sion­al ker­nel and coker­nel. The dif­fer­ence of these di­men­sions is called the in­dex, and, al­though the in­di­vidu­al di­men­sions may jump in a con­tinu­ous fam­ily, the dif­fer­ence is un­changed, and is thus a to­po­lo­gic­al in­vari­ant. The prob­lem of cal­cu­lat­ing this was posed by math­em­aticians in the school of I. Gel­fand, though Atiyah was at first un­aware of this. The Rus­si­ans had cal­cu­lated some simple ex­amples in terms of the to­po­lo­gic­al de­gree of a cer­tain map.

For Atiyah, by con­trast, the prob­lem arose from his vis­its to Bonn to talk with Hirzebruch. Hirzebruch’s work had shown that cer­tain com­bin­a­tions of char­ac­ter­ist­ic num­bers were ne­ces­sar­ily in­tegers. These had ori­gin­ally come from al­geb­ra­ic geo­metry where, with Hirzebruch’s tal­ent for cal­cu­la­tions, he had in­ter­preted them as al­tern­at­ing sums of di­men­sions of cer­tain sheaf co­homo­logy groups. Yet the num­bers were still in­tegers for al­most-com­plex man­i­folds — the purely to­po­lo­gic­al ver­sion of al­geb­ra­ic geo­metry where sheaf co­homo­logy made no sense. Not only that, but there were sim­il­ar res­ults for or­din­ary smooth man­i­folds. In par­tic­u­lar, Hirzebruch had shown that, for a man­i­fold whose Stiefel–Whit­ney class \( w_2(X) \) van­ishes, the par­tic­u­lar com­bin­a­tion of char­ac­ter­ist­ic classes called the \( \hat A \) genus is an in­teger. At the time, these subtle in­teg­ral­ity the­or­ems sat very nat­ur­ally in the lan­guage of \( K \)-the­ory, but there was no genu­ine ex­plan­a­tion for why they were true. As Atiyah com­men­ted, “We had the an­swer; we didn’t know what the prob­lem was” [e19].

When Atiyah ar­rived in Ox­ford, des­pite his pres­ti­gi­ous chair, there was very little money to in­vite people to stay. The few pounds ne­ces­sary to in­vite Serre across the Chan­nel were only be­grudgingly gran­ted. But in Janu­ary 1962, Atiyah’s first aca­dem­ic year in Ox­ford, Sing­er de­cided to spend time there. It was fam­ily con­sid­er­a­tions that forced him to change his pre­vi­ous sab­bat­ic­al plans, but he re­membered their friend­ship from the In­sti­tute in Prin­ceton in 1955 and called to see if he could come on his own money. He was of course wel­come, and then [e15]:

…on my second day at the Maths In­sti­tute you walked up to the fourth floor of­fice where I was warm­ing my­self by the elec­tric heat­er. After the usu­al form­al­it­ies, you asked “Why is the genus an in­teger for spin man­i­folds?”

“What’s up, Mi­chael? You know the an­swer much bet­ter than I.”

“There’s a deep­er reas­on,” you said.

By March, they had found a can­did­ate for what the an­swer must be — the in­dex of the Dir­ac op­er­at­or. This was an el­lipt­ic op­er­at­or only defined on man­i­folds which sat­is­fied the spin con­di­tion \( w_2(X)=0 \), and so fit­ted in with Hirzebruch’s the­or­em. In a way, they had re­dis­covered the op­er­at­or since phys­i­cists were already fa­mil­i­ar with it, and in fact Atiyah had at­ten­ded Dir­ac’s lec­tures in Cam­bridge as a stu­dent but did not fol­low the work very closely. In any case, there was a huge dif­fer­ence between the Eu­c­lidean sig­na­ture of Rieman­ni­an geo­metry and the Lorent­zi­an sig­na­ture of re­lativ­ity. Nowadays phys­i­cists are quite happy to work in the Rieman­ni­an do­main, but this was not the case in the 1950s.

The easi­est ex­ample of the in­dex the­or­em is the Gauss–Bon­net the­or­em, which ex­presses the Euler char­ac­ter­ist­ic of a com­pact man­i­fold as a curvature in­teg­ral which real­izes a to­po­lo­gic­al char­ac­ter­ist­ic class, the Euler class of the tan­gent bundle. The Fred­holm op­er­at­or here is \( d+d^* \), where \( d \) is the usu­al ex­ter­i­or de­riv­at­ive and \( d^* \) its form­al ad­joint, and one con­siders it map­ping even forms to odd ones. The ker­nel con­sists of the har­mon­ic even forms, and the coker­nel, which is the ker­nel of its ad­joint, of the har­mon­ic odd forms. Us­ing Hodge’s the­or­em, the in­dex is the sum of the even Betti num­bers minus the sum of the odd ones, which is the Euler char­ac­ter­ist­ic. Of course, \( d+d^* \) is an un­boun­ded op­er­at­or, so a con­sid­er­able amount of ana­lys­is is needed even to re­ph­rase this the­or­em rig­or­ously as an in­dex prob­lem.

An­oth­er ex­ample is the sig­na­ture op­er­at­or on dif­fer­en­tial forms. In­stead of de­com­pos­ing the dif­fer­en­tial forms in­to odd and even ones, there is an­oth­er way to do it in \( 4k \) di­men­sions, us­ing the Hodge star op­er­at­or of a Rieman­ni­an met­ric. The same op­er­at­or \( d+d^* \) maps one space to an­oth­er, and now the in­dex is the sig­na­ture of the man­i­fold: in \( 4k \) di­men­sions, the in­ter­sec­tion form on the middle-di­men­sion­al co­homo­logy is sym­met­ric, and the in­dex is the sig­na­ture of this form. Hirzebruch had ex­pressed this in­vari­ant as a par­tic­u­lar com­bin­a­tion of Pontry­agin num­bers in his 1953 pa­per.

Sing­er brought ana­lyt­ic­al ex­pert­ise in­to play, as well as dif­fer­en­tial geo­metry know-how, but they also had a lead when S. Smale passed through Ox­ford on his way back from Mo­scow. He told the pair about Gel­fand’s work on in­dices of op­er­at­ors, and fol­low­ing up on this Atiyah and Sing­er were able to con­sult the ana­lyt­ic­al pa­pers of M. S. Agran­ovich, A. S. Dyn­in and R. See­ley, and also rely on friends such as L. Niren­berg and L. Hör­mander. As Atiyah sub­sequently ad­mit­ted [e19],

I did ac­tu­ally make at­tempts to … read a few ser­i­ous books. They were the first books I'd ac­tu­ally tried to read since I was a stu­dent. After you've ceased be­ing a stu­dent you don’t usu­ally read text­books; you learn what you need to on the hoof.

The ad­vant­age they had over the Rus­si­ans was that they were con­cen­trat­ing on a par­tic­u­lar op­er­at­or, the Dir­ac op­er­at­or, and they knew what the an­swer should be. They also knew the an­swer for re­lated op­er­at­ors, such as the sig­na­ture op­er­at­or and the Dol­beau­lt op­er­at­ors on a com­plex man­i­fold. The in­dex the­or­em for each of these cases would give new proofs of the Hirzebruch sig­na­ture the­or­em and the Riemann–Roch the­or­em, re­spect­ively. Per­haps more im­port­antly, they had seen the prob­lem in the con­text of \( K \)-the­ory, and that was where the link really lay — the in­dex of an el­lipt­ic op­er­at­or only de­pends on its highest-or­der term, the prin­cip­al sym­bol, and this im­me­di­ately defines a \( K \)-the­ory class.

Atiyah and Sing­er dis­cussed the prob­lem in­tens­ively in Ox­ford, walk­ing to­geth­er in the Uni­versity Parks, or up the River Thames to the Trout Inn. A proof was fi­nally com­pleted in the Au­tumn of the same year as Atiyah vis­ited Har­vard, and the res­ults were presen­ted in a sem­in­ar run by Bott and Sing­er, sub­sequently ex­pounded in de­tail in the Prin­ceton sem­in­ar [e2]. The proof was based on Hirzebruch’s 1953 proof of the sig­na­ture the­or­em: both the sig­na­ture and the char­ac­ter­ist­ic num­ber it is sup­posed to equal are the same for two ori­ented man­i­folds which form the bound­ary of a third — such a pair is said to be cobord­ant. So, prov­ing the the­or­em can be re­duced to the eval­u­ation of spe­cial cases which gen­er­ate the cobor­d­ism classes. For the in­dex prob­lem, one has to de­scribe its change un­der cobor­d­ism, and this re­quired an ex­ten­sion of el­lipt­ic bound­ary-value tech­niques to sin­gu­lar in­teg­ral op­er­at­ors.

The proof was not ideal, though [e19]:

What was wrong with the first proof, be­sides be­ing … con­cep­tu­ally a bit un­at­tract­ive, be­cause you veri­fy things, was that it didn’t in­clude some gen­er­al­iz­a­tions that we had in mind.

These in­cluded the equivari­ant case, where a group acts on the man­i­fold, and the case of real op­er­at­ors rather than com­plex ones. In these cases, the in­dex is not simply an in­teger — it takes val­ues in the rep­res­ent­a­tion ring of the group in the first case, and is an in­teger mod 2 in the second. The case of fam­il­ies of el­lipt­ic op­er­at­ors was not dealt with in the first proof, and here the in­dex is an ele­ment in the \( K \)-the­ory of the space para­met­riz­ing the fam­ily.

So, over the next five years, Atiyah and Sing­er pro­duced new proofs, this time based on Grothen­dieck’s proof of the Riemann–Roch the­or­em us­ing em­bed­dings and pro­jec­tions, and in a series of An­nals pa­pers they dealt with all the oth­er rami­fic­a­tions, in­clud­ing fixed-point the­or­ems, with many ap­plic­a­tions. Their work also fed back­wards in­to the purely to­po­lo­gic­al world — it be­came im­me­di­ately ap­par­ent that the \( K \)-the­ory of a space \( X \) was ac­tu­ally the set of ho­mo­topy classes of maps from \( X \) to the space of Fred­holm op­er­at­ors. At an­oth­er stage, the au­thors ar­rived at Bott’s peri­od­icity the­or­em for the ho­mo­topy groups of the clas­sic­al groups, without be­ing aware that they had used it, and thereby dis­covered an ele­ment­ary proof of this res­ult. And then, the fixed-point for­mula gave a proof of the Weyl char­ac­ter for­mula for rep­res­ent­a­tions of a com­pact Lie group. By fo­cus­ing ini­tially on that single prob­lem of the Dir­ac op­er­at­or, they had opened up a the­ory which lay at the cross­roads of an enorm­ous num­ber of dis­cip­lines. In fact, it turned out that, for the in­dex the­or­em, the Dir­ac op­er­at­or is fun­da­ment­al, and coup­ling it to an ar­bit­rary vec­tor bundle yields all pos­sible in­dices.

The Abel Prize cita­tion for this work says it all:

Atiyah and Sing­er are re­ceiv­ing the prize for hav­ing dis­covered and proved the in­dex the­or­em, which links to­geth­er to­po­logy, geo­metry and ana­lys­is, and for play­ing an ex­traordin­ary role in build­ing new bridges between math­em­at­ics and the­or­et­ic­al phys­ics. The in­dex the­or­em was proved in the early 1960s and is one of the most im­port­ant math­em­at­ic­al res­ults of the twen­ti­eth cen­tury. It has had an enorm­ous im­pact on the fur­ther de­vel­op­ment of to­po­logy, dif­fer­en­tial geo­metry and the­or­et­ic­al phys­ics. The the­or­em also provides us with a glimpse of the beauty of math­em­at­ic­al the­ory, in that it ex­pli­citly demon­strates a deep con­nec­tion between math­em­at­ic­al dis­cip­lines that ap­pear to be com­pletely sep­ar­ate. Sir Mi­chael Atiyah and Is­ad­ore Sing­er have demon­strated math­em­at­ics at its very best and are worthy win­ners of the Abel Prize.

The five An­nals pa­pers began to ap­pear in 1968, but by 1969 Atiyah had left Ox­ford for Prin­ceton, to be­come a per­man­ent mem­ber at the In­sti­tute for Ad­vanced Study. However, in 1973, and hold­ing a Roy­al So­ci­ety Re­search Pro­fess­or­ship which freed him from ad­min­is­trat­ive du­ties, he re­turned. That same year, an­oth­er Chair was filled in Ox­ford — Ro­ger Pen­rose be­came the Rouse Ball Pro­fess­or of Math­em­at­ics.

Roger Penrose

Ro­ger Pen­rose, born on Au­gust 8, 1931, had been an un­der­gradu­ate at Uni­versity Col­lege, Lon­don, but was a con­tem­por­ary of Mi­chael Atiyah as a gradu­ate stu­dent in Cam­bridge. For one year, they had shared the same su­per­visor, Hodge, but then Pen­rose moved over to work with J. A. Todd, Atiyah’s dir­ect­or of un­der­gradu­ate stud­ies. So, while Atiyah was learn­ing about har­mon­ic forms and dif­fer­en­tial geo­metry, Pen­rose’s work was con­cerned with clas­sic­al in­vari­ant the­ory. He in­ven­ted a dia­gram­mat­ic way of keep­ing track of in­dices in tensori­al cal­cu­la­tions, not far re­moved from Feyn­man dia­grams. In fact, this pictori­al ana­lys­is of math­em­at­ics is a con­stant fea­ture of his work: some of the il­lus­tra­tions for his latest book were ex­hib­ited at the Roy­al Academy Sum­mer Ex­hib­i­tion in Lon­don in 2004!

Where­as geo­metry was his form­al top­ic of study (and con­tin­ued to the ex­tent that he wrote, with White­head and E. C. Zee­man in 1960, a pa­per about em­bed­ding man­i­folds in Eu­c­lidean space), un­der the tu­tel­age of D. Sciama in Cam­bridge he be­came more and more in­ter­ested in phys­ics. By the time he went to Ox­ford, he had made his name with S. W. Hawk­ing on sin­gu­lar­ity the­or­ems in re­lativ­ity — in 1965, us­ing to­po­lo­gic­al meth­ods, he found con­di­tions, which he called the ex­ist­ence of a trapped sur­face, that proved that a sin­gu­lar­ity must oc­cur in a grav­it­a­tion­al col­lapse. But he was also de­vel­op­ing an ap­proach to the equa­tions of math­em­at­ic­al phys­ics which few un­der­stood and was what he termed twis­tor the­ory.

Giv­ing a talk at Prin­ceton Uni­versity, just be­fore both he and Atiyah moved to Ox­ford, he spoke to his former class­mate, down­play­ing the to­po­lo­gic­al as­pects of the sin­gu­lar­ity the­or­ems. It seemed as if they would have little in com­mon in Ox­ford. But Free­man Dys­on had heard of twis­tors, and in Prin­ceton he had told Atiyah: “Twis­tors are a mys­tery … but per­haps you will un­der­stand them” [e9].

In Ox­ford, both Atiyah and Pen­rose quickly ac­cu­mu­lated gradu­ate stu­dents — something that Atiyah missed in Prin­ceton — and con­tin­ued their re­search, ap­par­ently pur­su­ing dif­fer­ent themes. Fairly soon, however, they got talk­ing about twis­tors. The Klein cor­res­pond­ence between lines in three-di­men­sion­al pro­ject­ive space and points of a four-di­men­sion­al quad­ric, they both knew from their Cam­bridge days, but Pen­rose’s use of com­plic­ated con­tour in­teg­rals to pro­duce solu­tions to zero rest-mass field equa­tions was very new to Atiyah. Pen­rose pa­tiently ex­plained the rules whereby one could change the in­teg­rand or the con­tour. And then [e9],

It was not long be­fore it dawned on me that Ro­ger was es­sen­tially strug­gling with sheaf co­homo­logy, but did not real­ize it. Once this was poin­ted out, Ro­ger and his stu­dents be­came fer­vent con­verts. After a few private sem­inars in my study they really took off. With­in a short peri­od of time Ro­ger’s group were more ex­pert with sheaf co­homo­logy than I had ever been.

Pen­rose’s new view­point was mo­tiv­ated by the de­sire to put the com­plex num­bers in­to the the­ory at the very be­gin­ning. There were a num­ber of reas­ons: one was that quantum the­ory de­mands the use of com­plex num­bers; an­oth­er was the ob­ser­va­tion that the ce­les­ti­al sphere is in fact a com­plex ob­ject. It is nat­ur­ally the Riemann sphere — the com­plex num­bers to­geth­er with a point at in­fin­ity — but, more im­port­antly, two re­lativ­ist­ic ob­serv­ers re­late their views of the ce­les­ti­al sphere by a holo­morph­ic trans­form­a­tion, or com­plex Möbi­us trans­form­a­tion. Also, he hoped that re­pla­cing the points of space­time by something more fun­da­ment­al might be a gate­way to non­loc­al­ity.

To de­scribe twis­tor the­ory, one be­gins with a com­plex four-di­men­sion­al vec­tor space \( T \) with a Her­mitian form of sig­na­ture \( (2,2) \). The vec­tors in \( T \) are the twis­tors. The two-di­men­sion­al sub­spaces of \( T \), or equi­val­ently the lines in the three-di­men­sion­al pro­ject­ive space \( P(T) \), are clas­sic­ally para­met­rized by the points of a four-di­men­sion­al pro­ject­ive quad­ric. The unit­ary group \( \mathit{SU}(2,2) \) of the Her­mitian form acts on the quad­ric as the group \( \mathit{SO}(4,2) \), which is the con­form­al group of Minkowski space. In fact, the equa­tion of the quad­ric has real coef­fi­cients, and the real points can be iden­ti­fied with a con­form­al com­pac­ti­fic­a­tion of Minkowski space.

What is achieved here is that the points of space­time are de­scribed by com­plex pro­ject­ive lines in com­plex pro­ject­ive three-space, and the mir­acle of twis­tor the­ory is that the ob­jects and con­struc­tions that one nat­ur­ally stud­ies in al­geb­ra­ic geo­metry trans­form in­to some of the well-known equa­tions of math­em­at­ic­al phys­ics. This is where those sheaf co­homo­logy groups come in. The Her­mitian form di­vides \( P(T) \) in­to two halves, \( P(T)^{\pm} \), and an ele­ment of the sheaf co­homo­logy group \( H^1(P(T)^{+},{\mathcal O}(-n)) \) defines a solu­tion to a field equa­tion of spin \( (n-2)/2 \). The cases \( n=3 \) or \( n=1 \) give the Dir­ac equa­tions with the two pos­sible chir­al­it­ies — the Rieman­ni­an ver­sion of which was so fun­da­ment­al to Atiyah’s work on the in­dex the­or­em. The case \( n=2 \) is the wave equa­tion.

The simplest case is the stat­ic solu­tion to the wave equa­tion — a har­mon­ic func­tion in three di­men­sions. Here, Pen­rose’s con­tour in­teg­ral in­volves a holo­morph­ic func­tion \( f(w,z) \) of two vari­ables, where \( z\ne 0 \). This may be changed by adding on a holo­morph­ic func­tion of \( w \) and \( z \), or \( z^{-2} \) times a holo­morph­ic func­tion of \( w \) and \( z^{-1} \). This pro­cess is es­sen­tially the Čech defin­i­tion of a sheaf co­homo­logy group. To ob­tain the value of the cor­res­pond­ing har­mon­ic func­tion \( \varphi \), the con­tour in­teg­ral is: \[ \varphi(x_1,x_2,x_3)=\int_C f\bigl((x_1+ix_2)z^2+2x_3z-(x_1-ix_2),\,z\bigr)\,dz . \] This for­mula was ac­tu­ally giv­en by E. T. Whit­taker in 1903, and the wave-equa­tion equi­val­ent a year later by H. Bate­man, but the new, gen­er­al point of view was to gen­er­ate means of solv­ing equa­tions far more dif­fi­cult than these.

This use of com­plex ana­lys­is was very at­tract­ive to Pen­rose [e13]:

I feel strongly that com­plex num­bers and com­plex ana­lyt­ic struc­tures are fun­da­ment­al for the way that the phys­ic­al world be­haves. I sup­pose that part of my reas­on for this goes way back to my math­em­at­ic­al train­ing. When I first learnt about com­plex ana­lys­is at uni­versity in Lon­don I was totally gob-smacked — it just seemed to me an in­cred­ible sub­ject.

The fact that Pen­rose’s stu­dents were able to ab­sorb the tech­niques of ho­mo­lo­gic­al al­gebra — ex­act co­homo­logy se­quences, spec­tral se­quences and so on — which the sheaf-co­homo­logy in­ter­pret­a­tion of twis­tor in­teg­rals opened up, was due to the co­hes­ive nature of the twis­tor-the­ory group. There were many gradu­ate stu­dents, but they shared a com­mon eth­os. Reg­u­lar all-day meet­ings were held on Fri­days, there was a privately cir­cu­lated hand-writ­ten Twis­tor news­let­ter, and a “Prob­lem Lib­rary” where stu­dents could take out a ques­tion for two weeks and re­turn it if they couldn’t solve it in the al­lot­ted time. New ideas very quickly spread through the group this way.

However suc­cess­ful the con­tour in­teg­rals were, Pen­rose’s hopes for twis­tor the­ory were more in Gen­er­al Re­lativ­ity and Ein­stein’s equa­tions. By 1975, he had par­tially achieved, with Atiyah’s help, this goal. His idea was to take an open set in the stand­ard three-di­men­sion­al pro­ject­ive space \( P(T) \) and de­form it as a com­plex man­i­fold, re­tain­ing the pro­ject­ive lines. Since the in­ter­sec­tion prop­er­ties of the lines in pro­ject­ive space de­term­ined the con­form­al struc­ture of Minkowski space, one should get a de­formed con­form­al struc­ture. The key math­em­at­ic­al in­gredi­ent was the as­ser­tion that the de­form­a­tion would keep the pro­ject­ive lines, and for this Atiyah provided the ref­er­ences to Kodaira’s pa­pers of the early 1960s which gave sheaf-the­or­et­ic con­di­tions for this to work. In the pa­per [e3], Pen­rose pro­duces a con­struc­tion of com­plexi­fied solu­tions to the Ein­stein va­cu­um equa­tions this way.

In terms of the ul­ti­mate goal, there was a catch. The met­rics have self-dual Weyl tensor, but in a Lorent­zi­an space­time the two halves of the Weyl tensor are com­plex-con­jug­ate, so self-du­al­ity im­plies flat­ness: only Minkowski space fits the bill, and the ob­vi­ous link with re­lativ­ity was lost. There was one use of this in Gen­er­al Re­lativ­ity, though, us­ing the \( H \)-space point of view of E. T. New­man on asymp­tot­ic space­times, but in many re­spects Pen­rose’s “non­lin­ear grav­iton” seemed a purely math­em­at­ic­al con­struc­tion with no ap­plic­a­tion. It was nev­er­the­less a solu­tion to a highly non­lin­ear dif­fer­en­tial equa­tion achieved by geo­met­ric­al means, and twis­tor the­ory would provide more of these.

Instantons

By 1976, it was not just Pen­rose’s work which led Atiyah to pay more at­ten­tion to what phys­i­cists were do­ing. Through Sing­er, he had learnt that their work on the in­dex the­or­em for the Dir­ac op­er­at­or was in­tim­ately tied up with ques­tions about an­om­alies in quantum field the­or­ies. In fact, there had been a par­al­lel de­vel­op­ment: the­or­et­ic­al phys­i­cists across the cor­ridor from Sing­er’s of­fice in MIT had been es­sen­tially re-de­riv­ing the in­dex the­or­em, but in the op­pos­ite dir­ec­tion from the math­em­aticians. Atiyah and Sing­er had star­ted with the in­teger in­dex and worked through dif­fer­ent ana­lyt­ic­al ap­proaches, the cobor­d­ism proof, the Riemann–Roch proof and then, in the early 1970s, the heat-equa­tion proof (to deal with non­loc­al bound­ary-value prob­lems); the phys­i­cists had star­ted at the oth­er end.

By the end of 1976, Sing­er was aware of the in­terest of phys­i­cists in the Yang–Mills equa­tions and, in par­tic­u­lar, the fi­nite-ac­tion solu­tions on \( \mathbb R^4 \), called Yang–Mills in­stan­tons. These were equa­tions for con­nec­tions on vec­tor bundles, something that dif­fer­en­tial geo­met­ers had been fa­mil­i­ar with for years, but not these equa­tions nor the no­tion of gauge equi­val­ence. In early 1977, Sing­er came to Ox­ford again for sev­er­al months, and began to give weekly lec­tures on Yang–Mills the­ory: the ba­sic prob­lem, the ex­amples of G. 't Hooft and R. Jackiw, the group of gauge trans­form­a­tions, con­form­al in­vari­ance and trans­lat­ing the prob­lem to the sphere.

At the same time, Richard Ward, a South Afric­an stu­dent of Pen­rose, had been look­ing at these same equa­tions from the twis­tor point of view. He was sched­uled to give a sem­in­ar on his work; Pen­rose sug­ges­ted to Atiyah that he might be in­ter­ested, and in­deed [e10]:

…at the last minute I de­cided to go. You know, you go to a lot of sem­inars, and I sup­pose ninety per­cent of the time you get something out of them, and some­times it’s a bit bor­ing, but every now and then something really in­ter­est­ing hap­pens … I was really ter­ribly ex­cited by what I had heard, and wanted to un­der­stand it in my own lan­guage, and then I went away and spent a hard week­end try­ing to fol­low through the im­plic­a­tions … and then I saw how one could use it for get­ting glob­al solu­tions on the four-sphere.

Ward’s in­ter­pret­a­tion of the self-dual Yang–Mills equa­tions was that com­plex solu­tions cor­res­pon­ded to holo­morph­ic vec­tor bundles on pro­ject­ive twis­tor space. Atiyah’s ex­cite­ment stemmed from vari­ous sources. One was that his first pa­per was about holo­morph­ic bundles, and his stu­dent R. Schwar­zen­ber­ger had writ­ten his thes­is about such bundles on pro­ject­ive space. But he was also aware that there was con­sid­er­able cur­rent work by G. Hor­rocks in New­castle and W. Barth in Er­lan­gen in this area. An­oth­er link re­lated to a dis­cus­sion Atiyah had had some time earli­er, with stu­dents of Pen­rose, on what twis­tor the­ory meant if you re­placed Minkowski space by Eu­c­lidean space, whose con­form­al com­pac­ti­fic­a­tion was the four-sphere. The pic­ture he had found was very simple: it meant con­sid­er­ing the four-di­men­sion­al com­plex vec­tor space \( T \) as a two-di­men­sion­al vec­tor space over the qua­ternions. A one-di­men­sion­al com­plex sub­space then gen­er­ates a one-di­men­sion­al qua­ternion­ic one, and so there is a pro­jec­tion map from \( P(T) \) to the qua­ternion­ic pro­ject­ive line, which is the four-sphere.

These facts fit­ted the in­stan­ton prob­lem very well, since the Yang–Mills con­nec­tion ex­tends by con­form­al in­vari­ance of the equa­tions to the sphere, and defines dir­ectly a holo­morph­ic struc­ture on its pull­back to the pro­ject­ive space. Con­versely, by Ward’s the­or­em, this gives back the con­nec­tion. The fi­nite-ac­tion bound­ary con­di­tions trans­late very neatly in­to the ex­ist­ence of a glob­al holo­morph­ic bundle on pro­ject­ive space. To­geth­er with Ward, Atiyah used a con­struc­tion of bundles due to Serre to get solu­tions to the equa­tions, but it was still not clear how gen­er­al these ex­amples were.

Sing­er’s sem­in­ar each week ab­sorbed these new points of view, and also one of the writer, con­cern­ing the in­fin­ites­im­al de­form­a­tions of the in­stan­ton equa­tions. This was a simple ap­plic­a­tion of the in­dex the­or­em, coupled to a dif­fer­en­tial-geo­met­ric van­ish­ing the­or­em de­pend­ent on the pos­it­ive scal­ar curvature of the sphere. It gave the ex­pec­ted di­men­sion of the mod­uli space for in­stan­tons with to­po­lo­gic­al charge \( k \) as \( 8k-3 \). As Sing­er then poin­ted out, an ad­apt­a­tion of Kur­an­ishi’s ar­gu­ments for mod­uli of com­plex struc­tures showed that there was a genu­ine smooth mod­uli space of this di­men­sion. The di­men­sion had also been cal­cu­lated in­de­pend­ently by the Rus­si­an phys­i­cist A. S. Schwartz. Since this di­men­sion was big­ger than the al­geb­ra­ic-geo­met­ric con­struc­tions to date, there clearly had to be an­oth­er way to ob­tain all in­stan­tons.

The al­geb­ra­ic geo­met­ers had already found, in a dif­fer­ent lan­guage, such a way, for the pre­vi­ous sum­mer Hor­rocks had pro­duced a lin­ear-al­gebra con­struc­tion of cer­tain holo­morph­ic bundles on pro­ject­ive spaces, and Barth had used this to show that, if \( E \) was any rank-2 holo­morph­ic bundle with van­ish­ing first Chern class on pro­ject­ive three-space, and sat­is­fied a cer­tain sheaf-co­homo­lo­gic­al con­di­tion, then one could con­struct it us­ing Hor­rocks' con­crete tech­niques. The con­di­tion was the van­ish­ing of the sheaf co­homo­logy group writ­ten as \( H^1(P^3,E(-2)) \). For the holo­morph­ic bundle com­ing from an in­stan­ton, this space can be in­ter­preted, by a modi­fic­a­tion of Pen­rose’s ori­gin­al con­tour in­teg­rals, as the space of solu­tions to a cer­tain Laplace-type equa­tion on the sphere, and again the pos­it­iv­ity of the curvature of the sphere forces it to van­ish. Thus, us­ing Barth’s res­ult, every solu­tion to the self-dual Yang–Mills equa­tions on the sphere can be ex­pressed from lin­ear-al­geb­ra­ic data.

These pieces of the jig­saw puzzle were fi­nally as­sembled by Atiyah and the writer be­fore go­ing off to have lunch at St. Cath­er­ine’s Col­lege on Novem­ber 22, 1977. On our re­turn to the Math­em­at­ic­al In­sti­tute, we found a let­ter from Yu. Man­in giv­ing es­sen­tially the same con­struc­tion, with V. G. Drin­feld. A joint pa­per was pub­lished and the meth­od be­came known as the ADHM con­struc­tion of in­stan­tons.

This in­ter­ac­tion between math­em­at­ics and phys­ics was a sig­ni­fic­ant event, es­pe­cially for Atiyah [e6]:

Around this time I was in fact giv­ing lec­tures in many parts of the world on the geo­metry of gauge the­or­ies. I think it is fair to say that the pa­pers had caused quite a flurry of in­terest on the math­em­at­ics/phys­ics in­ter­face. It is re­por­ted that Polyakov had de­scribed (ADHM) as the first time ab­stract mod­ern math­em­at­ics had been of any use!

In time, most con­cepts in math­em­at­ics be­come ob­vi­ous. Nowadays, the ADHM con­struc­tion can be seen, in the form­al­ism of hy­per­kähler mo­ment maps, as a sort of Four­i­er trans­form of the ori­gin­al Yang–Mills equa­tions, and fits in­to a num­ber of such con­struc­tions. And here is a phys­i­cist’s view [e11]: “The ar­cane ADHM con­struc­tion of Yang–Mills in­stan­tons can be very nat­ur­ally un­der­stood in the frame­work of D-brane dy­nam­ics in string the­ory.” Or, as E. Wit­ten has writ­ten [e5],

Learn­ing about the ADHM con­struc­tion has served me well re­peatedly — es­pe­cially in 1995 when it helped in un­der­stand­ing the prob­lem of small in­stan­tons, and the be­ha­viour of Type I fivebranes.

Atiyah had met Wit­ten, then a Ju­ni­or Fel­low at Har­vard, in the Spring of 1977, and his sub­sequent in­ter­ac­tions with phys­ics were vir­tu­ally al­ways fed by con­ver­sa­tions between the two.

Simon Donaldson

In Oc­to­ber 1980, after a re­com­mend­a­tion that he was “the best stu­dent in ten years,” Si­mon Don­ald­son began his math­em­at­ic­al life in Ox­ford; six years later, he had won a Fields Medal.

He was born on Au­gust 20, 1957, and ob­tained his first de­gree at Pem­broke Col­lege, Cam­bridge. Of his ex­per­i­ences in Ox­ford at this time, he has said [e14]:

The early 1980s was a golden age for geo­metry in Ox­ford, or at least it seems so to me and prob­ably to all who were lucky to be a part of the group led by Atiyah at that time. This was a size­able group — among the fac­ulty were Graeme Segal, Nigel Hitchin, Bri­an Steer, Glenys Luke, George Wilson and (some­what later) Si­mon Sala­mon and Dan Quil­len. Re­search stu­dents in­cluded Frances Kir­wan, Mi­chael Mur­ray, Mi­chael Pen­ning­ton, Jacques Hur­tu­bise, John Roe (con­tem­por­ar­ies of the writer) and a little later, Yat-Sun Poon, Hen­rik Ped­er­sen, Peter Kron­heimer and Peter Braam — with in­ter­weav­ing re­search in­terests. There were also many in­ter­ac­tions with the equally large and act­ive group of math­em­at­ic­al phys­i­cists work­ing with Ro­ger Pen­rose. For us re­search stu­dents the weeks (at least dur­ing the short Ox­ford terms) re­volved around Atiyah’s “Geo­metry and Ana­lys­is” sem­in­ar, which met each Monday at 3pm … The most mem­or­able of these sem­inars were those giv­en by Atiyah him­self, which were in­vari­ably vir­tu­oso per­form­ances.

Since the flurry of in­terest over in­stan­tons, re­search in geo­metry was fol­low­ing a num­ber of dif­fer­ent paths. The year 1977 saw an­oth­er vis­it­or to Ox­ford, Raoul Bott, on his way back from a stay at the Tata In­sti­tute in Bom­bay. He had been ex­posed to work on the mod­uli spaces of stable bundles on Riemann sur­faces, a sub­ject vir­tu­ally in­ven­ted there some years earli­er. Talk­ing to Atiyah, he wondered if the ma­gic­al Yang–Mills the­ory would do any­thing for this prob­lem. The solu­tions to the Yang–Mills equa­tions were es­sen­tially trivi­al in two di­men­sions — flat con­nec­tions or con­nec­tions with con­stant curvature — but what they wanted to do was de­rive the known for­mu­las for the co­homo­logy of the mod­uli space of stable bundles by Morse-the­or­et­ic­al means us­ing the Yang–Mills func­tion­al. A key ob­ser­va­tion was the mo­ment map in­ter­pret­a­tion, in an in­fin­ite-di­men­sion­al set­ting. The space of all con­nec­tions on a bundle over a closed sur­face is form­ally a sym­plect­ic man­i­fold, and the ac­tion of the group of gauge trans­form­a­tions pre­serves the sym­plect­ic form. It has a mo­ment map, which is the curvature of the con­nec­tion, and so the mod­uli space of flat con­nec­tions is form­ally the quo­tient of the zero-set of the mo­ment map by the ac­tion of the group — a sym­plect­ic quo­tient, or re­duced-phase space, which in­her­its a sym­plect­ic form.

With a choice of com­plex struc­ture on the sur­face, this view­point be­comes an in­fin­ite-di­men­sion­al ana­logue of the ac­tion of a group on a pro­ject­ive vari­ety, and there it was known that sta­bil­ity of points un­der the ac­tion of the com­plex group was re­lated to the mo­ment map for its max­im­al com­pact sub­group. The fam­ous the­or­em of Narasim­han and Se­shadri from 1964, that a stable holo­morph­ic bundle on a Riemann sur­face has a nat­ur­al flat con­nec­tion, could now be seen in a very nat­ur­al light through the use of mo­ment maps. So, Atiyah, Bott and some of Atiyah’s stu­dents, not­ably Frances Kir­wan, were ap­ply­ing them­selves act­ively to fi­nite- and in­fin­ite-di­men­sion­al mo­ment-map prob­lems.

The writer, by con­trast, was in­ter­ested in de­vel­op­ing fur­ther the mani­fest­a­tions of twis­tor the­ory in Rieman­ni­an sig­na­ture. Hawk­ing and G. W. Gib­bons had in 1976 giv­en simple con­struc­tions of com­plete solu­tions to Ein­stein’s equa­tions in pos­it­ive-def­in­ite sig­na­ture, and these turned out to be self-dual, which meant that Pen­rose’s twis­tor meth­ods could be ap­plied to them — Gen­er­al Re­lativ­ity could not fully be­ne­fit from twis­tor the­ory, but Rieman­ni­an geo­met­ers could. The twis­tor spaces of Hawk­ing’s met­rics were easy to con­struct, and so what came in­to be­ing seemed at the time like a co­her­ent class of ob­jects, with plenty of ex­amples: one should look at four-man­i­folds with a self-dual con­form­al struc­ture and self-dual con­nec­tions over them. These were some sort of qua­ternion­ic ana­logues of Riemann sur­faces and holo­morph­ic line bundles. That Pen­rose’s twis­tor the­ory ap­plied to these meant that ul­ti­mately one was do­ing com­plex-ana­lyt­ic geo­metry, which put the area on a sol­id foot­ing, to­geth­er with a range of avail­able tech­niques.

There were, however, in­dic­a­tions in oth­er dir­ec­tions. One was the res­ult of C. Taubes [e4] about the ex­ist­ence of self-dual con­nec­tions on non-self-dual man­i­folds. The im­pact of this was in some ways com­par­able to the Hirzebruch in­teg­ral­ity the­or­ems that mo­tiv­ated the gen­er­al in­dex the­or­em. Taubes' ana­lyt­ic­al ex­ist­ence the­or­em came out in 1982, but pre­lim­in­ary ver­sions were cir­cu­lat­ing much earli­er than that. He showed that, on any four-man­i­fold with pos­it­ive-def­in­ite in­ter­sec­tion form, one could find self-dual \( \mathit{SU}(2) \) con­nec­tions. His con­struc­tion yiel­ded solu­tions of to­po­lo­gic­al charge \( k \), whose curvature was con­cen­trated around \( k \) points.

An­oth­er as­pect which poin­ted away from self-dual spaces was the role of spe­cial con­nec­tions on holo­morph­ic bundles over a Kähler man­i­fold. A choice of Her­mitian met­ric on a holo­morph­ic bundle nat­ur­ally defines a con­nec­tion, and one is in­ter­ested in such met­rics where the curvature of the con­nec­tion, which is a mat­rix of two-forms, is or­tho­gon­al to the Kähler form. In two com­plex di­men­sions, this is the same as the anti-self-dual Yang–Mills equa­tions. It was con­jec­tured by the writer at a Tanigu­chi Sym­posi­um in 1979 that such con­nec­tions should ex­ist on stable holo­morph­ic bundles, a con­jec­ture also ad­vanced by oth­ers, in par­tic­u­lar S. Kobay­ashi. There was evid­ence for the con­jec­ture; the Narasim­han–Se­shadri the­or­em was one. There were also con­crete ex­amples: the in­stan­ton con­nec­tions on the four-sphere, pulled back to pro­ject­ive space.

It seemed then that self-dual spaces did not have to be the unique set­ting for study­ing the Yang–Mills equa­tions.

Si­mon Don­ald­son be­came the writer’s re­search stu­dent in 1980, and was giv­en the task of at­tempt­ing to prove the con­jec­ture about sta­bil­ity. He ab­sorbed vari­ous ana­lyt­ic­al ap­proaches dur­ing his first year, study­ing Eells and Sampson’s work on har­mon­ic maps and Yau’s work on the Calabi con­jec­ture. After ob­serving that there was a mo­ment-map de­scrip­tion of these equa­tions, he trans­ferred to Atiyah’s group and en­lis­ted in the mo­ment-map activ­ity there. His first suc­cess was a new, more ana­lyt­ic­al but more ele­ment­ary, proof of the the­or­em of Narasim­han and Se­shadri, us­ing the mo­ment-map form­al­ism; but then, in the Au­tumn of 1981, he had a rad­ic­ally new idea which turned the sub­ject of in­stan­tons on its head.

As Don­ald­son re­calls [e8],

I stud­ied Taubes' pa­per in de­tail in 1980–81: it fit­ted in with my think­ing about Hitchin’s prob­lem in the fol­low­ing way. In study­ing the non­lin­ear heat equa­tion men­tioned above the es­sen­tial thing was to ob­tain ana­lyt­ic­al com­pact­ness the­or­ems which would al­low one to get some kind of lim­it. This is closely re­lated to un­der­stand­ing the com­pact­ness of in­stan­ton mod­uli spaces: now the al­gebro-geo­met­ric lit­er­at­ure con­tained vari­ous ex­amples of mod­uli spaces of stable bundles, and one can ob­serve in these ex­amples that the mod­uli spaces have nat­ur­al com­pac­ti­fic­a­tions in which one ad­joins points “at in­fin­ity,” made up of con­fig­ur­a­tions of points in the un­der­ly­ing com­plex sur­face. It was there­fore nat­ur­al to make the hy­po­thes­is, as­sum­ing the con­jec­tured re­la­tion between in­stan­tons and holo­morph­ic bundles, that in­stan­ton mod­uli spaces over gen­er­al 4-man­i­folds should be com­pac­ti­fied by ad­join­ing con­fig­ur­a­tions of points.

This hy­po­thes­is was sup­por­ted by Taubes' con­struc­tion — his in­stan­tons were near the bound­ary of the mod­uli space. The stand­ard ex­ample of a mod­uli space of in­stan­tons was the case of the four-sphere with charge 1. In this case, the \( 8k-3=5 \)-di­men­sion­al mod­uli space is an or­bit of the con­form­al group of the sphere, \( \mathit{SO}(5,1) \) — it is nat­ur­ally hy­per­bol­ic 5-space, and its bound­ary is the sphere it­self. Moreover, the ex­pli­cit en­ergy dens­it­ies of the in­stan­tons can eas­ily be seen to con­verge to a Dir­ac delta func­tion at a point as one ap­proaches the bound­ary, so this mod­el fit­ted in with Don­ald­son’s ideas — the four-man­i­fold bounds the mod­uli space. In the case where the in­ter­sec­tion form of a man­i­fold \( X \) is pos­it­ive def­in­ite, the in­dex the­or­em pre­dicts the di­men­sion of the charge-one mod­uli space to be 5 again, so it looked as if the bound­ary of the mod­uli space would be the four-man­i­fold it­self in this case. But Don­ald­son knew enough about cobor­d­ism the­ory to real­ize that, for ex­ample, the com­plex pro­ject­ive plane did not bound a smooth man­i­fold and he real­ized that it was the re­du­cible con­nec­tions — those with holonomy \( U(1) \) and not the full \( \mathit{SU}(2) \) — which gave sin­gu­lar points in the in­teri­or of the mod­uli space. The man­i­fold did not it­self bound, but was giv­en an ex­pli­cit cobor­d­ism to a dis­joint uni­on of com­plex pro­ject­ive spaces.

His con­clu­sion was that when the in­ter­sec­tion mat­rix of \( X \) is pos­it­ive def­in­ite it must be re­duced over the in­tegers to the stand­ard di­ag­on­al form. As he ad­mits [e8],

The point I wish to make is that the chain of reas­on­ing was to a large ex­tent a product of na­iv­eté, the ini­tial im­petus be­ing the de­sire to test the com­pac­ti­fic­a­tion hy­po­thes­is for in­stan­ton mod­uli spaces. Moreover, it was not im­me­di­ately clear to me what use the ar­gu­ment should be put to: I did not even know that there were any non-stand­ard quad­rat­ic forms!

At the time, his su­per­visor Atiyah went to vis­it the In­sti­tut des Hautes Études Sci­en­ti­fiques in Par­is, and met Mi­chael Freed­man who had been work­ing on the to­po­lo­gic­al the­ory of four-man­i­folds. He began to ex­plain Don­ald­son’s work [e10]:

I asked him “Would this sort of res­ult be of in­terest? How would it fit in­to…”

“Oh,” he said, “that would be spec­tac­u­lar, totally un­be­liev­able.”

And it just so happened that, be­cause Freed­man had fin­ished off the prob­lem on the to­po­lo­gic­al side, he ex­pec­ted that with a little bit more work he would fin­ish off the smooth case, and to be told that ex­actly the op­pos­ite was true was really, you know, a bit of a shock to him.

Freed­man had shown that any un­im­od­u­lar quad­rat­ic form could ap­pear as the in­ter­sec­tion mat­rix of a to­po­lo­gic­al four-man­i­fold. He was awar­ded a Fields Medal at the same time as Don­ald­son in 1986. By that time, their res­ults had shown that \( \mathbb R^4 \) it­self has in­fin­itely many dif­fer­ent dif­fer­en­ti­able struc­tures.

After the re­mark­able achieve­ment of his thes­is, Don­ald­son re­turned to the sta­bil­ity con­jec­ture, and soon suc­ceeded in prov­ing it for Kähler sur­faces (it was fol­lowed up in more gen­er­al­ity by S.-T. Yau and K. Uh­len­beck). He needed it for al­geb­ra­ic sur­faces, to be­gin to ap­ply his four-man­i­fold the­ory fur­ther, since there are con­crete ways of con­struct­ing stable holo­morph­ic bundles on some of these, in par­tic­u­lar, el­lipt­ic fibra­tions. Such fibra­tions were also a source of in­ter­est­ing con­jec­tures — Kodaira had in 1970 pro­duced ex­amples of com­plex sur­faces which were ho­mo­topy equi­val­ent to a \( K3 \) sur­face but not ob­vi­ously dif­feo­morph­ic to one.

One fea­ture in all this work is the fact, men­tioned earli­er, that the di­men­sion of the ker­nel of an el­lipt­ic op­er­at­or can change in a smooth fam­ily. In pro­du­cing a smooth mod­uli space, Don­ald­son needed to prove the van­ish­ing of the null space of a par­tic­u­lar op­er­at­or. For the sphere, this came from pos­it­iv­ity of curvature, and in the early days of Don­ald­son’s proof it was not clear wheth­er he was prov­ing something about just a man­i­fold or a man­i­fold with a met­ric sat­is­fy­ing a curvature con­di­tion. The same was true of early ver­sions of Taubes' ex­ist­ence the­or­em. Don­ald­son got around this for his the­or­em by de­form­ing the self-du­al­ity equa­tions, though sub­sequently it was shown that one can de­form the met­ric. This jump­ing in di­men­sion was made to work pos­it­ively a little later. Fix­ing a sur­face in the four-man­i­fold, one may study the Dir­ac op­er­at­or on the sur­face coupled to the self-dual con­nec­tion. The set of con­nec­tions where the di­men­sion of the null space jumps defines a cycle in the mod­uli space and, al­though it is not com­pact, in­ter­sec­tion num­bers can be defined. These gave rise to the Don­ald­son poly­no­mi­als in 1988, which gave an al­geb­ra­ic struc­ture to his res­ults and en­abled power­ful the­or­ems about four-man­i­folds to be proved.

In 1985, at the age of 28, Don­ald­son was ap­poin­ted to the Wal­lis Chair of Math­em­at­ics in Ox­ford, and he rap­idly built up a large num­ber of gradu­ate stu­dents, deal­ing with them ul­ti­mately in a man­ner not un­like that of Pen­rose.

Geometry and physics

The story of the in­stan­ton helped to ce­ment the ties between math­em­at­ics in Ox­ford and the new de­vel­op­ments in the­or­et­ic­al phys­ics around the world. The ob­vi­ous next prob­lem was to study mag­net­ic mono­poles in Eu­c­lidean three-space — these were stat­ic solu­tions to the Yang–Mills–Higgs equa­tions, and cous­ins of the in­stan­tons. Ward, by then at Trin­ity Col­lege, Dub­lin, had pro­duced some in­ter­est­ing ex­amples us­ing el­lipt­ic func­tions in 1981, but the writer began to de­vel­op the the­ory us­ing the twis­tor ap­proach, which led to the study of holo­morph­ic line bundles on an al­geb­ra­ic curve. The phys­i­cist W. Nahm had pro­duced an ana­logue of the ADHM con­struc­tion in this case, which led to a sys­tem of or­din­ary dif­fer­en­tial equa­tions which were al­geb­ra­ic­ally in­teg­rable in the same sense that the equa­tions of a spin­ning top are in­teg­rable, and led again to the same curve. Nahm’s ap­proach also al­lowed Don­ald­son to prove that the mod­uli space of \( \mathit{SU}(2) \) mono­poles of charge \( k \) had a far more con­crete de­scrip­tion than the in­stan­ton case — it was the space of ra­tion­al func­tions \( f(z)=p(z)/q(z) \) of de­gree \( k \) map­ping in­fin­ity to zero.

At the same time, the Rieman­ni­an ver­sion of twis­tor the­ory was be­ing ex­ten­ded to high­er di­men­sions and, in par­tic­u­lar, to the the­ory of hy­per­kähler man­i­folds — Rieman­ni­an struc­tures based on the qua­ternions. The same hy­per­kähler geo­metry was be­ing used by phys­i­cists in the su­per­sym­met­ric sigma-mod­el and, for a while, twis­tor the­ory and su­per­sym­metry were run­ning side by side. The math­em­at­ic­al out­come of a col­lab­or­a­tion between the writer and the phys­i­cists M. Roček, U. Lind­ström and A. Karl­hede was the hy­per­kähler quo­tient con­struc­tion. This fit­ted in well with Atiyah’s mo­ment-map philo­sophy, and it be­came clear that the self-du­al­ity equa­tions on \( \mathbb R^4 \) and the mono­pole equa­tions on \( \mathbb R^3 \) could both be ex­pressed as zer­os of a hy­per­kähler mo­ment map. This meant that the mod­uli spaces (in the in­stan­ton case, an \( \mathit{SU}(2) \) bundle over the \( (8k-3) \)-di­men­sion­al space) had a hy­per­kähler struc­ture. Us­ing this fact and the eval­u­ation of the met­ric in the charge-2 case, Atiyah and the writer in­vest­ig­ated the scat­ter­ing prop­er­ties of two-mono­poles.

These were all in­ter­ac­tions between geo­metry and phys­ics, but was it genu­ine phys­ics? Wit­ten re­called hear­ing Atiyah give a lec­ture dur­ing his first vis­it to Ox­ford in 1978 [e17]

I re­mem­ber him be­gin­ning the first lec­ture ex­plain­ing that the trouble with work­ing on prob­lems posed by phys­i­cists is that, once the prob­lem is solved, one might be told that the prob­lem wasn’t quite the right one. This must have been at least partly a re­sponse to my im­pa­tience, at the time, with any­thing that didn’t shed light on quantum be­ha­viour of gauge the­or­ies.

There was one geo­met­er in Ox­ford, however, who had thought a lot about quantum the­ory — Graeme Segal. Segal had been one of Atiyah’s first stu­dents, work­ing on equivari­ant \( K \)-the­ory, and then oth­er equivari­ant gen­er­al­ized co­homo­logy the­or­ies. He was a col­lab­or­at­or on the second of the An­nals pa­pers on the in­dex the­or­em. Well known as an al­geb­ra­ic to­po­lo­gist, he ar­rived in Mo­scow in the early 1970s to give some lec­tures and met S. Novikov, who told him, “So you are a to­po­lo­gist? Here we think that al­geb­ra­ic to­po­logy is dead.” Novikov had won a Fields Medal in 1970, largely for his work in to­po­logy, but was at the time in­volved in solitons and in­teg­rable sys­tems.

Segal’s work after that changed course and, in par­tic­u­lar, he wrote with his stu­dent A. Press­ley a highly in­flu­en­tial book on loop groups — the group of maps from a circle to a Lie group — and their rep­res­ent­a­tions. With G. Wilson, he used the loop-group prop­er­ties to give a rig­or­ous ver­sion of Mi­wa and Sato’s ap­proach to the KdV equa­tion.

In the mid-1980s, Segal pro­duced a pre­print, “The defin­i­tion of con­form­al field the­ory,” which was only privately cir­cu­lated but nev­er­the­less be­came quite in­flu­en­tial. Its mo­tiv­a­tion was to link up what the phys­i­cists were do­ing with a num­ber of pure-math­em­at­ic­al top­ics, such as the Griess–Fisc­her Mon­ster group, rep­res­ent­a­tions of the dif­feo­morph­ism group of the circle, rep­res­ent­a­tions of loop groups, and mod­uli of Riemann sur­faces. He chose to do this by de­fin­ing a con­form­al field the­ory in an ax­io­mat­ic way, as a func­tor from a cer­tain cat­egory to the cat­egory of Hil­bert spaces, sat­is­fy­ing a list of prop­er­ties. The work fi­nally ap­peared, still in a re­l­at­ively in­form­al mode, in [e18].

To many math­em­aticians, a cat­egory was a huge thing: the cat­egory whose ob­jects are sets and morph­isms maps between them; or the one whose ob­jects are fi­nite-di­men­sion­al vec­tor spaces and morph­isms, lin­ear maps. Segal had earli­er been in­ter­ested in smal­ler ones, a group for ex­ample (which is a cat­egory with one ob­ject, the morph­isms be­ing ele­ments of the group); he had dis­cussed the ho­mo­topy the­ory of clas­si­fy­ing spaces of such small cat­egor­ies.

The ob­jects of Segal’s cat­egory are ori­ented dis­joint uni­ons of circles, and the morph­isms, Riemann sur­faces with bound­ary. So, the Riemann sur­face has an in­go­ing bound­ary com­pon­ent and an out­go­ing one, and it rep­res­ents a morph­ism from one to the oth­er. Morph­isms are com­posed by glu­ing the Riemann sur­faces to­geth­er, ex­tend­ing the con­form­al struc­ture. Segal’s point of view was that a con­form­al field the­ory is a func­tor from this cat­egory to the cat­egory of pro­ject­ive in­fin­ite-di­men­sion­al Hil­bert spaces.

Mean­while, in 1982, Wit­ten had ap­plied quantum-the­or­et­ic­al ideas to a simple geo­met­ric­al prob­lem, to pro­duce a rad­ic­al new view­point [e5]. The geo­met­ric prob­lem was Morse the­ory — the study of a com­pact man­i­fold with a smooth func­tion \( f \) on it, with nonde­gen­er­ate crit­ic­al points. Wit­ten viewed the even forms on the man­i­fold as a bo­son­ic Hil­bert space, and the odd forms as a fer­mi­on­ic one, with the op­er­at­ors \( d+d^* \) and \( i(d-d^*) \) as su­per­sym­metry op­er­at­ors go­ing from the bo­son­ic space to the fer­mi­on­ic one. Con­jug­at­ing by \( e^{tf} \), he ar­gued that the ei­gen­func­tions of the Hamilto­ni­an \( dd^*+d^*d \) should, for large \( t \), be con­cen­trated around the crit­ic­al points; but there was “tun­nel­ing” from one state to an­oth­er, giv­en by “in­stan­tons.” This lan­guage re­flec­ted the ori­gin­al role of the Yang–Mills in­stan­tons in quantum field the­ory. The in­stan­tons in this case are the gradi­ent flow lines of the func­tion \( f \), go­ing from one crit­ic­al point to an­oth­er. He saw this as a quantum-mech­an­ic­al sys­tem in one-space/one-time di­men­sions.

This pa­per in­tro­duced a dif­fer­ent cat­egory, whose the ob­jects are the crit­ic­al points of the func­tion \( f \), and the gradi­ent flow lines between them are the morph­isms. Here, the ob­jects are zero-di­men­sion­al man­i­folds, and the morph­isms one-di­men­sion­al ones, but with no ad­di­tion­al struc­ture like the con­form­al struc­ture of the Riemann sur­face in Segal’s case. There was clearly a gen­er­al concept here, of a to­po­lo­gic­al quantum field the­ory — the ob­jects \( d \)-man­i­folds, the morph­isms \( (d+1) \)-cobor­d­isms, and the the­ory a func­tor to fi­nite-di­men­sion­al Hil­bert spaces.

Atiyah and Segal ax­io­mat­ic­ally form­al­ized this [e20]:

Be­cause math­em­aticians are frightened by the Feyn­man in­teg­ral and are un­fa­mil­i­ar with all the jar­gon of phys­i­cists, there seemed to me to be a need to ex­plain to math­em­aticians what a to­po­lo­gic­al quantum field the­ory really was, in user-friendly terms. I gave a simple ax­io­mat­ic treat­ment (something math­em­aticians love) and lis­ted the ex­amples that arise from phys­ics. The task of the math­em­atician is then to con­struct, by any meth­od pos­sible, a the­ory that fits the ax­ioms. I like to think of this as ana­log­ous to the Ei­len­berg–Steen­rod ax­ioms of co­homo­logy, where one can use sim­pli­cial, Čech or de Rham meth­ods to con­struct the the­ory.

The ax­ioms re­quire for each ori­ented \( d \)-di­men­sion­al man­i­fold \( \Sigma \) a com­plex vec­tor space \( Z(\Sigma) \); for each \( (d+1) \)-di­men­sion­al man­i­fold with bound­ary \( \Sigma \), there is a dis­tin­guished vec­tor in \( Z(\Sigma) \). The no­men­clature re­flects the par­ti­tion func­tion of quantum field the­ory. With the op­pos­ite ori­ent­a­tion, one gets the dual space; for the empty set, \( Z(\emptyset)=\mathbb C \). The vec­tor space for the dis­joint uni­on of \( \Sigma_1 \) and \( \Sigma_2 \) is the tensor product \( Z(\Sigma_1)\otimes Z(\Sigma_2) \). There are two more ax­ioms: one is as­so­ci­ativ­ity for com­pos­ing cobor­d­isms, and the oth­er, a ho­mo­topy con­di­tion that the morph­ism defined by \( \Sigma \times I \) is the iden­tity.

Thus did Atiyah re­spond to Wit­ten’s chal­lenge to move up­wards to the quantum level in do­ing geo­metry. He con­jec­tured that Flo­er’s new ho­mo­logy groups should be the Hil­bert spaces of a TQFT, that Don­ald­son’s the­ory should be one where \( d=3 \), and that the new knot poly­no­mi­als of Vaughan Jones should be an­oth­er for \( d=2 \).

In the sum­mer of 1988, the In­ter­na­tion­al Con­gress of Math­em­at­ic­al Phys­i­cists was held in Swansea, and at din­ner in An­nie’s res­taur­ant (an event now im­mor­tal­ized by a plaque) Wit­ten talked to Atiyah and Segal about Segal’s ideas on con­form­al field the­ory and “mod­u­lar func­tors.” He then real­ized that “the right the­ory to get the Jones poly­no­mi­als was a TQFT whose Hil­bert space is the fi­nite-di­men­sion­al space of con­form­al blocks of a two-di­men­sion­al WZW the­ory, and the Lag­rangi­an of this the­ory was the Chern–Si­mons func­tion­al.”

The con­form­al blocks are es­sen­tially the space of sec­tions of a cer­tain holo­morph­ic line bundle on the mod­uli space of stable bundles over a Riemann sur­face. Some­what later, Wit­ten would give also an in­ter­pret­a­tion of Don­ald­son the­ory as a TQFT.

Atiyah’s com­par­is­on of his TQFT ax­ioms with the ax­io­mat­ic ap­proach to co­homo­logy sug­gests that there should be many dif­fer­ent ways of es­tab­lish­ing this TQFT, but at the time of writ­ing (2007) there is only one — a com­bin­at­or­i­al one — which truly works. Oth­er ap­proaches, us­ing more al­geb­ra­ic geo­metry, are close to work­ing and the writer in 1990 made a con­tri­bu­tion to this by de­fin­ing a pro­ject­ively flat con­nec­tion de­signed to identi­fy the Hil­bert space for two dif­fer­ent con­form­al struc­tures.

But oth­er events took place in 1990. Wit­ten won a Fields Medal for his work on the Jones poly­no­mi­al, and Atiyah and Segal both left Ox­ford for Cam­bridge. Atiyah be­came Mas­ter of Trin­ity Col­lege, Dir­ect­or of the newly formed Isaac New­ton In­sti­tute for Math­em­at­ic­al Sci­ences, and Pres­id­ent of the Roy­al So­ci­ety. Segal had been elec­ted to the Lown­dean Chair of As­tro­nomy and Geo­metry after the death of the to­po­lo­gist J. F. Adams. This writer too left in that year, to take a Chair at the Uni­versity of War­wick. The quant­iz­a­tion of geo­metry was to be pur­sued else­where.