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Celebratio Mathematica

Michael F. Atiyah

Papers on index theory (1963–1984)

Commentary by M. Atiyah

In the Spring of 1962, my first year at Ox­ford, Sing­er de­cided to spend part of his sab­bat­ic­al there. This turned out to be par­tic­u­larly for­tu­nate for both of us and led to our long col­lab­or­a­tion on the in­dex the­ory of el­lipt­ic op­er­at­ors. This had its ori­gins in my work on \( K \)-the­ory with Hirzebruch and the at­tempt to ex­tend the Hirzebruch–Riemann–Roch the­or­em in­to dif­fer­en­tial geo­metry. We had already shown that the in­teg­ral­ity of the Todd genus of an al­most com­plex man­i­fold and the \( \hat{A} \)-genus of a spin man­i­fold could be el­eg­antly ex­plained in terms of \( K \)-the­ory. For an al­geb­ra­ic vari­ety the Hirzebruch–Riemann–Roch the­or­em went one step fur­ther and iden­ti­fied the Todd genus with the arith­met­ic genus or Euler char­ac­ter­ist­ic of the sheaf co­homo­logy. Also the \( L \)-genus of a dif­fer­en­tial man­i­fold, as proved by Hirzebruch, gave the sig­na­ture of the quad­rat­ic form of middle-di­men­sion­al co­homo­logy and, by Hodge the­ory, this was the dif­fer­ence between the di­men­sions of the rel­ev­ant spaces of har­mon­ic forms. As Hirzebruch him­self had real­ized, it was nat­ur­al there­fore to look for a sim­il­ar ana­lyt­ic­al in­ter­pret­a­tion of the \( \hat{A} \)-genus. The co­homo­lo­gic­al for­mula and the as­so­ci­ated char­ac­ter for­mula clearly in­dic­ated that one should use the spin rep­res­ent­a­tions. I was strug­gling with this prob­lem when Sing­er ar­rived. For­tu­nately Sing­er’s strengths were pre­cisely in dif­fer­en­tial geo­metry and ana­lys­is, the areas where I was weak­est. With his help we soon re­dis­covered the Dir­ac op­er­at­or! My know­ledge of phys­ics was very slim, des­pite hav­ing at­ten­ded a course on quantum mech­an­ics by Dir­ac him­self; Sing­er had a bet­ter back­ground in the area but, in any case, we were deal­ing with Rieman­ni­an man­i­folds and not Minkowski space, so that phys­ics seemed far away. In a sense, his­tory was re­peat­ing it­self be­cause Hodge, in de­vel­op­ing his the­ory of har­mon­ic forms, had been strongly mo­tiv­ated by Max­well’s equa­tions. Sing­er and I were just go­ing one step fur­ther in pur­su­ing the Rieman­ni­an ver­sion of the Dir­ac equa­tion. Also, as with Hodge, our start­ing point was really al­geb­ra­ic geo­metry. However, spinors were (and re­main) more mys­ter­i­ous than dif­fer­en­tial forms, and they had nev­er be­fore been used in dif­fer­en­tial geo­metry.

Once we had grasped the sig­ni­fic­ance of spinors and the Dir­ac equa­tion, it be­came evid­ent that the \( \hat{A} \)-genus had to be the dif­fer­ence of the di­men­sions of pos­it­ive and neg­at­ive har­mon­ic spinors. Prov­ing this then be­came our main ob­ject­ive. By good for­tune Smale passed through Ox­ford at this time and, when we ex­plained our ideas to him, he drew our at­ten­tion to a pa­per of Gel­fand on the gen­er­al prob­lem of com­put­ing the in­dex of el­lipt­ic op­er­at­ors. Fol­low­ing up this pa­per we dis­covered a num­ber of pa­pers by ana­lysts (Agran­ovic, Dyn­in, See­ley) de­voted to the in­dex prob­lem. In par­tic­u­lar we learnt the rel­ev­ance of pseudo-dif­fer­en­tial op­er­at­ors to the prob­lem. In all this Sing­er was a great help. My ana­lyt­ic­al back­ground was very weak and I re­mem­ber hav­ing to be in­struc­ted on the sig­ni­fic­ance of Four­i­er trans­forms.

Sing­er and I had some great ad­vant­age over the ana­lysts in­vest­ig­at­ing the prob­lem. We were in­vest­ig­at­ing a par­tic­u­lar case, the Dir­ac op­er­at­or, and we already “knew” the an­swer. Also, this case, ex­ist­ing in all (even) di­men­sions, en­com­passed all the glob­al to­po­lo­gic­al com­plic­a­tions. Moreover, we had ar­rived at the prob­lem start­ing from \( K \)-the­ory and this turned out to be just the right tool to study the in­dex prob­lem. Fi­nally the Dir­ac op­er­at­or was in a sense the most gen­er­al case, all oth­ers be­ing es­sen­tially de­form­able to it.

With con­sid­er­able help from our ana­lyt­ic­al friends, such as Louis Niren­berg, Sing­er and I even­tu­ally pro­duced a proof of the gen­er­al in­dex the­or­em dur­ing my stay at Har­vard in the Fall in 1962. This was based on the use of bound­ary-value prob­lems and fol­lows the cobor­d­ism ap­proach of Hirzebruch’s proof of the sig­na­ture the­or­em. We an­nounced our res­ults in the Bul­let­in note [1]. I real­ized at the time the sig­ni­fic­ance of the in­dex the­or­em and that it rep­res­en­ted the high-point of my work, but it would have been hard to pre­dict that the sub­ject would con­tin­ue to oc­cupy me in vari­ous forms for the next twenty years. I would also have been ex­tremely sur­prised if I had been told that this work would in due course be­come im­port­ant in the­or­et­ic­al phys­ics.

The proof of the in­dex the­or­em was only briefly sketched in [1]. The full de­tails were presen­ted in the sem­in­ar which I ran with Bott and Sing­er in the Fall of 1962. In due course a more com­pre­hens­ive ver­sion of this ap­peared as the An­nals of Math­em­at­ics Study, based on the Sem­in­ar at the In­sti­tute for Ad­vanced Study run by Borel and Pal­ais. I wrote an Ap­pendix [1965: The in­dex the­or­em for man­i­folds with bound­ary] for this de­scrib­ing the ex­ten­sion of the in­dex prob­lem for man­i­folds with bound­ary. This was joint work with Bott and is also de­scribed in [2]. In fact un­der­stand­ing the role of bound­ary con­di­tions for el­lipt­ic op­er­at­ors was by no means routine. Bott and I struggled with the prob­lem for some time, be­fore we saw the light. It was clear, for to­po­lo­gic­al reas­ons, that a good bound­ary con­di­tion should some­how “trivi­al­ize” the sym­bol of the op­er­at­or at the bound­ary, so as to define a re­l­at­ive \( K \)-the­ory class. De­riv­ing this trivi­al­iz­a­tion from the bound­ary con­di­tions turned out to in­volve a very thor­ough un­der­stand­ing of the Bott peri­od­icity the­or­em, and this led as a by-product to our ele­ment­ary proof [e1].

The open­ing of the new Cour­ant In­sti­tute build­ing and the In­ter­na­tion­al Con­gress at Mo­scow in 1966 provided op­por­tun­it­ies for ex­pos­it­ory sur­vey talks [8], [15] on the in­dex the­or­em and its re­la­tion to the to­po­logy of the lin­ear groups (i.e., to \( K \)-the­ory).

At the Woods Hole con­fer­ence in the sum­mer of 1964, Bott and I learnt of a con­jec­ture of Shimura con­cern­ing a gen­er­al­iz­a­tion of the Lef­schetz fixed-point for­mula for holo­morph­ic maps. After much ef­fort we con­vinced ourselves that there should be a gen­er­al for­mula of this type for maps pre­serving any el­lipt­ic op­er­at­or (or, more gen­er­ally, any el­lipt­ic com­plex). As a first test we ap­plied it to el­lipt­ic curves with com­plex mul­ti­plic­a­tion and, as we had the lead­ing ex­perts with us at Woods Hole (Cas­sells, Tate, etc.), we asked them to veri­fy it for us. Un­for­tu­nately our for­mula failed the test. For­tu­nately we did not be­lieve the ex­perts: the for­mula seemed too beau­ti­ful to be wrong, and so it proved. We were es­pe­cially con­vinced when one day we sud­denly real­ized that the fam­ous Her­mann Weyl char­ac­ter for­mula was a par­tic­u­lar case of our gen­er­al for­mula. In due course we found the ne­ces­sary proof and the res­ult was briefly presen­ted in [5] with full de­tails ap­pear­ing later in [6], [13].

Per­haps the most in­ter­est­ing ap­plic­a­tion of the gen­er­al Lef­schetz for­mula con­cerned the ac­tion of a fi­nite group on the middle-di­men­sion­al co­homo­logy of a man­i­fold. This yiel­ded sur­pris­ingly strong the­or­ies and with Mil­nor’s help we were able in [13] to prove that \( h \)-cobord­ant lens spaces are iso­met­ric. The for­mu­lae in this case were very re­min­is­cent of those for Re­idemeister tor­sion of lens spaces, so Bott and I spent some time try­ing to re­late Re­idemeister tor­sion to el­lipt­ic op­er­at­or the­ory. Moreover our first proof in [5] of the Lef­schetz for­mula, re­lied on the \( \zeta \)-func­tions ori­gin­ally in­tro­duced by Minakshisundaram and Pleijel. Al­though we even­tu­ally found a sim­pler proof, we found the \( \zeta \)-func­tions very ap­peal­ing and tried to ex­ploit them fur­ther. Al­though we were un­suc­cess­ful at the time, \( \zeta \)-func­tions came in­to their own later on and, in par­tic­u­lar, Ray and Sing­er found how to use them to at­tack Re­idemeister tor­sion.

The first proof of the in­dex the­or­em for el­lipt­ic op­er­at­ors, as presen­ted in [1], was based on cobor­d­ism and it suffered from cer­tain lim­it­a­tions. For some years Sing­er and I searched for a bet­ter proof modeled more on Grothen­dieck’s proof of the gen­er­al­ized Hirzebruch–Riemann–Roch the­or­em. Even­tu­ally we found such a proof, based on em­bed­ding a man­i­fold in Eu­c­lidean space and then trans­fer­ring the prob­lem to one on the Eu­c­lidean space by a suit­able “dir­ect im­age” con­struc­tion. This proof, giv­en in [11], worked purely in a \( K \)-the­ory con­text and avoided ra­tion­al co­homo­logy. It there­fore lent it­self to vari­ous sig­ni­fic­ant gen­er­al­iz­a­tions de­veloped in the re­main­ing pa­pers [10], [19], [27], [26] of the series. One gen­er­al­iz­a­tion, the “equivari­ant” in­dex the­or­em, dealt with com­pact group ac­tions pre­serving an el­lipt­ic op­er­at­or. This ex­ten­ded the Lef­schetz for­mula of my pa­pers with Bott to the case of non-isol­ated fixed points (of iso­met­rics), and res­ted on the pri­or de­vel­op­ment of equivari­ant the­ory. This had been car­ried out by Graeme Segal in his thes­is and the rel­ev­ant ap­plic­a­tions to in­dex the­ory were de­veloped in [10]. In par­tic­u­lar the “loc­al­iz­a­tion the­or­em” re­lat­ing geo­met­ric loc­al­iz­a­tion (fixed points) to al­geb­ra­ic loc­al­iz­a­tion (via ideal the­ory of the char­ac­ter ring \( R(G) \)) was a soph­ist­ic­ated ver­sion of the Lef­schetz fixed-point prin­ciple. In vari­ous forms this was to re­appear in sub­sequent work, not­ably that of Quil­len on the co­homo­logy of fi­nite groups.

The in­dex the­or­em for fam­il­ies of el­lipt­ic op­er­at­ors [27] was an easy con­sequence of the new proof and could be seen as a first at­tempt to gen­er­al­ize the Grothen­dieck Riemann–Roch the­or­em. In its real form [26] it in­cluded vari­ous mod 2 in­dex the­or­ems, such as the com­pu­ta­tion of the di­men­sion (mod 2) of the ker­nel of a real skew-ad­joint el­lipt­ic op­er­at­or. Des­pite the many in­ter­est­ing new proofs of the in­dex the­or­em that have been pro­duced in re­cent years (heat equa­tion, su­per­sym­metry, etc.), no oth­er proof en­com­passes these re­fined mod 2 in­dex the­or­ems, and the proof in [26] re­mains the only one avail­able.

Al­though [27] was ana­log­ous to Grothen­dieck–Riemann–Roch for a fibre map, we were still a long way from a fully fledged ana­logue deal­ing with ar­bit­rary smooth maps. A ba­sic reas­on for this was that in al­geb­ra­ic geo­metry Grothen­dieck had the ad­vant­age of two \( K \)-groups, one \( K^0 \) based on vec­tor bundles and the oth­er \( K_0 \) based on co­her­ent sheaves. Form­ally these be­haved like co­homo­logy and ho­mo­logy re­spect­ively and in par­tic­u­lar \( K_0 \) was nat­ur­ally co­v­ari­ant. I felt that, in the dif­fer­en­ti­able con­text, we were miss­ing the ana­logue of \( K_0 \). In [22], pos­ing as a func­tion­al ana­lyst, I put for­ward some pre­lim­in­ary ideas on how to define \( K_0 \) by us­ing an ab­stract no­tion of el­lipt­ic op­er­at­or. I was pleased when years later these ideas were taken up and de­veloped in­to very sat­is­fact­ory the­or­ies by Brown–Douglas–Fill­more and Kas­parov. Moreover these the­or­ies are rather nat­ur­al in the gen­er­al con­text of \( C^* \)-al­geb­ras (not ne­ces­sar­ily com­mut­at­ive) and have led to con­sid­er­able activ­ity in this field.

In [17] Sing­er and I car­ried out a sys­tem­at­ic in­vest­ig­a­tion of mod 2 in­dices in the frame­work of Clif­ford al­geb­ras. Very much to our sur­prise we found an en­tirely new proof of the (real) peri­od­icity the­or­em emerge, based on Kuiper’s proof of the con­tract­ib­il­ity of the unit­ary group of Hil­bert space. The con­nec­tion of this with oth­er proofs re­mains to this day some­what mys­ter­i­ous and so far it does not ap­pear to have been ex­ploited, but I un­der­stand from Quil­len that it may find a nat­ur­al niche in re­la­tion to Connes’ cyc­lic ho­mo­logy the­ory.

One of the in­ter­est­ing ap­plic­a­tions of the in­dex the­or­em for fam­il­ies of el­lipt­ic op­er­at­ors con­cerns the sig­na­ture of vi­bra­tions, and this is the top­ic of [18]. Chern, Hirzebruch, and Serre had shown that the sig­na­ture of a vi­bra­tion is mul­ti­plic­at­ive (i.e., the product of sig­na­tures of base and fibre) provided the fun­da­ment­al group of the base acts trivi­ally on the co­homo­logy of fibre. The in­dex the­or­em for fam­il­ies cla­ri­fied the role of the fun­da­ment­al group and non-mul­ti­plic­at­ive ex­amples oc­cur nat­ur­ally in al­geb­ra­ic geo­metry. As I was to real­ize much later from Wit­ten these ques­tions are re­lated to grav­it­a­tion­al an­om­alies and [18] provides the simplest ex­amples of such an­om­alies.

My gen­er­al in­terest in dif­fer­en­tial equa­tions (both el­lipt­ic and hy­per­bol­ic) to­geth­er with my back­ground in al­geb­ra­ic geo­metry led nat­ur­ally to the little note [23], where I ex­plained how Hironaka’s the­or­em on the res­ol­u­tion of sin­gu­lar­it­ies gave an easy proof of the Hör­mander–Lo­jasiewicz the­or­em on the di­vi­sion of dis­tri­bu­tions.

One of the most sur­pris­ing ap­plic­a­tions of the equivari­ant in­dex the­or­em was the res­ult proved in my joint pa­per [21] with Hirzebruch show­ing that spin man­i­folds with non-zero \( \hat{A} \)-genus can­not have ef­fect­ive circle ac­tions. This arose out of re­lated res­ults proved my by stu­dent C. Kos­niowski for the holo­morph­ic case. It is also re­min­is­cent of the Lich­ner­ow­icz the­or­em that \( \hat{A}\neq 0 \) im­plies that there is no met­ric of pos­it­ive scal­ar curvature. These ques­tions have since been de­veloped much fur­ther by Gro­mov and Lawson, and very re­cently there have been new re­lated res­ults by Landweber and Stong which have stim­u­lated Wit­ten to de­vel­op a re­mark­able link between quantum field the­ory, man­i­folds, and mod­u­lar forms.

Dur­ing a vis­it to Har­vard in 1970, Mum­ford asked me wheth­er some clas­sic­al res­ults on Riemann sur­faces, con­cern­ing square roots \( L \) of the ca­non­ic­al line-bundle \( K \), could be fit­ted in­to the in­dex-the­ory frame­work. In fact, they fit­ted ex­tremely well in­to the mod-2 in­dex for­mu­lae for spin man­i­folds that Sing­er and I had ex­plored. Nor­mally our sights were on high­er-di­men­sions (\( \gt 4 \)) and I found it sur­pris­ing that non-trivi­al phe­nom­ena of this type already oc­curred in di­men­sion 2. My ac­count of this top­ic ap­peared in [25] and this in turn led Mum­ford to pro­duce a par­al­lel al­geb­ra­ic proof. More re­cently such 2-di­men­sion­al ques­tions fig­ure prom­in­ently in su­per­string the­ory.

In [25] the di­men­sion mod 2 of the space of holo­morph­ic sec­tions of \( L \) is a de­form­a­tion in­vari­ant (as we de­form the Riemann sur­face) and is the mod-2 in­dex of the el­lipt­ic the­ory. An ana­log­ous mod-2 in­dex is the es­sen­tial in­gredi­ent in my joint pa­per [42] with Elmer Rees. This ap­pears as a mod-2 “semi-char­ac­ter­ist­ic” for the sheaf co­homo­logy of a rank-2 vec­tor bundle (with \( c_1=0 \)) on com­plex pro­ject­ive 3-space. With its in­ter­play of to­po­logy and al­geb­ra­ic geo­metry via sheaf co­homo­logy, it seemed a par­tic­u­larly ap­pro­pri­ate pa­per to ded­ic­ate to Serre (on his 50th birth­day), as an in­dic­a­tion of how much I learnt from him in earli­er years, in terms of both con­tent and style.

My col­loqui­um lec­tures [56] to the AMS in 1973 (just after my per­man­ent re­turn to Ox­ford!) were nev­er pub­lished, but they provide per­haps a use­ful his­tor­ic­al sur­vey of the Riemann–Roch story and its evol­u­tion in­to the in­dex the­or­em. Of course, this should now be up­dated, since much has happened in the in­ter­ven­ing dec­ade, par­tic­u­larly in re­la­tion to the­or­et­ic­al phys­ics, but that will have to await an­oth­er day and pos­sibly an­oth­er pen.

The Lef­schetz for­mula of my pa­pers with Bott gave the an­swer as a sum of loc­al con­tri­bu­tions from the fixed points. For a circle ac­tion the Lef­schetz num­ber is a char­ac­ter of the circle, i.e., a fi­nite Four­i­er series, but the loc­al con­tri­bu­tions have poles. In par­tic­u­lar the Her­mann Weyl char­ac­ter for­mula is of this type, and we know in that case the loc­al con­tri­bu­tions can be in­ter­preted as dis­tri­bu­tion­al char­ac­ters of in­fin­ite-di­men­sion­al in­duced rep­res­ent­a­tions. These facts led Sing­er and me to look for a gen­er­al con­text in which such loc­al con­tri­bu­tions could be in­ter­preted in an ap­pro­pri­ate sense as loc­al “dis­tri­bu­tion­al in­dices”. We had in mind pos­sible ap­plic­a­tions to the Har­ish-Chandra the­ory of rep­res­ent­a­tions of non-com­pact semi-simple groups and the Blattner con­jec­ture (giv­ing the re­stric­tion of the char­ac­ter to a max­im­al com­pact sub­group). The ideas led us to in­tro­duce the no­tion of an op­er­at­or which was el­lipt­ic trans­vers­ally to the or­bits of a \( G \)-ac­tion. Such an op­er­at­or then had an in­dex which was a dis­tri­bu­tion on \( G \), and the prob­lem as usu­al was to give a to­po­lo­gic­al con­struc­tion for this in­dex. This is the prob­lem stud­ied and par­tially solved in [36], the lec­ture notes of a sem­in­ar I gave in 1971 at the In­sti­tute for Ad­vanced Study. Gen­er­al meth­ods re­duce the prob­lem down to the case of a tor­us act­ing lin­early on a vec­tor space. Deal­ing with this case turns out to be un­ex­pec­tedly in­tric­ate and heavy use has to be made of com­mut­at­ive al­gebra in­clud­ing the the­ory of Co­hen–Ma­caulay mod­ules. There are in­ter­est­ing ap­plic­a­tions to situ­ations where there are only fi­nite iso­tropy groups, but our hope of at­tack­ing the Har­ish-Chandra the­ory this way nev­er ma­ter­i­al­ized. However, sub­sequent work by Schmid and oth­ers has had some suc­cess with Lef­schetz for­mula meth­ods. Moreover, I re­turned later (with Schmid) with a quite dif­fer­ent ap­proach to the Har­ish-Chandra the­ory (see be­low). On the whole there­fore [36] has not made much of an im­pact, prob­ably be­cause there are really no nat­ur­al ex­amples of tra­versally el­lipt­ic op­er­at­ors (oth­er than or­din­ary el­lipt­ic ones). Moreover the com­mut­at­ive al­gebra in [36] was prob­ably un­fa­mil­i­ar and un­pal­at­able to po­ten­tial users (e.g., to­po­lo­gists or ana­lysts).

The joint pa­per [33] arose out of the at­tempts by Bott and my­self to un­der­stand the re­mark­able res­ults of the young In­di­an math­em­atician V. K. Pat­odi on the heat-equa­tion ap­proach to the in­dex the­or­em, a top­ic which was to blos­som later in the con­tact with the­or­et­ic­al phys­ics. In our work on the Lef­schetz for­mula for el­lipt­ic com­plexes, Bott and I had de­scribed the Zeta-func­tion ap­proach to the in­dex the­or­em but had com­men­ted on its com­pu­ta­tion­al dif­fi­culty. Sing­er and McK­ean took the pro­cess a step fur­ther, in the heat-equa­tion ver­sion, by con­cen­trat­ing on the Rieman­ni­an geo­metry. They spec­u­lated on the pos­sib­il­ity of re­mark­able can­cel­la­tions lead­ing dir­ectly to the Gauss–Bon­net form for the Euler char­ac­ter­ist­ic. This was proved some years later by Pat­odi. Even more sig­ni­fic­ant was Pat­odi’s ex­ten­sion of the res­ult to deal with the Riemann–Roch the­or­em on Kähler man­i­folds. Pat­odi’s ap­proach was rather dir­ect, and a con­sid­er­able tour de force. But the al­geb­ra­ic ma­nip­u­la­tions were dif­fi­cult to un­der­stand and it was there­fore very in­ter­est­ing when Gilkey pro­duced an al­tern­at­ive in­dir­ect ap­proach de­pend­ing on a simple char­ac­ter­iz­a­tion of the Pontry­agin forms of a Rieman­ni­an man­i­fold. On the oth­er hand, while Gilkey’s res­ult was beau­ti­fully simple, and eas­ily led to the in­dex the­or­em, it ap­peared to be enorm­ously com­plic­ated to prove. In fact Gilkey had dis­covered it while per­form­ing al­geb­ra­ic com­pu­ta­tions on the com­puter.

This was the situ­ation when Bott and Pat­odi joined me at the In­sti­tute for Ad­vanced Study in 1971–72. After con­sid­er­able ef­fort we even­tu­ally real­ized that Gilkey’s res­ults was a very easy con­sequence of the Bi­an­chi iden­tit­ies, to­geth­er with the main the­or­em on tensori­al in­vari­ants of the or­tho­gon­al group. Gilkey’s proof had ap­peared com­plic­ated only be­cause he had not taken this route.

Be­sides prov­ing Gilkey’s res­ult, and giv­ing its ap­plic­a­tion to the in­dex the­or­em, [33] also in­cluded a leis­urely ac­count of the heat equa­tion asymp­tot­ics based on See­ley’s ap­proach. Moreover we gave, in the Ap­pendix, a new proof of the the­or­em on or­tho­gon­al in­vari­ants. We were im­pelled to do this by our dif­fi­culty in un­der­stand­ing the no­tori­ous “Capelli iden­tity”.

My second col­lab­or­a­tion with Pat­odi, this time with Sing­er as the third part­ner, con­cerned the sig­na­ture the­or­em for man­i­folds with bound­ary and led to the series of pa­pers [31], [38], [37], [43] The prob­lem of gen­er­al­iz­ing the Hirzebruch sig­na­ture the­or­em to man­i­folds with bound­ary had long been an in­triguing ques­tion. There had been many clues, not­ably the work of Hirzebruch on sig­na­ture de­fects of cusps of Hil­bert mod­u­lar sur­faces. While Pat­odi was with me in Prin­ceton I had sug­ges­ted this prob­lem to him, hop­ing that he could ap­ply his vir­tu­os­ity with the heat equa­tion to the prob­lem. In fact he suc­ceeded on these lines, but his meth­od was again highly com­pu­ta­tion­al and tied to the ex­tens­ive use of dif­fer­en­tial forms. As such it did not ap­ply to the Dir­ac op­er­at­or and its gen­er­al­iz­a­tions which I felt should be in­cluded. Sing­er and I there­fore tried to ana­lyse the prob­lem in its more gen­er­al form. Even­tu­ally we saw that the nat­ur­al for­mu­la­tion was that of an in­dex prob­lem with a “glob­al” bound­ary con­di­tion. This was con­cep­tu­ally a ma­jor break­through but there were sev­er­al cru­cial obstacles still to be over­come when I left the In­sti­tute and re­turned per­man­ently to Ox­ford at the end of 1972. Shortly after my re­turn I solved the out­stand­ing tech­nic­al prob­lems. One of these in­volved us­ing stand­ard for­mu­lae from a clas­sic­al text­book on heat con­duc­tion, which I found rather amus­ing. More im­port­ant was the psy­cho­lo­gic­al ef­fect of feel­ing that my re­turn to Ox­ford had star­ted off well, and that the dif­fi­cult de­cision to leave the In­sti­tute for Ad­vanced Study would not turn out to have been dis­astrous.

In many ways the pa­pers on spec­tral asym­metry were per­haps the most sat­is­fy­ing ones I was in­volved with. The way they stretched over dif­fer­en­tial geo­metry, to­po­logy, and ana­lys­is with a nod in the dir­ec­tion of num­ber the­ory ap­pealed greatly to me. At the time these pa­pers had only a mod­est im­pact but, a few years later when con­tact was made with the­or­et­ic­al phys­ics, they be­came ex­tremely pop­u­lar. In par­tic­u­lar Wit­ten’s work on glob­al an­om­alies brought our \( \eta \)-in­vari­ant in­to prom­in­ence, in a way which we could nev­er have fore­seen.

Sadly these pa­pers were the end of my col­lab­or­a­tion with Pat­odi. He re­turned to the Tata In­sti­tute and we con­tin­ued to cor­res­pond for a while but later his health de­teri­or­ated, and a kid­ney trans­plant be­came the only hope. I was in­volved in the med­ic­al dis­cus­sions con­cern­ing this and at one stage it was planned to bring him to Ox­ford for the pur­pose. However, this was even­tu­ally deemed un­ne­ces­sary and plans were made in Bom­bay, but sadly Pat­odi died of some pre-op­er­a­tion com­plic­a­tions. It was a tra­gic loss both per­son­al and math­em­at­ic­al. Pat­odi was a math­em­atician of real ori­gin­al­ity and power. He was charm­ingly mod­est, with a friendly and cap­tiv­at­ingly simple dis­pos­i­tion.

Al­though el­lipt­ic dif­fer­en­tial equa­tions con­sti­tuted the centre of my math­em­at­ic­al in­terest for many years, there was an in­ter­est­ing col­lab­or­a­tion with Bott and Gård­ing on hy­per­bol­ic dif­fer­en­tial equa­tions which led to the pa­pers [58], [24], [34] Our col­lab­or­a­tion began in an un­usu­al way. Bott was stay­ing at the time in Ox­ford and Gård­ing came from Lund for a few weeks. He said he had a prob­lem for us. There was this im­port­ant but ob­scure pa­per of Pet­rovsky which in­volved some ho­mo­logy of al­geb­ra­ic vari­et­ies. Bott and I were es­sen­tially con­trac­ted to un­der­stand and ex­plain Pet­rovsky’s pa­per. We were at the time very ig­nor­ant about hy­per­bol­ic equa­tions, but we had Lars Gård­ing a world ex­pert and ex­cel­lent tu­tor. In re­turn we in­struc­ted him in to­po­logy, and so the col­lab­or­a­tion began with mu­tu­al edu­ca­tion. In due course we man­aged to un­der­stand Pet­rovsky, then to mod­ern­ize and gen­er­al­ize, lead­ing even­tu­ally to our rather lengthy joint pa­per. Al­to­geth­er it was an en­joy­able and in­struct­ive epis­ode.

An­oth­er part of my edu­ca­tion on ana­lys­is had of course been go­ing on for some time with Sing­er (and at oth­er times Hör­mander) as tu­tor. In par­tic­u­lar I learnt from Sing­er, who had a strong back­ground in func­tion­al ana­lys­is, about von Neu­mann al­geb­ras of type II with their pe­cu­li­ar real-val­ued di­men­sions. We real­ized that \( K \)-the­ory and in­dex the­ory could be gen­er­al­ized in this dir­ec­tion, but it was not clear at first if such a gen­er­al­iz­a­tion would really be of any in­terest. However in one par­tic­u­larly simple case, that of von Neu­mann al­geb­ras, were quite nat­ur­al and led to con­crete non-trivi­al res­ults. Later on in the hands of Alain Connes, the world ex­pert on the sub­ject, these simple ideas were enorm­ously ex­ten­ded and de­veloped in­to a whole the­ory of lin­ear ana­lys­is for fo­li­ations.

For many years I had taken a gen­er­al in­terest in the rep­res­ent­a­tion the­ory of non-com­pact semi-simple groups. In fact this was such a ma­jor in­dustry at Prin­ceton, and it had so many rami­fic­a­tions, that it was im­possible to ig­nore it. On the oth­er hand the work of Har­ish-Chandra was for­bid­dingly tech­nic­al and I con­stantly hoped that more geo­met­ric­al meth­ods might lead to a con­cep­tu­al sim­pli­fic­a­tion. In par­tic­u­lar I was at­trac­ted to the idea of real­iz­ing the dis­crete series rep­res­ent­a­tions by solu­tions of the Dir­ac equa­tion. This seemed a nat­ur­al gen­er­al­iz­a­tion of the case of com­pact groups where the Borel–Weil the­or­em could also, as I knew, be re­in­ter­preted in terms of the Dir­ac op­er­at­or. I had a num­ber of dis­cus­sions on those top­ics with Wil­fred Schmid, after which I real­ized that my pa­per [44] could be used as an ex­ist­ence the­or­em for square-in­teg­rable har­mon­ic spinors on suit­able ho­mo­gen­eous spaces. I wrote to Schmid, ex­plain­ing this idea, and he soon saw how one could really de­vel­op much of the the­ory in de­tail from this start­ing point. However, to avoid re­ly­ing on Har­ish-Chandra’s work it was ne­ces­sary to find an al­tern­at­ive dir­ect proof of the fun­da­ment­al the­or­em on the loc­al in­teg­rabil­ity of the ir­re­du­cible char­ac­ters. In 1975 Schmid and I both spent a term at the In­sti­tute for Ad­vanced Study, and dur­ing that time we worked out a reas­on­ably sat­is­fact­ory proof of the loc­al in­teg­rabil­ity based on a care­ful study of in­vari­ant dif­fer­en­tial op­er­at­ors and con­jugacy classes. We planned to write this up as a joint pa­per, but first we de­cided to write up quickly the work on the dis­crete series. Un­for­tu­nately this took much longer than planned, end­ing up as a much more sub­stan­tial pa­per [45] than ori­gin­ally planned. The oth­er pro­ject, on the loc­al in­teg­rabil­ity, got post­poned in­def­in­itely but the es­sen­tial ideas were ex­plained in a series of lec­tures [53] I gave at Ox­ford in the Spring of 1976.

An­oth­er set of lec­tures [40] giv­en at a sum­mer school in Italy in 1975, give a leis­urely ac­count of in­dex the­ory. Al­though not con­tain­ing any new ma­ter­i­al, these lec­ture notes re­main per­haps a use­ful quick in­tro­duc­tion to the sub­ject.

The \( \eta \)-in­vari­ant which was in­tro­duced in my joint pa­per with Pat­odi and Sing­er had been, in part, mo­tiv­ated by Hirzebruch’s work on cusp sin­gu­lar­it­ies and in par­tic­u­lar his res­ult ex­press­ing the sig­na­ture de­fect in terms of val­ues of \( L \)-func­tions of real quad­rat­ic fields. Sing­er and I saw how to pre­serve Hirzebruch’s for­mula from our gen­er­al res­ults and we planned to ex­tend this to deal with the con­jec­tured for­mula for totally real fields of any de­gree. However, there were some ana­lyt­ic­al dif­fi­culties in the gen­er­al case which needed care­ful treat­ment, so that mat­ter was post­poned. Many years later with the as­sist­ance of Har­old Don­nelly we com­pleted the pro­gramme res­ult­ing in the pa­pers [49], [50]. In­de­pend­ently, and al­most sim­ul­tan­eously, sim­il­ar res­ults were ob­tained by W. Müller in East Ger­many.

Works

[1]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors on com­pact man­i­folds,” Bull. Amer. Math. Soc. 69 (1963), pp. 422–​433. Re­prin­ted in Sémin­aire Bourbaki. 15e an­née: 1962/63 (1964) and Sémin­aire Bourbaki 8 (1995). MR 0157392 Zbl 0118.​31203 article

[2]M. F. Atiyah and R. Bott: “The in­dex prob­lem for man­i­folds with bound­ary,” pp. 175–​186 in Dif­fer­en­tial Ana­lys­is: Pa­pers presen­ted at the in­ter­na­tion­al col­loqui­um (Bom­bay, 7–14 Janu­ary 1964). Tata In­sti­tute of Fun­da­ment­al Re­search Stud­ies in Math­em­at­ics 2. Ox­ford Uni­versity Press (Lon­don), 1964. MR 0185606 Zbl 0163.​34603 incollection

[3]M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors on com­pact man­i­folds” in Sémin­aire Bourbaki. 15e an­née: 1962/63. Sémin­aire Bourbaki. Secrétari­at Math­ématique (Par­is), 1964. Ex­posé no. 253. Re­print of art­icle in Bull. Am. Math. Soc. 69 (1963). See also Sémin­aire Bourbaki 8 (1995). Zbl 0124.​31102 incollection

[4]M. F. Atiyah: “The in­dex the­or­em for man­i­folds with bound­ary,” pp. 337–​351 in Sem­in­ar on the Atiyah–Sing­er in­dex the­or­em. Edi­ted by R. S. Pal­ais. An­nals of Math­em­at­ics Stud­ies 57. Prin­ceton Uni­versity Press, 1965. Ap­pendix I. Atiyah’s sole con­tri­bu­tion to Sem­in­ar on the Atiyah–Sing­er in­dex the­or­em (1965). Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 3. See also sim­il­arly-titled art­icle in Dif­fer­en­tial ana­lys­is (1965). incollection

[5]M. F. Atiyah and R. Bott: “A Lef­schetz fixed point for­mula for el­lipt­ic dif­fer­en­tial op­er­at­ors,” Bull. Am. Math. Soc. 72 : 2 (1966), pp. 245–​250. MR 0190950 Zbl 0151.​31801 article

[6]M. F. Atiyah and R. Bott: “A Lef­schetz fixed point for­mula for el­lipt­ic com­plexes, I,” Ann. Math. (2) 86 : 2 (1967), pp. 374–​407. MR 0212836 Zbl 0161.​43201 article

[7]M. F. Atiyah: “A Lef­schetz fixed-point for­mula for el­lipt­ic dif­fer­en­tial op­er­at­ors,” pp. 38–​39 in Sim­posio in­ternazionale di geo­metria al­gebrica (Rome, 30 Septem­ber–5 Oc­to­ber 1965), published as Rend. Mat. Ap­pl. V : 25. Issue edi­ted by G. Castel­nuovo. Cre­monese (Rome), 1967. See also Bull. Amer. Math. Soc. 72:2 (1966). Zbl 0149.​41201 incollection

[8]M. F. Atiyah: “Al­geb­ra­ic to­po­logy and el­lipt­ic op­er­at­ors,” Comm. Pure Ap­pl. Math. 20 (1967), pp. 237–​249. MR 0211418 Zbl 0145.​43804 article

[9]M. F. Atiyah and G. B. Segal: “The in­dex of el­lipt­ic op­er­at­ors, II,” Us­pehi Mat. Nauk 23 : 6 (144) (1968), pp. 135–​149. Rus­si­an trans­la­tion of art­icle in Ann. Math. 87:3 (1968). MR 0236953 article

[10]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

[11]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, I,” Ann. Math. (2) 87 : 3 (1968), pp. 484–​530. Rus­si­an trans­la­tion pub­lished in Usp. Mat. Nauk 23:5(143) (1968). MR 0236950 Zbl 0164.​24001 article

[12]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, III,” Ann. Math. (2) 87 : 3 (1968), pp. 546–​604. Rus­si­an trans­la­tion pub­lished in Us­pehi Mat. Nauk 24:1(145) (1969). MR 0236952 Zbl 0164.​24301 article

[13]M. F. Atiyah and R. Bott: “A Lef­schetz fixed point for­mula for el­lipt­ic com­plexes, II: Ap­plic­a­tions,” Ann. Math. (2) 88 : 3 (November 1968), pp. 451–​491. MR 0232406 Zbl 0167.​21703 article

[14]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, I,” Us­pehi Mat. Nauk 23 : 5 (143) (1968), pp. 99–​142. Rus­si­an trans­la­tion of art­icle from Ann. Math. 87:3 (1968). MR 0232402 article

[15]M. F. Atiyah: “Glob­al as­pects of the the­ory of el­lipt­ic dif­fer­en­tial op­er­at­ors,” pp. 57–​64 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­at­ics (Mo­scow, 16–26 Au­gust 1966). Mir (Mo­scow), 1968. MR 0233378 Zbl 0204.​41902 incollection

[16]M. F. Atiyah: “Hy­per­bol­ic dif­fer­en­tial equa­tions and al­geb­ra­ic geo­metry (after Pet­rowsky),” pp. 87–​99 in Sémin­aire Bourbaki 1966/1967. W. A. Ben­jamin (New York and Am­s­ter­dam), 1968. Ex­posé no. 319. Re­pub­lished in 1995. Zbl 0201.​12501 incollection

[17]M. F. Atiyah and I. M. Sing­er: “In­dex the­ory for skew-ad­joint Fred­holm op­er­at­ors,” Inst. Hautes Études Sci. Publ. Math. 37 : 1 (January 1969), pp. 5–​26. MR 0285033 Zbl 0194.​55503 article

[18]M. F. Atiyah: “The sig­na­ture of fibre-bundles,” pp. 73–​84 in Glob­al ana­lys­is: Pa­pers in hon­or of K. Kodaira. Edi­ted by S. Iy­anaga and D. C. Spen­cer. Prin­ceton Math­em­at­ic­al Series 29. Uni­versity of Tokyo Press, 1969. MR 0254864 Zbl 0193.​52302 incollection

[19]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, III,” Us­pehi Mat. Nauk 24 : 1 (145) (1969), pp. 127–​182. Rus­si­an trans­la­tion of art­icle in Ann. Math. 87:3 (1968). MR 0256417 article

[20]M. F. Atiyah: “To­po­logy of el­lipt­ic op­er­at­ors,” pp. 101–​119 in Glob­al ana­lys­is (Berke­ley, CA, 1–26 Ju­ly 1968). Edi­ted by S.-S. Chern and S. Smale. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 16. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1970. MR 0264700 Zbl 0207.​22601 incollection

[21]M. Atiyah and F. Hirzebruch: “Spin-man­i­folds and group ac­tions,” pp. 18–​28 in Es­says on to­po­logy and re­lated top­ics: Mé­m­oires dédiés à Georges de Rham. Edi­ted by A. Hae­fli­ger and R. Narasim­han. Spring­er (New York), 1970. MR 0278334 Zbl 0193.​52401 incollection

[22]M. F. Atiyah: “Glob­al the­ory of el­lipt­ic op­er­at­ors,” pp. 21–​30 in Func­tion­al ana­lys­is and re­lated top­ics (Tokyo, 1969). Edi­ted by S. T. Kur­oda. Uni­versity of Tokyo Press, 1970. MR 0266247 Zbl 0193.​43601 incollection

[23]M. F. Atiyah: “Res­ol­u­tion of sin­gu­lar­it­ies and di­vi­sion of dis­tri­bu­tions,” Comm. Pure Ap­pl. Math. 23 : 2 (1970), pp. 145–​150. MR 0256156 Zbl 0188.​19405 article

[24]M. F. Atiyah, R. Bott, and L. Gård­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, I,” Acta Math. 124 : 1 (July 1970), pp. 109–​189. A Rus­si­an trans­la­tion was pub­lished in Usp. Mat. Nauk 26:2(158). MR 0470499 Zbl 0191.​11203 article

[25]M. F. Atiyah: “Riemann sur­faces and spin struc­tures,” Ann. Sci. École Norm. Sup. (4) 4 : 1 (1971), pp. 47–​62. MR 0286136 Zbl 0212.​56402 article

[26]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, V,” Ann. Math. (2) 93 : 1 (1971), pp. 139–​149. Rus­si­an trans­la­tion pub­lished in Us­pehi Mat. Nauk 27:4(166) (1972). MR 0279834 article

[27]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, IV,” Ann. Math. (2) 93 : 1 (1971), pp. 119–​138. Rus­si­an trans­la­tion pub­lished in Us­pehi Mat. Nauk 27:4(166) (1972). MR 0279833 article

[28]M. F. At’ja, R. Bott, and L. Gord­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, I,” Usp. Mat. Nauk 26 : 2(158) (1971), pp. 25–​100. Rus­si­an trans­la­tion of an art­icle in Acta Math. 124:1 (1970). MR 0606062 Zbl 0208.​13201 article

[29]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, IV,” Us­pehi Mat. Nauk 27 : 4(166) (1972), pp. 161–​178. Rus­si­an trans­la­tion of Ann. Math. 93:1 (1971). MR 0385933 article

[30]M. F. Atiyah and I. M. Sing­er: “The in­dex of el­lipt­ic op­er­at­ors, V,” Usp. Mat. Nauk 27 : 4(166) (1972), pp. 179–​188. Rus­si­an trans­la­tion of Ann. Math. 93:1 (1971). Zbl 0237.​58018 article

[31]M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry,” Bull. Lon­don Math. Soc. 5 : 2 (1973), pp. 229–​234. MR 0331443 Zbl 0268.​58010 article

[32]M. Atiyah, R. Bott, and V. K. Pat­odi: “On the heat equa­tion and the in­dex the­or­em,” Matem­atika, Mosk­va 17 : 6 (1973), pp. 3–​48. Rus­si­an trans­la­tion of an art­icle in In­vent. Math. 19:4 (1973). Zbl 0364.​58016 article

[33]M. Atiyah, R. Bott, and V. K. Pat­odi: “On the heat equa­tion and the in­dex the­or­em,” In­vent. Math. 19 : 4 (1973), pp. 279–​330. Ded­ic­ated to Sir Wil­li­am Hodge on his 70th birth­day. Er­rata were pub­lished in In­vent. Math. 28:3 (1975). A Rus­si­an trans­la­tion was pub­lished in Matem­atika 17:6 (1973). MR 0650828 Zbl 0257.​58008 article

[34]M. F. Atiyah, R. Bott, and L. Gård­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, II,” Acta Math. 131 : 1 (December 1973), pp. 145–​206. A Rus­si­an trans­la­tion was pub­lished in Usp. Mat. Nauk 39:3(237). MR 0470500 Zbl 0266.​35045 article

[35]M. F. Atiyah: The in­dex of el­lipt­ic op­er­at­ors, 1973. Notes dis­trib­uted at sev­enty-ninth an­nu­al meet­ing of the Amer­ic­an Math­em­at­ic­al So­ci­ety, Dal­las, TX, 25–28 Janu­ary 1973. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 3, pp. 475–498. Re­pub­lished in Fields Medal­lists’ lec­tures (1997). Zbl 0263.​58012 misc

[36]M. F. Atiyah: El­lipt­ic op­er­at­ors and com­pact groups. Lec­ture Notes in Math­em­at­ics 401. Spring­er (Ber­lin), 1974. MR 0482866 Zbl 0297.​58009 book

[37]M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry, II,” Math. Proc. Cam­bridge Philos. Soc. 78 : 3 (1975), pp. 405–​432. MR 0397798 Zbl 0314.​58016 article

[38]M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry, I,” Math. Proc. Cam­bridge Philos. Soc. 77 : 1 (1975), pp. 43–​69. MR 0397797 Zbl 0297.​58008 article

[39]M. F. Atiyah: “Ei­gen­val­ues and Rieman­ni­an geo­metry,” pp. 5–​9 in Man­i­folds (Tokyo, 1973). Edi­ted by A. Hat­tori and N. Sūgakkai. Uni­versity of Tokyo Press, 1975. MR 0372928 Zbl 0319.​53030 incollection

[40]M. F. Atiyah: “Clas­sic­al groups and clas­sic­al dif­fer­en­tial op­er­at­ors on man­i­folds,” pp. 5–​48 in Dif­fer­en­tial op­er­at­ors on man­i­folds (Var­enna, Italy, 24 Au­gust–2 Septem­ber 1975). Edi­ted by E. Vesentini. Cre­monese (Rome), 1975. MR 0650830 incollection

[41]M. Atiyah, R. Bott, and V. K. Pat­odi: “Er­rata to: ‘On the heat equa­tion and the in­dex the­or­em’,” In­vent. Math. 28 : 3 (1975), pp. 277–​280. Er­rata for art­icle in In­vent. Math. 19:4 (1973). MR 0650829 Zbl 0301.​58018 article

[42]M. F. Atiyah and E. Rees: “Vec­tor bundles on pro­ject­ive 3-space,” In­vent. Math. 35 : 1 (1976), pp. 131–​153. MR 0419852 article

[43]M. F. Atiyah, V. K. Pat­odi, and I. M. Sing­er: “Spec­tral asym­metry and Rieman­ni­an geo­metry, III,” Math. Proc. Cam­bridge Philos. Soc. 79 : 1 (1976), pp. 71–​99. MR 0397799 Zbl 0325.​58015 article

[44]M. F. Atiyah: “El­lipt­ic op­er­at­ors, dis­crete groups and von Neu­mann al­geb­ras,” pp. 43–​72 in Col­loque “Ana­lyse et To­po­lo­gie” en l’hon­neur de Henri Cartan (Or­say, 1974). As­térisque 32–​33. So­ciété Math­ématique de France (Par­is), 1976. MR 0420729 Zbl 0323.​58015 incollection

[45]M. Atiyah and W. Schmid: “A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups,” In­vent. Math. 42 : 1 (1977), pp. 1–​62. Re­pub­lished in Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (1980). Er­rata pub­lished in In­vent. Math. 54:2 (1979). MR 0463358 Zbl 0373.​22001 article

[46]M. Atiyah and W. Schmid: “Er­rat­um: ‘A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups’,” In­vent. Math. 54 : 2 (1979), pp. 189–​192. Er­rat­um for art­icle in In­vent. Math. 42:1 (1977). See also Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (1980). MR 550183 article

[47]M. Atiyah and W. Schmid: “A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups,” pp. 317–​378 in Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (Liège, 5–17 Septem­ber 1977). Edi­ted by J. A. Wolf, M. Ca­hen, and M. de Wilde. Math­em­at­ic­al Phys­ics and Ap­plied Math­em­at­ics 5. D. Re­idel (Dordrecht), 1980. Re­pub­lished from In­vent. Math. 42:1 (1977). Zbl 0466.​22012 incollection

[48]M. Atiyah and W. Schmid: “Er­rat­um to the art­icle ‘A geo­met­ric con­struc­tion of the dis­crete series for semisimple Lie groups’” in Har­mon­ic ana­lys­is and rep­res­ent­a­tions of semisimple Lie groups (Liège, 5–17 Septem­ber 1977). Edi­ted by J. A. Wolf, M. Ca­hen, and M. de Wilde. Math­em­at­ic­al Phys­ics and Ap­plied Math­em­at­ics 5. D. Re­idel (Dordrecht), 1980. Er­rat­um for art­icle in same volume. Re­pub­lished from In­vent. Math. 54:2 (1979). Zbl 0466.​22013 incollection

[49]M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Geo­metry and ana­lys­is of Shim­izu \( L \)-func­tions,” Proc. Nat. Acad. Sci. U.S.A. 79 : 18 (1982), pp. 5751. MR 674920 Zbl 0503.​12007 article

[50]M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Eta in­vari­ants, sig­na­ture de­fects of cusps, and val­ues of \( L \)-func­tions,” Ann. Math. (2) 118 : 1 (1983), pp. 131–​177. An ad­dendum was pub­lished in Ann. Math. 119:3 (1984). MR 707164 Zbl 0531.​58048 article

[51]M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Sig­na­ture de­fects of cusps and val­ues of \( L \)-func­tions: The non­split case. Ad­dendum to: ‘Eta in­vari­ants, sig­na­ture de­fects of cusps, and val­ues of \( L \)-func­tions’,” Ann. Math. (2) 119 : 3 (1984), pp. 635–​637. Ad­dendum to an art­icle in Ann. Math. 118:1 (1983). MR 744866 article

[52]M. F. At’ya, R. Bott, and L. Gård­ing: “La­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors with con­stant coef­fi­cients, II,” Usp. Mat. Nauk 39 : 3(237) (1984), pp. 171–​224. Rus­si­an trans­la­tion of an art­icle in Acta Math. 131:1 (1973). MR 747794 Zbl 0568.​35058 article

[53]M. F. Atiyah: “Char­ac­ters of semi-simple Lie groups,” pp. 489–​558 in Col­lec­ted works, vol. 4: In­dex the­ory 2. Ox­ford Sci­ence Pub­lic­a­tions. Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 1988. Lec­tures giv­en at Math­em­at­ic­al In­sti­tute, Ox­ford, 1976. incollection

[54]M. Atiyah: Col­lec­ted works, vol. 4: In­dex the­ory 2. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford and New York), 1988. MR 951895 book

[55]M. Atiyah: Col­lec­ted works, vol. 3: In­dex the­ory 1. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford and New York), 1988. MR 951894 Zbl 0724.​53001 book

[56]M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors,” pp. 475–​498 in Col­lec­ted works, vol. 3: In­dex the­ory 1. Ox­ford Sci­ence Pub­lic­a­tions. Ox­ford Uni­versity Press, 1988. Col­loqui­um Lec­tures (Dal­las, 1973), Amer­ic­an Math­em­at­ic­al So­ci­ety.

[57]V. Vassiliev: “The math­em­at­ic­al leg­acy of ‘La­cunas for hy­per­bol­ic dif­fer­en­tial equa­tions’,” pp. xxiii–​xxviii in Raoul Bott: Col­lec­ted pa­pers, vol. 2: Dif­fer­en­tial op­er­at­ors. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston, MA), 1994. Com­ment­ary on two-part art­icle pub­lished in Acta Math. 124 (1970) and Acta Math. 131 (1973). MR 1290364 incollection

[58]M. F. Atiyah: “Hy­per­bol­ic dif­fer­en­tial equa­tions and al­geb­ra­ic geo­metry (after Pet­rowsky),” pp. 87–​99 in Sémin­aire Bourbaki 10: An­nées 1966/67–1967/68. So­ciété Math­ématique de France (Par­is), 1995. Ex­posé no. 319. Re­pub­lic­a­tion of 1968 ori­gin­al. MR 1610456 incollection

[59]M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors on com­pact man­i­folds,” pp. 159–​169 in Sémin­aire Bourbaki, vol. 8, An­nées 1962–63, 1963–64. As­térisque. Sémin­aire Bourbaki. So­ciété Math­ématique de France (Par­is), 1995. Ex­posé no. 253. Re­print of art­icle in Bull. Am. Math. Soc. 69 (1963). See also Sémin­aire Bourbaki. 15e an­née: 1962/63 (1964). MR 1611539 incollection

[60]M. F. Atiyah: “The in­dex of el­lipt­ic op­er­at­ors,” pp. 115–​127 in Fields Medal­lists’ lec­tures. Edi­ted by M. F. Atiyah and D. Iag­ol­nitzer. World Sci­entif­ic Series in 20th Cen­tury Math­em­at­ics 5. World Sci­entif­ic (River Edge, NJ), 1997. Re­pub­lic­a­tion of notes dis­trib­uted at AMS con­fer­ence (1973). MR 1622942 incollection