Celebratio Mathematica

Is­ad­ore Manuel Sing­er (“Is” or “I.M.”) was born in De­troit on 3 May 1924. His par­ents, Freda Rose and Si­mon Sing­er, came to Canada from Po­land in 1917, and mar­ried there. Freda was an orphan, but was taught by three uncles, a writer, a law­yer, and a print­er. Later she was a seam­stress. Si­mon was a print­er in Warsaw, and ac­cord­ing to fam­ily lore he set the first pub­lished story of his cous­in, the cel­eb­rated au­thor Is­rael Joshua Sing­er (broth­er of the No­bel laur­eate Isaac Bashev­is Sing­er).

Once in Canada, Si­mon and Freda in­ten­ded to slip over the bor­der in­to the US, cross­ing from Wind­sor to De­troit. Upon ar­riv­ing in De­troit, Si­mon dis­covered that the print­ers were on strike, and they offered him the roy­al sum of \$5 a day to join the strike. Si­mon wrote back home to Po­land to say that it was true that the streets of Amer­ica were paved in gold!

As a young child, Is learned to read while watch­ing and help­ing his fath­er type­set, even be­fore at­tend­ing kinder­garten, and he de­clares that his moth­er was the smartest per­son he ever met.

Is’ young­er broth­er, Sid­ney Sing­er, was born in 1929. He at­ten­ded Wayne State Uni­versity and then earned his Ph.D. in phys­ics and math­em­at­ics at the Uni­versity of Illinois. He went on to a long ca­reer at Los Alam­os Na­tion­al Labor­at­ory.

Is re­mem­bers the De­pres­sion years well; times were tough and his fath­er was out of a job for a year. Those years in com­par­at­ive poverty left Is with com­pas­sion for those in need.

Al­though a straight A stu­dent, Is found school bor­ing, and pre­ferred to play sand­lot base­ball and oth­er sports (he was an avid De­troit Ti­gers fan). But in his ju­ni­or year in high school he had ex­cel­lent teach­ers in Eng­lish lit­er­at­ure (they read War and Peace, and Is’ old­est daugh­ter is named Nata­sha after the char­ac­ter in the book) and chem­istry. He found the peri­od­ic table of ele­ments es­pe­cially fas­cin­at­ing. He re­mem­bers de­liv­er­ing Forverts (Yid­dish Daily For­ward) be­fore school and then read­ing nov­els at night.

Fas­cin­ated by Ein­stein, he re­called in his lec­ture for the 1979 Ein­stein centen­ni­al con­fer­ence ([e3], p. 39):

I gave my first series of talks on re­lativ­ity forty years ago to a high school sci­ence club. I can’t ima­gine what my peers thought of me at the time; it took me twenty years to un­der­stand what I was talk­ing about.

Near­ing gradu­ation from high school, Is ex­pec­ted to go to Wayne State in De­troit and live at home. But he also ap­plied to the Uni­versity of Michigan, and in a com­pet­i­tion won a fel­low­ship pay­ing \$50 a semester which ex­actly covered tu­ition. His moth­er told him that he couldn’t pass up \$50, so he ma­tric­u­lated at UM in 1941.

His first year at Michigan was not aca­dem­ic­ally chal­len­ging but Is’ in­terest was piqued when his cal­cu­lus in­struct­or ex­plained “how New­ton de­rived the el­lipt­ic or­bits of plan­ets from his in­verse square law” ([e11], p. 25). “Awe­some!” he re­mem­bers think­ing. In high school, he had been equally in­ter­ested in Eng­lish lit­er­at­ure and phys­ics, and re­calls that when he took a po­etry class at UM, the oth­er stu­dents in­stinct­ively un­der­stood what he did not, where­as in phys­ics it was the oth­er way around. So at UM he soon settled on phys­ics, and though he was al­ways ob­liged to work part-time, he took a heavy load of courses, in­clud­ing math and phys­ics, and fin­ished in just two and a half years.

When World War II began, Is joined the Army Sig­nal Corps for they al­lowed him to stay in col­lege un­til he com­pleted his de­gree, though they re­quired him also to take courses in elec­tric­al en­gin­eer­ing. One reas­on that he fin­ished his un­der­gradu­ate de­gree so quickly was that he was skep­tic­al the Army would let him study a full four years without call­ing him up to act­ive duty. He sub­sequently came to be­lieve that those fast-paced years at Michigan were good train­ing for his time at MIT.

On his way from the east coast to the Phil­ip­pines in 1944, he stopped in Chica­go to in­quire about ex­ten­sion courses at the Uni­versity of Chica­go, where he learned that he could take ex­ten­sion classes in geo­metry and group the­ory re­motely, both use­ful for un­der­stand­ing phys­ics bet­ter. Thus, while he ran a com­mu­nic­a­tions school for the Phil­ip­pine Army dur­ing his war years, he de­voted his off-hours not to poker with his fel­low sol­diers, but to his Uni­versity of Chica­go courses.1

In ([e3], p. 38), Is re­min­isced about the con­sequences of his ex­ten­sion stud­ies for his fu­ture ca­reer:

My in­clin­a­tion to­ward in­tel­lec­tu­al pur­suits was en­cour­aged by my Eng­lish and chem­istry teach­ers in Cent­ral High School, De­troit. I ma­jored in phys­ics at the Uni­versity of Michigan (1941–44) but re­gret not avail­ing my­self of their mar­velous math­em­at­ics de­part­ment. Puzzled by both re­lativ­ity and quantum mech­an­ics, I tried to mas­ter both these sub­jects while serving in the U.S. Army. Hav­ing failed, I con­cluded I needed a year of math­em­at­ics be­fore I re­turned to phys­ics, and I be­came a gradu­ate stu­dent at the Uni­versity of Chica­go in 1947. That year stretched to 30.

With the war over, Is ap­plied to the math de­part­ment at Chica­go, telling them he wanted more math­em­at­ic­al train­ing be­fore switch­ing to phys­ics. They ad­mit­ted him in Janu­ary, 1947, and he re­ceived a Mas­ters De­gree in 1948. Struck with the beauty of math­em­at­ics, he aban­doned phys­ics and de­cided to work to­ward his Ph.D. in math in­stead.

Al­though S.-S. Chern and Irving Segal were both ment­ors to Is at Chica­go, Chern did not ar­rive un­til Is’ last year, 1949–50, so Segal be­came his Ph.D. ad­viser. Is de­scribes Segal as a “won­der­ful and lousy teach­er at the same time”. Won­der­ful be­cause he was teach­ing great ma­ter­i­al in func­tion­al ana­lys­is, but lousy be­cause he be­came stuck about halfway through most lec­tures. However “lousy” meant that Is would go home and work hard on the sticky points, of­ten work­ing things out for him­self. This pro­ced­ure meant that he was im­mersed in something akin to the Moore meth­od in teach­ing, where stu­dents must work out con­cepts and prin­ciples for them­selves as much as pos­sible. Segal typ­ic­ally worked late in­to the even­ing in his of­fice, and was thus of­ten avail­able to Is for math talk, and not just about thes­is work. So the “Moore meth­od” worked quite well in Is’ case.

Upon fin­ish­ing his Ph.D., Is re­ceived two of­fers, one from the In­sti­tute for Ad­vanced Study (IAS) and one from MIT, the lat­ter a slightly bet­ter of­fer at \$50 more per year.

Is went to MIT as a Moore In­struct­or, a pro­gram with out­side sup­port be­gun un­der Ted Mar­tin in 1948 which al­lowed for a lower teach­ing load.2

The day Is ar­rived at MIT and met War­ren Am­brose is an oft-told story, nowhere bet­ter than in this trib­ute to Am­brose writ­ten by Is him­self in 1966 [15].

I have not for­got­ten my first day at MIT. In 1950 Moore in­struct­ors had to teach sum­mer school. On a sunny af­ter­noon early in Ju­ly, I crossed the bridge in search of Build­ing 2. Math headquar­ters was on the second floor; it still is. I in­tro­duced my­self to Ruth Good­win, who handled all sec­ret­ari­al ser­vices, and I asked to see the chair­man. When I gave Ruth my name, a chap sit­ting across from her, head bur­ied in the Bo­ston Globe lowered his pa­per and said: “Sing­er, I’m Am­brose. There is a sem­in­ar in Lie groups in five minutes. You can see Mar­tin later. Come.”

I did, and met John Moore, Bar­rett O’Neill, and George White­head, who be­came lifelong friends. After get­ting my teach­ing as­sign­ment from Ted Mar­tin, Am­brose told me the sem­in­ar met at mid­night in the Hayes-Bick­ford cof­fee shop. He would pick me up at 11:45. Kay White­head joined us for these even­ing ses­sions; the cof­fee was deadly, the con­ver­sa­tion lively. Am­brose gave me a tour of Bo­ston that first night and by the time he dropped me off, we were close friends.

One day Am­brose said, “Sing­er, you listened to Chern’s lec­tures. What did he say?” At Chica­go, I had pass­ively taken notes of S. S. Chern’s course, while writ­ing a dis­ser­ta­tion in an­oth­er sub­ject. What with in­ter­pret­ing my notes, read­ing Chern’s pa­pers, pour­ing over Élie Cartan, and Am­brose in­sist­ing on ab­so­lute clar­ity in every de­tail, we learned dif­fer­en­tial geo­metry to­geth­er.

Am­brose de­signed the Geo­metry of Man­i­folds course, and we taught it in al­tern­ate years. It is pretty much the same today as it was then: stand­ard man­i­fold the­ory the first term and in­struct­or’s choice of top­ics the second. Our stu­dents wrote some well-known gradu­ate texts based on this course: Bish­op–Crit­tenden, Hicks, and Warner.

With cus­tom­ary zeal, Am­brose changed the un­der­gradu­ate pro­gram in pure math­em­at­ics. Where­as in 1948 An­dré Weil ex­plained dif­fer­en­tial forms to the fac­ulty at the Uni­versity of Chica­go, less than a dec­ade later we were us­ing them in un­der­gradu­ate dif­fer­en­tial geo­metry.

Am­brose taught the Le­besgue in­teg­ral in the ana­lys­is course for ju­ni­ors and seni­ors “be­cause it’s sim­pler than the Riemann in­teg­ral.” For al­most twenty years Am­brose was the guid­ing spir­it of pure math­em­at­ics at MIT. His ef­forts were key in mak­ing it a great de­part­ment. Am­brose and I reg­u­larly drove around Bo­ston late at night talk­ing about math­em­at­ics and life. We knew every street; and to this day, with the in­ev­it­able traffic jam, I’ll re­mem­ber an al­tern­ate route and, of­ten enough, a spe­cial mo­ment: yes, here is where we fi­nally un­der­stood holonomy.

Am­brose taught me jazz. I had been an en­thu­si­ast of Dixie­land and swing. But by 1950 bop was dom­in­ant. Charlie Park­er was king, and every­one else his stu­dent. We heard all the great jazz mu­si­cians at all the jazz joints. There was a cas­u­al­ness and dir­ect­ness then that makes it dif­fi­cult for me to hear jazz live today. It’s too ri­gid and form­al now. Ima­gine watch­ing Bird take on a long line of young sax play­ers, listen­ing to each in­tently and then play­ing their vari­ation as it should be played. Teach­ing at its best, I felt. Ima­gine hav­ing cof­fee between sets with Bil­lie and thank­ing her for her early re­cords that meant so much to many of us. Dick Kadis­on and I used to hear Ella night after night at Bird­land when we took a break from work. Am­brose ex­plained jazz to me and talked about it in the fifties the way Wyn­ton Mar­s­al­is does now.

Though math­em­at­ics was won­der­ful, I was heav­ily burdened. My old­est son was blinded at birth and, as I learned later, brain dam­aged. I could not have sur­vived as a math­em­atician without Am­brose’s steady sup­port and the steady sup­port of my very good friend, Dick Kadis­on. Those who knew Am­brose know that ex­press­ing grat­it­ude was for­bid­den. He walked rap­idly away the one time I tried. Oc­ca­sion­ally, I can provide spe­cial help to a young math­em­atician. I think of Am­brose and feel that by the time I have helped a hun­dred, I’ll have be­gun to pay my debt to him.

I loved Am­brose for his ab­so­lute hon­esty, his gen­er­os­ity, his wit, his en­ergy, and above all for his ten­der­ness, which he tried so hard to hide. I am sad that so few math­em­aticians knew what a great man he was. But I was happy for him when he found his wife, Jean­nette, with whom he could be him­self for twenty years.

Re­flect­ing on the biggest dif­fer­ence between the Uni­versity of Chica­go and MIT, Is has said that at Chica­go teach­ing of un­der­gradu­ates was done by a des­ig­nated fac­ulty hired for the pur­pose. So Is saw very little of the un­der­grads. At MIT, by con­trast, the teach­ing of un­der­gradu­ates was shared by all fac­ulty.

Am­brose and a few oth­ers were very eager to re­vital­ize the cur­riculum, and they called upon Is to teach the fac­ulty, and then the stu­dents, the latest math­em­at­ics from Chica­go, es­pe­cially the per­spect­ives on dif­fer­en­tial geo­metry that Is learned from Chern [15].

Out of this in­tense work in geo­metry came the fam­ous Am­brose–Sing­er the­or­em re­lat­ing holonomy to curvature. For a nice de­scrip­tion of this the­or­em, see Hung-Hsi Wu’s trib­ute in the same No­tices [15].

After two years at MIT, Is went to UCLA for 1952–54, where he was wont to tease Ted Mar­tin, then chair at UCLA, by say­ing that MIT had sent him out to the minor leagues for fur­ther train­ing. Is then spent a year at Columbia Uni­versity, fol­lowed by a mo­ment­ous year at IAS with Atiyah, Raoul Bott, Fritz Hirzebruch, J.-P. Serre and Arnold Sha­piro, be­fore he even­tu­ally re­turned to MIT in 1956.

The Kadis­on–Sing­er prob­lem, posed in 1959 by Dick Kadis­on and Is [1], was mo­tiv­ated by Dir­ac’s work in quantum mech­an­ics. It was a prob­lem in func­tion­al ana­lys­is about wheth­er cer­tain ex­ten­sions of cer­tain lin­ear func­tion­als on cer­tain \( \operatorname{C}^* \)-al­geb­ras were unique. The unique­ness was proven in 2013 [e12], al­though the proof was greeted skep­tic­ally by Dick and Is.

Is got to know Mi­chael Atiyah at IAS in 1955, and when he wished to go to Eng­land on a sab­bat­ic­al, he asked Mi­chael, who had just ar­rived in Ox­ford, if there was a spot for him. Mi­chael im­me­di­ately said yes.

After ar­riv­ing in Ox­ford in Janu­ary 1962, and while warm­ing him­self at a little heat­er, Is looked up to see Mi­chael in the door­way. An im­mor­tal three-line con­ver­sa­tion fol­lowed:

Mi­chael: Why is the \( A \)-roof genus an in­teger for spin man­i­folds?

Is: Mi­chael, why are you ask­ing me that, you know the an­swer?

Mi­chael: Yes, I do know the an­swer, but I feel there is a deep­er reas­on.

Is had oth­er work on his mind (a joint pro­ject with Shlomo Stern­berg on the in­fin­ite groups of Élie Cartan [3], but Atiyah’s ques­tion in­trigued him. He spent the early spring, sit­ting on garden benches, think­ing about the prob­lem. By March he un­der­stood the is­sues and went to Mi­chael’s of­fice and said that in­tegers arise as the count of the num­ber of solu­tions of the Dir­ac op­er­at­or gen­er­al­ized to spin man­i­folds. And in fact in all the oth­er cases where in­tegers oc­cur (e.g., the sig­na­ture of a man­i­fold or the arith­met­ic genus), it is be­cause of count­ing solu­tions of geo­met­ric op­er­at­ors.

Mi­chael ran with that, pro­posed some for­mu­las, and their col­lab­or­a­tion began in earn­est. Nine months later they had a com­plete proof of the Atiyah–Sing­er in­dex the­or­em!

Atiyah gives his per­spect­ive on the in­dex the­or­em in his com­ment­ary be­gin­ning in the third volume of his col­lec­ted works [e8], ex­cerp­ted here:

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 co­hom­mo­logy. 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! […]

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 Gel’fand on the gen­er­al prob­lem of com­put­ing the in­dex of el­lipt­ic op­er­at­ors. […]

Sing­er and I had some great ad­vant­ages over the ana­lysts in­vest­ig­at­ing the prob­lem. We were in­vest­ig­at­ing a par­tic­u­lar case, the Dir­ac op­er­at­or, and we already ‘knew’ the an­swer. Also this case…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. […]

[…] Sing­er and I even­tu­ally pro­duced a proof of the gen­er­al in­dex the­or­em dur­ing my stay at Har­vard in the Fall in 1962. […] 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 presen­ted in de­tail in an Atiyah–Bott–Sing­er sem­in­ar at Har­vard in Fall 1962, was an­nounced in the Bul­let­in in 1963 [2], and was writ­ten up in de­tail in the Borel–Pal­ais sem­in­ar at the In­sti­tute for Ad­vanced Study in 1963 [e1]. This ver­sion was a proof us­ing cobor­d­isms, where one showed that both the ana­lyt­ic and to­po­lo­gic­al in­dex are in­vari­ant un­der cobor­d­isms, and then check­ing that both in­dices are equal on the gen­er­at­ors of the rel­ev­ant cobor­d­ism groups, as had been done by Hirzebruch in his proof of the Hirzebruch–Riemann–Roch the­or­em.

Bott ex­plains this in his math re­view:

The au­thors here de­scribe their solu­tion to the in­dex prob­lem for el­lipt­ic op­er­at­ors on closed man­i­folds. Their res­ult may also be thought of as a beau­ti­ful and far-reach­ing gen­er­al­iz­a­tion of Hirzebruch’s Riemann–Roch the­or­em — both in state­ment and in the spir­it of the proof.

The au­thors for­mu­late the in­dex prob­lem in the fol­low­ing gen­er­al set­ting. Let \( E \) and \( F \) be vec­tor-bundles over the com­pact man­i­fold \( X \), where everything is \( C^{\infty} \) throughout. It then makes sense to speak of a dif­fer­en­tial op­er­at­or \( D \) from \( E \) to \( F \), and such an op­er­at­or in­duces a lin­ear map \[ D:\Gamma(E)\to \Gamma(F) \] from the sec­tions of \( E \) to those of \( F \). When \( D \) is el­lipt­ic, both the ker­nel and the coker­nel of \( D \) are fi­nite-di­men­sion­al, and the dif­fer­ence of these di­men­sions is by defin­i­tion the in­dex of \( D \). Al­tern­ately, one has the equal­ity in­dex \[ \sigma(D)=\sum (-1)^i\operatorname{dim}H^i(X;K), \] where \( K \) is the ker­nel sheaf of \( D \), as fol­lows from rudi­ment­ary sheaf the­ory, and the loc­al “onto” prop­erty of el­lipt­ic op­er­at­ors: \begin{align*} H^0 (X;K) &\simeq \operatorname{ker}D,\\ H^1(X;K) &\simeq \operatorname{coker}D,\\ H^i(X;K) &=0,\quad i\geq2. \end{align*}

The “in­dex prob­lem” is to give a de­scrip­tion of this in­teger in terms of the to­po­lo­gic­al data im­pli­cit in the el­lipt­ic op­er­at­or. To de­scribe these one needs the fol­low­ing in­ter­pret­a­tion of the highest-or­der terms in \( D \). Let \( T(X) \) be the co­tan­gent bundle of \( X \), and let \( T_0(X) \) be the sub­set of nonzero vec­tors in \( T(X) \). The pro­jec­tion \( T(X)\to X \) will be de­noted by \( \pi \). With this un­der­stood, the highest-or­der terms of \( D \) are seen to define a def­in­ite ho­mo­morph­ism \[ \sigma(D):\pi^{\ast} E\to \pi^{\ast}F \] of the pulled-back bundles on \( T(X) \). Fur­ther, \( D \) is el­lipt­ic if and only if this ho­mo­morph­ism, called the sym­bol of \( D \), is an iso­morph­ism on \( T_0(X) \). This iso­morph­ism — or rather its stable ho­mo­topy class \( [\sigma (D)] \) — is to be thought of as the to­po­lo­gic­al “twist” of the el­lipt­ic op­er­at­or \( D \).

One may re­late \( [\sigma (D)] \) in vari­ous ways to more stand­ard to­po­lo­gic­al ob­jects. Maybe the simplest con­struc­tion is the fol­low­ing one. Let \( B(X) \) and \( S(X) \), re­spect­ively, stand for the unit ball and unit sphere bundle of \( T(X) \) en­dowed with some fixed Rieman­ni­an struc­ture. Let \[ W(X)=B(X)\cup^{}_{S(X)} B(X) \] be the man­i­fold ob­tained by glue­ing two cop­ies of\( B(X) \) to­geth­er along their bound­ary, i.e., \( W(X) \) is the doubled man­i­fold con­struc­ted from \( B(X) \). Now one uses \( \sigma(D) \) to con­struct a bundle \( E \cup_{\sigma(D)} F \) on \( W(X) \) by tak­ing \( \pi^*E \) on one copy of \( B(X) \), \( \pi^*F \) on the oth­er, and glue­ing them to­geth­er over \( S(X) \) by means of the iso­morph­ism \( \sigma(D) \). (Al­tern­at­ively and really equi­val­ently, one may use \( \sigma(D) \) to con­struct a dif­fer­ence ele­ment in \( K(B(X),S(X)) \) and that is the point of view taken in the pa­per un­der re­view.)

A for­mula quite equi­val­ent to the in­dex for­mula of the pa­per now is of the form \[\operatorname{index}(D)=\int_{W(X)} \operatorname{ch}(E \cup^{}_{\sigma(D)} F)\wedge \pi^*A(X). \] Here \( \operatorname{ch} \) de­notes the Chern char­ac­ter and \( A(X) \) is a dif­fer­en­tial form on \( X \) which we will not spe­cify here, but which is an ex­pli­cit poly­no­mi­al in the char­ac­ter­ist­ic classes of \( X \).

The in­dex for­mula eas­ily yields the res­ults of the pi­on­eers in this field such as Agran­ovič, Dyn­in, Gel’fand, See­ley, Vol’pert, etc. The for­mula is also seen to gen­er­al­ize the Riemann–Roch the­or­em of Hirzebruch to (not ne­ces­sar­ily) al­geb­ra­ic com­plex man­i­folds and bundles. (To this one need only fol­low up our earli­er ex­pres­sion for in­dex \( (D) \) as an Euler char­ac­ter­ist­ic.)

The proof of the in­dex for­mula, as well as its vari­ous con­sequences in spe­cial cases, is clearly sum­mar­ized in the note. For the sake of ana­lysts we re­mark only that the gen­er­al set­ting of op­er­at­ors on vec­tor-bundles — as op­posed to sys­tems — is es­sen­tial for the proof.

The proof us­ing cobor­d­ism the­ory suffered be­cause it did not work in the equivari­ant case, as the rel­ev­ant equivari­ant cobor­d­ism groups were not all known. Atiyah writes [e9]:

For some years Sing­er and I searched for a bet­ter proof mod­elled 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 [4] worked purely in a \( K \)-the­ory con­text and avoided ra­tion­al co­homo­logy. It there­fore lent it­self to vari­ous sig­ni­fic­ant gen­er­al­iz­a­tions de­veloped in the re­main­ing pa­pers of the series [e2], [5], [6], [7]. 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 non­isol­ated fixed points (of iso­met­ries), 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 [e2].

An In­di­an math­em­atician, V. K. Pat­odi, ob­tained re­mark­able res­ults ap­ply­ing the heat equa­tion to the in­dex the­or­em. Pat­odi joined Is and Atiyah in a sig­ni­fic­ant series of pa­pers [8], [9], [10], [11], es­tab­lish­ing a sig­na­ture the­or­em for man­i­folds with bound­ary, lead­ing to the \( \eta \)-in­vari­ant. In [e8] Atiyah re­marked:

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…. 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.

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.

Vic­tor Guille­min’s re­view of [8] gives a clear state­ment:

This pa­per an­nounces some very ex­cit­ing res­ults con­cern­ing the so-called \( \eta \)-in­vari­ant of a self-ad­joint (not ne­ces­sar­ily pos­it­ive) el­lipt­ic op­er­at­or. For such an op­er­at­or one can form: \[ \eta_A(s)=\sum_{\lambda \neq 0} \operatorname{sign}\lambda \, |\lambda|^{-s}, \] with the sum taken over all ei­gen­val­ues in the spec­trum of \( A \). This turns out to be a well defined mero­morph­ic func­tion of \( s \). One of the main res­ults an­nounced here is that \( \eta_A(s) \) is fi­nite at the ori­gin; hence the \( \eta \)-in­vari­ant of \( A \), \[ \eta (A) =\eta_A(0), \] is defined. In par­tic­u­lar take \( X \) to be a com­pact ori­ented Rieman­ni­an man­i­fold of di­men­sion \( 4k-1 \) and let \( A \) be the op­er­at­or act­ing on ex­ter­i­or dif­fer­en­tial forms of even de­gree giv­en by \[ A(\phi)=(-1)^{p+1}d* \phi +(-1)^p *d \phi, \] “\( * \)” be­ing the usu­al Hodge star op­er­at­or. Sup­pose that \( X \) is the bound­ary of an ori­ented Rieman­ni­an man­i­fold \( Y \) that is iso­met­ric near the bound­ary to the cyl­in­der \( X \times [0,1] \). Let \( p_i(Y) \) de­note the \( i \)-th Pon­trja­gin form of \( Y \). Then the au­thors link the \( \eta \)-in­vari­ant of \( A \) to the sig­na­ture of \( Y \) by the for­mula \[ \operatorname{sign}(Y)-\int_Y L(p) = (-1)^{k+1} \eta (A), \] where \( L \) is the Hirzebruch \( L \)-poly­no­mi­al. For \( Y \) bound­ary­less, the right-hand side is zero, and one re­trieves the usu­al Hirzebruch sig­na­ture for­mula. The au­thors also an­nounce an in­dex for­mula with bound­ary term for the Dir­ac op­er­at­or in­volving an \( \eta \)-in­vari­ant. They con­clude by giv­ing a brief out­line of the proof.

The \( \eta \)-in­vari­ant was partly mo­tiv­ated by Hirzebruch’s res­ult ex­press­ing the sig­na­ture de­fect in terms of val­ues of \( L \)-func­tions of real quad­rat­ic fields. Is and Atiyah with Har­old Don­nelly ex­ten­ded Hirzebruch’s for­mula to the case of totally real fields of any de­gree [14].

In 1977, with Atiyah and Nigel Hitchin, Is showed that the mod­uli space of \( \mathrm{SU}(2) \) in­stan­tons (with Pontry­agin in­dex equal to one) over \( X^4 \) is hy­per­bol­ic 5-space [12], [13]. Then Clif­ford Taubes [e5] ex­ten­ded this res­ult to the case of 4-man­i­folds \( X^4 \) with pos­it­ive def­in­ite in­ter­sec­tion forms; he showed that the 5-di­men­sion­al mod­uli space has as its com­pac­ti­fic­a­tion \( X^4 \) con­sist­ing of sin­gu­lar in­stan­tons at each point of \( X \). This, to­geth­er with work of Kar­en Uh­len­beck paved the way for Si­mon Don­ald­son’s spec­tac­u­lar in­vari­ants [e7] dis­tin­guish­ing many smooth 4-man­i­folds which are ho­mo­topy equi­val­ent and thus homeo­morph­ic by Freed­man’s work [e6]. A re­mark­able co­rol­lary (first ob­served by Freed­man and the au­thor ([e10], Chapter 10) is that \( \mathbb{R}^4 \) has un­count­ably many smooth struc­tures, some which em­bed smoothly in stand­ard \( \mathbb{R}^4 \) and some which do not. The 1977 pa­per was the fore­run­ner of these start­ling res­ults.

For per­son­al reas­ons Is went to Berke­ley in 1977. MIT was not happy about this de­cision, so they ar­ranged for Is to re­tire but to be brought back eas­ily if he was will­ing. Every year MIT would call and in 1984 he fi­nally said yes, for they caught him at a time when he was par­tic­u­larly an­noyed at de­part­ment­al polit­ics at Berke­ley (in­ter­view with au­thor, Decem­ber 1, 2017).

Is’ wife Rose­marie and daugh­ters Emily and An­na­belle did not want to leave Berke­ley, but things worked out well for both daugh­ters: Emily is now bi­otech ed­it­or of MIT Tech­no­logy Re­view and An­na­belle got her Ph.D. at UC­SF in brain and cog­nit­ive sci­ences and is now a postdoc at MIT.

While at Berke­ley, Is star­ted his now fam­ous geo­metry-math-gauge the­ory-phys­ics sem­in­ar, partly due to a wish from some of the young­er people at Berke­ley, in­clud­ing three of his best Ph.D. stu­dents — Dan Freed, Dan Friedan, and John Lott — who wrote theses in this area.3

He also joined forces dur­ing this peri­od with S.-S. Chern and Calv­in Moore to put to­geth­er a plan for a new in­sti­tute, the Math­em­at­ic­al Sci­ences Re­search In­sti­tute, to be loc­ated on the West Coast on the grounds of the Uni­versity of Cali­for­nia (though re­tain­ing its leg­al in­de­pend­ence) and to serve as a coun­ter­part to the In­sti­tute for Ad­vanced Study. With the com­bined weight of their repu­ta­tions, they eas­ily won an en­dorse­ment from the UC Chan­cel­lor, Mark Hey­man, and then per­suaded the Na­tion­al Sci­ence Found­a­tion to fund MSRI, des­pite the like­li­hood that a new in­sti­tute would re­duce the num­ber/size of the tra­di­tion­al in­di­vidu­al grants. Chern and Moore be­came the first Dir­ect­or and Deputy Dir­ect­or of MSRI, re­spect­ively, but Is re­turned to MIT.

Mean­while, the im­pact of the in­dex the­or­em con­tin­ued to grow. Is de­scribed its con­sequences for oth­er fields ([e3], p. 38):

In 1962, while a Sloan Fel­low…Mi­chael Atiyah and I dis­covered an in­dex the­or­em that com­bined to­po­logy, geo­metry, and el­lipt­ic par­tial dif­fer­en­tial op­er­at­ors in a new way. Our res­ults ex­ten­ded, uni­fied, and gave in­sights to some older the­or­ies. Ex­plor­ing the con­sequences of this start­ling com­bin­a­tion of dif­fer­ent fields has kept us busy since then. Deep ap­plic­a­tions con­tin­ue to be dis­covered. Some of the most in­ter­est­ing…are in high-en­ergy the­or­et­ic­al phys­ics. Math­em­aticians and phys­i­cists now real­ize that un­der­ly­ing gauge the­or­ies is the math­em­at­ic­al struc­ture of fiber bundles. As a res­ult, glob­al geo­metry has ap­plic­a­tion to ele­ment­ary particle phys­ics, and at the same time phys­i­cists are ask­ing many new geo­met­ric ques­tions. It is ex­cit­ing to search for the an­swers.

Is was elec­ted to the Na­tion­al Academy of Sci­ences in 1968, and served as Chair of their Com­mit­tee on Sci­ence, En­gin­eer­ing and Pub­lic Policy (COSEPUP). One of the first is­sues he tackled was what to do with nuc­le­ar waste. An­oth­er is­sue he brought up to the com­mit­tee was the threat to pri­vacy from the com­ing com­puter/in­ter­net re­volu­tion, but Is was way ahead of his time in this re­spect, and the rest of the com­mit­tee was less in­ter­ested.

While serving on the com­mit­tee, Is went to see Jerry Weis­ner, then pres­id­ent of MIT, who con­grat­u­lated him on be­ing Chair of COSEPUP. Is ex­pressed doubt about his com­mit­tee role, aver­ring that it would take away from his re­search. Jerry replied that the ex­per­i­ence would be valu­able and would give Is a dif­fer­ent and much broad­er view of what he (Is) would want to do. Is real­ized that Jerry was right, and in­stead of fo­cus­ing merely on the par­tic­u­lar tech­nic­al as­pects of his work, he learned to step back and ask, Do I want to do this? Why do I want to do this? Where does it fit in­to my view of math­em­at­ics? — and this re­flec­tion of­ten changed things.

Is also served on the Dav­id Com­mit­tee, or­gan­ized by Ken Hoff­man (the then-re­cent chair of the MIT math de­part­ment) who per­suaded Ed Dav­id, Nix­on’s Sci­ence Ad­viser (who had resigned in 1973 out of frus­tra­tion), to be chair.

Is was a mem­ber of the White House Sci­ence Coun­cil from 1982–88. His ex­per­i­ence in Wash­ing­ton left him with a fa­vor­able im­pres­sion of the sci­entif­ic tal­ent summoned to duty in the cap­it­ol. For ex­ample, he stated that the choice of ex­perts for sci­ence policy re­flec­ted great­er sound­ness of judg­ment, in many cases, than did ten­ure de­cisions in aca­demia. He also had great re­spect for the mil­it­ary, arising from his ser­vice in World War II.

Though Is later served on the Gov­ern­ing Board of the Na­tion­al Re­search Coun­cil (1995–99), it must be said that math­em­aticians are not fre­quently called to serve in policy-mak­ing roles in DC. Is re­mem­bers the White House Sci­ence Ad­viser, Jay Keyworth, com­ing in one morn­ing after a long meet­ing the night be­fore re­gard­ing some par­tic­u­larly in­tract­able prob­lem and say­ing (with a smile, one as­sumes): “Is, your sug­ges­tion solves the prob­lem. How can you be so smart, you’re just a math­em­atician!”.

For many years at MIT Is only taught gradu­ate classes and a few high-level un­der­gradu­ate classes. Then he de­cided to be a TA for the ba­sic cal­cu­lus class simply be­cause he had nev­er taught lower-di­vi­sion math courses be­fore. He looks back on this ex­per­i­ence fondly, and feels it made him feel part of MIT’s en­com­passing mis­sion to edu­cate un­der­gradu­ates.

Is is an MIT “pat­ri­ot”: He sums up MIT with one word, “gen­er­os­ity”, for they give fac­ulty free­dom to do their re­search and re­sources with which to do it. This was par­tic­u­larly true when Is was des­ig­nated a “Uni­versity Pro­fess­or”, re­port­ing dir­ectly to the Prov­ost.

He still plays ten­nis twice a week — three times in sum­mer — and at a very high level. “He really runs, at age 85!” mar­vels his col­league from Har­vard, Mi­chael Hop­kins. His former stu­dent, Daniel Burns, notes, “Is has al­ways had a very spe­cif­ic, jaunty walk. He bounces around on the balls of his feet. It’s part of his pan­ache, like wear­ing cravats rather than ties. I have an idea that he likes move­ment like this, and it’s re­lated to why he likes ten­nis.” Sing­er him­self says simply, “I play ten­nis to learn. I think that’s true of everything I am in­ter­ested in — I do it to learn. In math­em­at­ics and phys­ics, when I’ve learned enough, I can ex­plain it to oth­ers.”