Celebratio Mathematica

Raoul H. Bott

Raoul Harry Bott: 24 September 1923 – 20 December 2005

by Michael Atiyah

Raoul Bott was one of the out­stand­ing geo­met­ers of our time, and his in­flu­ence on math­em­at­ics owed much to the warmth of his per­son­al­ity.

Early background

Photo by Bachrach Photography

Raoul was born in Bud­apest, Hun­gary, but un­til the age of 16 years he lived in Slov­akia. He was a typ­ic­al child of the Aus­tro-Hun­gari­an world, edu­cated in a vari­ety of lan­guages: Hun­gari­an, Slov­ak, Ger­man, and Eng­lish (learned from his Eng­lish gov­ernesses). Raoul’s moth­er was Hun­gari­an and Jew­ish, where­as his fath­er was Aus­tri­an and Cath­ol­ic. Des­pite the fact that his par­ents’ mar­riage broke up shortly after he was born, and that he saw very little of his fath­er, his moth­er brought him up as a Cath­ol­ic. Though lapsing as a teen­ager, he re­mained a Cath­ol­ic throughout his life.

Raoul’s moth­er re­mar­ried, to the chief ex­ec­ut­ive of the loc­al sug­ar fact­ory, a po­s­i­tion of some im­port­ance in the com­munity. The fam­ily was well off, liv­ing a com­fort­able middle-class life; the par­ents traveled ex­tens­ively and the young­er chil­dren were edu­cated at home. Even­tu­ally, in 1932, they moved to Brat­is­lava, the cap­it­al of Slov­akia. Here Raoul fi­nally went to a prop­er school, where he had to mas­ter Slov­ak; but he was a me­diocre stu­dent and dis­tin­guished him­self only in singing and Ger­man. His main in­terests at the time were mu­sic, which re­mained a pas­sion all his life, and mak­ing elec­tric­al ex­per­i­ments in the base­ment, a fore­taste of his de­cision many years later to study elec­tric­al en­gin­eer­ing.

Raoul’s moth­er died of can­cer when he was just 12 years old — a dev­ast­at­ing blow that brought him closer to his step­fath­er. He also had a net­work of uncles, aunts, and cous­ins who provided an ex­ten­ded fam­ily. In 1938–1939, with the Ger­man designs on Czechoslov­akia be­com­ing in­creas­ingly clear, his step­fath­er ac­ted promptly and moved to Eng­land, with Raoul fol­low­ing in June 1939, where he met the new step­moth­er from a wealthy Jew­ish fam­ily in Bud­apest whom his step­fath­er had just mar­ried.

Raoul was now sent to a board­ing school, but for­tu­nately for him he es­caped (for the time be­ing) the tra­di­tion­al rig­ors of an Eng­lish pub­lic school. In­stead, he went to a pro­gress­ive private school, which be­lieved in free­dom, self-ex­pres­sion, and coedu­ca­tion. Raoul re­membered his time there as one of the truly form­at­ive peri­ods of his life: “In one stroke it made me a life­time anglo­phile as well as a great ad­mirer of the op­pos­ite sex.”

His step­par­ents had only a trans­it visa for Eng­land, and they left for Canada early in 1940, with Raoul fol­low­ing shortly there­after. In Canada he had to have a fur­ther year of school­ing and, as he said, “the harsh fate of go­ing to a Brit­ish pub­lic school, which I had mi­ra­cu­lously so avoided in Eng­land, caught up with me in Canada.” In the au­tumn of 1941 he en­rolled at Mc­Gill Uni­versity as an elec­tric­al en­gin­eer­ing stu­dent. Here he had in­spir­ing teach­ers; he worked hard and gradu­ated in April 1945. At that stage he de­cided to en­list in the army, but with the end of the Pa­cific War his mil­it­ary ca­reer was cut short, and he re­turned to Mc­Gill in the au­tumn for a mas­ter’s de­gree. Dur­ing this time Raoul was very un­sure what path he should fol­low, and he even tried to take up medi­cine be­fore be­ing sym­path­et­ic­ally but firmly dis­cour­aged by the school’s dean of medi­cine. This re­jec­tion triggered a prompt re­sponse from Raoul — he de­cided there and then to be­come a math­em­atician.

With the en­cour­age­ment of his teach­ers at Mc­Gill, he went to the Carne­gie In­sti­tute of Tech­no­logy (now Carne­gie Mel­lon Uni­versity) in Pitt­s­burgh to work on ap­plied math­em­at­ics un­der Pro­fess­or John L. Synge. Des­pite Bott’s sketchy and form­ally in­ad­equate math­em­at­ic­al back­ground, he was ac­cep­ted as a Ph.D. stu­dent and, in the spring of 1949 when he re­ceived his de­gree, he fi­nally found him­self on the verge of be­com­ing a math­em­atician. His thes­is, writ­ten un­der the dir­ec­tion of Dick Duffin, led, also in 1949, to his first (joint) pa­per, [1]. Al­though the “Bott–Duffin the­or­em” came to be well known among elec­tric­al en­gin­eers, it was Bott’s last con­tact with that pro­fes­sion. However, the pa­per it­self had at­trac­ted the at­ten­tion of the great Her­mann Weyl, ori­gin­ally from Göt­tin­gen, Ger­many, but now a col­league of Al­bert Ein­stein’s at the In­sti­tute for Ad­vanced Study in Prin­ceton, NJ. Weyl in­vited Bott to come to the in­sti­tute, a move that im­me­di­ately opened his eyes to the wider vista of math­em­at­ics and trans­formed his ca­reer.

Princeton and Michigan, 1949–1960

Bott spent the first dec­ade after earn­ing his Ph.D. between the In­sti­tute for Ad­vanced Study and the Uni­versity of Michigan in Ann Ar­bor. His teach­ing ap­point­ment to the Michigan fac­ulty was pre­ceded by two years, 1949–1951, as a vis­it­or at the in­sti­tute, and he re­turned there for a sab­bat­ic­al in 1955–1956. He de­scribed both of his stays in Prin­ceton as de­cis­ive in his math­em­at­ic­al de­vel­op­ment.

In 1949, the in­sti­tute in­tro­duced him to an en­tirely new math­em­at­ic­al world. There were the gi­ants of the time — Al­bert Ein­stein, John von Neu­mann, and Weyl, all refugees like him­self from Nazi Ger­many — to­geth­er with lead­ing Amer­ic­an math­em­aticians such as Os­wald Veblen and Mar­ston Morse. Between them, and un­der their in­flu­ence, the whole can­vas of math­em­at­ics at the highest level was be­ing ex­plored, and the young Bott was totally en­rap­tured. He soon dropped the rather ele­ment­ary and mundane work he had been do­ing at Carne­gie on elec­tric­al cir­cuits and began ab­sorb­ing the new ideas that sur­roun­ded him in Prin­ceton.

Both Prin­ceton Uni­versity and the in­sti­tute were pi­on­eers at the time in de­vel­op­ing the new field of to­po­logy, which was rap­idly matur­ing in­to the ma­jor en­ter­prise it would be­come later. Morse had made his name (at the in­sti­tute and earli­er at Har­vard Uni­versity) in the ap­plic­a­tion of to­po­logy to the study of crit­ic­al points of func­tions (star­ted by Henri Poin­caré) and in to­po­logy’s ex­ten­sion to the cal­cu­lus of vari­ations. Both pur­suits en­abled the de­riv­ing of in­form­a­tion about closed geodesics on Rieman­ni­an man­i­folds. This was a field that at­trac­ted Bott and re­mained a dom­in­at­ing theme throughout his life.

Bott’s most im­port­ant work for many years centered on the ap­plic­a­tion of Morse the­ory to the to­po­logy of Lie groups and their ho­mo­gen­eous spaces. Lie groups, in par­tic­u­lar the clas­sic­al mat­rix groups, had ori­gin­ated in the pi­on­eer­ing work of the Nor­we­gi­an math­em­atician Sophus Lie at the end of the 19th cen­tury; by the mid-20th cen­tury, Lie groups had be­come of cent­ral im­port­ance. Weyl had trans­formed their rep­res­ent­a­tion the­ory, and they were play­ing an im­port­ant role both in dif­fer­en­tial geo­metry and quantum phys­ics.

Raoul Bott, McGill University, Montreal, 1942
Photograph courtesy of the Bott family

In the 1950s, the time was ripe for bring­ing to­geth­er the new fields of to­po­logy and Lie groups, and Bott was the right man at the right time to bridge this gap. Oth­ers con­trib­uted to the al­geb­ra­ic side of the story, but the link with ana­lys­is through Morse the­ory was due to Bott and his Michigan col­lab­or­at­or Hans Samel­son. It was Bott’s good for­tune that, when he went to Michigan in 1951 with his head full of new ideas, he found in Samel­son a kindred spir­it a little older in years and know­ledge­able about Lie groups. Between them they wrote many pa­pers about the to­po­logy of Lie groups, and in par­tic­u­lar about their loop spaces in [4]. For­tu­nately, the in­tro­duc­tion of loop spaces by J.-P. Serre in his 1951 thes­is had re­vo­lu­tion­ized to­po­logy by provid­ing a sys­tem­at­ic ap­proach to cal­cu­lat­ing ho­mo­topy groups. Bott and Samel­son mastered Serre’s work and com­bined it with Morse the­ory in spec­tac­u­lar fash­ion.

In tra­di­tion­al Morse the­ory it was cus­tom­ary to as­sume that the crit­ic­al points of a func­tion were isol­ated, since this would be the gen­er­ic case. However, Bott real­ized that “in nature” things are not gen­er­ic and that crit­ic­al points of­ten arise along sub-man­i­folds. But he also real­ized how to in­cor­por­ate such situ­ations in­to the the­ory, and this was ap­plied to great ef­fect in the study of geodesics on Lie groups. For ex­ample, the closed geodesics on the group \( \mathrm{SU}(2) \), the three-di­men­sion­al sphere, come nat­ur­ally in con­tinu­ous fam­il­ies para­met­er­ized by the equat­ori­al 2-sphere. The cul­min­a­tion of this work of Bott and Samel­son was the fam­ous peri­od­icity the­or­ems dis­covered by Bott, which he pub­lished in [2], but these de­serve a sec­tion of their own.

The periodicity theorems

Cal­cu­lat­ing the ho­mo­topy groups of spheres and re­lated spaces such as Lie groups, had be­come the fun­da­ment­al goal of ho­mo­topy the­or­ists. In the early days, and by fairly crude geo­met­ric meth­ods, this was only pos­sible for low di­men­sions. Serre’s thes­is had, in prin­ciple, provided power­ful al­geb­ra­ic ma­chinery for more ex­tens­ive cal­cu­la­tions, but these were tricky and del­ic­ate. By 1955 ho­mo­topy the­or­ists had got­ten as far as \( \pi_{10}(\mathrm{SU}(n)) \), for \( n \) large, and found it to be cyc­lic of or­der 3. But early in 1957 Friedrich Hirzebruch and Ar­mand Borel had con­cluded, through their quite in­de­pend­ent com­pu­ta­tions, that the or­der of this group was a power of 2.

Such ex­pli­cit con­tra­dic­tions are a chal­lenge to math­em­aticians, and Bott felt that this one was right up his al­ley. He was sure that his meth­ods would settle the is­sue, so he sat down with his friend Arnold Sha­piro and cal­cu­lated over an en­tire week­end. By Sunday even­ing they had ad­ju­dic­ated in fa­vor of Borel and Hirzebruch. The ho­mo­topy the­or­ists were wrong, re­luct­antly con­ceded de­feat, and sub­sequently found their er­ror. Serre, who was watch­ing this battle from the side­lines, re­marked “Quel dom­mage!” (What a pity!), ob­serving tongue-in-cheek what a tri­umph it would have been for to­po­logy to be the first sub­ject to demon­strate the in­con­sist­ency of math­em­at­ics!

This epis­ode sug­ges­ted to Bott that in fact all the high even-di­men­sion­al ho­mo­topy groups \( \pi_{2k}(\mathrm{SU}(n)) \), for \( n \) lar­ger than \( k \), should be zero. Fur­ther ex­am­in­a­tion of the evid­ence then sug­ges­ted to him that the (stable) ho­mo­topy groups of all the clas­sic­al groups should be peri­od­ic, with peri­od 2 in the unit­ary case and peri­od 8 in the or­tho­gon­al and sym­plect­ic cases. Moreover, he felt con­fid­ent that his Morse-the­ory tech­niques would yield a proof. By the sum­mer of 1957 he had found the proof, which was then pub­lished [2].

This pa­per was a bomb­shell. The res­ults were beau­ti­ful, far-reach­ing, and totally un­ex­pec­ted. By us­ing ana­lys­is, Bott had proven res­ults way out of reach of con­ven­tion­al cal­cu­la­tions. His repu­ta­tion was made, and shortly af­ter­ward (in 1960) he moved to Har­vard, where he re­mained for the rest of his life.

At this stage I have to make the move from be­ing the of­fi­cial writer of this mem­oir to be­com­ing an act­ive par­ti­cipant in the drama. I had got­ten to know Raoul at the In­sti­tute for Ad­vanced Study in Prin­ceton, when I went there in 1955 after re­ceiv­ing my Ph.D. We were to go on to be­come lifelong friends and col­lab­or­at­ors, pub­lish­ing no less than 13 joint pa­pers on a wide vari­ety of top­ics and over many years. But our sub­stant­ive col­lab­or­a­tions really took off from the peri­od­icity the­or­ems and their de­vel­op­ment in­to \( K \)-the­ory.

Among the many new top­ics flour­ish­ing in the 1950s, al­geb­ra­ic geo­metry was shar­ing the stage with to­po­logy, again due in large part to J.-P. Serre, who had ap­plied sheaf the­ory with Henri Cartan first in ana­lyt­ic geo­metry and then in al­geb­ra­ic geo­metry. The cul­min­a­tion of this work was the fam­ous gen­er­al­iz­a­tion of the clas­sic­al Riemann–Roch the­or­em proved by Hirzebruch in Decem­ber 1953. Dur­ing the first of the in­flu­en­tial Arbeit­sta­gun­gen or­gan­ized an­nu­ally by Hirzebruch in Bonn, Al­ex­an­dre Grothen­dieck ex­pounded on his spec­tac­u­lar and beau­ti­ful gen­er­al­iz­a­tion of the Hirzebruch the­or­em. This in­volved the in­tro­duc­tion of the \( K \)-groups of an al­geb­ra­ic vari­ety, groups whose defin­i­tion was very ab­stract but that yet were simple and ef­fect­ive. Be­cause I was, and re­mained, a reg­u­lar at­tendee of the Bonn Arbeit­sta­gung, I ab­sorbed Grothen­dieck’s \( K \)-the­ory and, when I heard of Bott’s peri­od­icity the­or­em, I even­tu­ally real­ized how to com­bine the two ideas. This led to “to­po­lo­gic­al” \( K \)-the­ory, which I de­veloped jointly with Hirzebruch and which res­ted in a fun­da­ment­al way on Bott’s peri­od­icity the­or­em.

We needed Bott’s help at this early stage, and he re­spon­ded with a pa­per [3] writ­ten, as he said later, “in flu­ent French.” He went on ([19]):

Alas, the French is not mine, and I am ashamed to see that there is no ref­er­ence to the kind trans­lat­or. Math­em­at­ic­ally [the pa­per] deals with the “new” \( K \)-the­or­et­ic for­mu­la­tion of the peri­od­icity the­or­em. Grothen­dieck’s \( K \)-the­ory and his bril­liant func­tori­al proof of Riemann–Roch in the al­geb­ra­ic cat­egory had a tre­mend­ous ef­fect on all our think­ing. Nev­er­the­less, the ideas of Atiyah and Hirzebruch, in­ter­pret­ing the peri­od­icity the­or­ems as “Kun­neth” for­mu­lae in an “ex­traordin­ary co­homo­logy the­ory,” came as a com­plete sur­prise to me! In one swoop my spe­cial com­pu­ta­tions had be­come a po­ten­tial tool in all as­pects of to­po­logy.

The peri­od­icity the­or­em in the real case, with the peri­od be­ing 8, was subtler than the com­plex case when the peri­od­icity was just 2. Bott and Sha­piro had real­ized that this could best be un­der­stood through the struc­ture of the Clif­ford al­geb­ras, which had the same peri­od­icity in purely al­geb­ra­ic form. In [5], I joined forces with Bott (Sha­piro, sadly, hav­ing died) to cla­ri­fy the way the al­gebra and the to­po­logy were linked. This has since proved use­ful in in­dex the­ory.

\( K \)-the­ory and its fur­ther de­vel­op­ments, in­clud­ing the in­dex the­or­em, were at the cen­ter of my sub­sequent col­lab­or­a­tions with Bott. But we were so close in our math­em­at­ic­al tastes that, over the years, every time we met a new joint ven­ture would start, as will be­come clear in the rest of this mem­oir.

Index theory

Raoul Bott with Michael Atiyah, Oxford, 1975
Photograph courtesy of the Bott family

In 1962–1963 Is­ad­ore Sing­er and I were work­ing on the in­dex the­or­em for el­lipt­ic dif­fer­en­tial op­er­at­ors on com­pact man­i­folds [e1]. This ef­fort had many rami­fic­a­tions and was so close to Bott’s in­terests that he soon joined our en­ter­prise and played an act­ive role in the many dis­cus­sions that took place at Har­vard and the Mas­sachu­setts In­sti­tute of Tech­no­logy (MIT) dur­ing my two vis­its there in 1962 and 1964. He also spent the year with me in Ox­ford dur­ing 1965–1966.

Bott’s first con­tri­bu­tion to this area, pub­lished in 1965, was a dir­ect geo­met­ric veri­fic­a­tion of the in­dex the­or­em for holo­morph­ic vec­tor bundles over ho­mo­gen­eous spaces [9]. This arose nat­ur­ally from his ex­per­i­ence with Lie groups. An­oth­er key con­tri­bu­tion was to the in­dex prob­lems for man­i­folds with bound­ary [6], which re­quired a deep­er un­der­stand­ing of bound­ary-value prob­lems. As a byproduct we also pro­duced an ele­ment­ary proof of the peri­od­icity the­or­em [7]. I re­mem­ber plan­ning a talk on this top­ic at MIT that was rather ab­stract. It was only Bott’s in­sist­ence, at the el­ev­enth hour, on search­ing out the es­sen­tials that made the talk genu­inely “ele­ment­ary.”

The first ma­jor ex­ten­sion of the in­dex the­or­em, pub­lished in 1966, con­cerned the in­ter­play between el­lipt­ic op­er­at­ors and fixed points of maps [10]. It was in­spired by ques­tions that Bott and I were asked by G. Shimura at a con­fer­ence in Woods Hole in 1964. The gen­er­al for­mula that even­tu­ally emerged was sim­il­ar in ap­pear­ance to the fam­ous Lef­schetz fixed-point the­or­em, which relates the num­ber of fixed points of a map to its ac­tion on co­homo­logy. The el­lipt­ic ver­sion is a re­fine­ment in which the map is re­quired to be com­pat­ible with the el­lipt­ic op­er­at­or, and each fixed point con­trib­utes not an in­teger but a com­plex num­ber cal­cu­lated from the lin­ear­ized ac­tion at the fixed point.

The fixed-point case has sev­er­al beau­ti­ful ap­plic­a­tions, in­clud­ing a de­riv­a­tion of the Weyl for­mula for the char­ac­ters of the ir­re­du­cible rep­res­ent­a­tions of Lie groups. A very dif­fer­ent ap­plic­a­tion es­tab­lished an old con­jec­ture of Paul Smith’s. This as­serts that if a cyc­lic group of odd prime or­der acts on a sphere with just two fixed points, then the lin­ear ac­tions on the tan­gent spaces at these points are iso­morph­ic. It was this wide range of ap­plic­a­tions that made our fixed-point the­or­em one of Bott’s fa­vor­ite res­ults.

Bott’s fi­nal con­tri­bu­tion in this area was to help cla­ri­fy the heat-equa­tion ap­proach to the in­dex for­mula. Earli­er al­geb­ra­ic com­pu­ta­tions had been very com­plic­ated and dif­fi­cult to un­der­stand. With the use of clas­sic­al in­vari­ant the­ory, Bott, V. K. Pat­odi, and I were able to present in 1973 a con­cep­tu­ally simple proof [12]. Bott’s ex­pert­ise both in Rieman­ni­an geo­metry and in­vari­ant the­ory were cru­cial in­gredi­ents. The heat-equa­tion proof of the in­dex the­or­em has turned out to be very pro­duct­ive and, in par­tic­u­lar, es­tab­lished a close link with work in the­or­et­ic­al phys­ics. In sub­sequent years these links with phys­ics were greatly strengthened, and they lay be­hind much of Bott’s later work.

Equivariant cohomology

Bott’s in­terest in Morse the­ory, to­geth­er with his ex­pert­ise in Lie groups, made it nat­ur­al for him to be aware of the role of sym­metry at crit­ic­al points of func­tions or at fixed points of maps. One out­come of this was the sys­tem­at­ic ex­ploit­a­tion of equivari­ant co­homo­logy in dif­fer­en­tial geo­metry.

Our joint pa­per [17] arose from our at­tempt to un­der­stand pa­pers by Ed­ward Wit­ten on Morse the­ory, writ­ten from a phys­ics per­spect­ive [e2], and by Duister­maat and Heck­man [e3] on the ex­act­ness of sta­tion­ary-phase ap­prox­im­a­tion. The meth­ods were not really ori­gin­al but our present­a­tion brought sev­er­al strands to­geth­er and has sub­sequently been in­flu­en­tial.

An­oth­er of our joint pa­pers [16] also used equivari­ant co­homo­logy but in an in­fin­ite-di­men­sion­al con­text in­spired by phys­ics. The out­come was a new de­riv­a­tion of the co­homo­logy of the mod­uli space of vec­tor bundles over a com­pact Riemann sur­face. This top­ic has been at the cen­ter of much activ­ity on the fron­ti­er between phys­ics and geo­metry in re­cent years, and our pa­per has stim­u­lated an ex­tens­ive de­vel­op­ment.

Other results

My col­lab­or­a­tions with Bott also in­cluded a di­gres­sion in­to the ques­tion of la­cunas for hy­per­bol­ic dif­fer­en­tial equa­tions [11], an old top­ic go­ing back to Chris­ti­aan Huy­gens and de­veloped by Ig­or Pet­rovsky in the 1940s. The work of Pet­rovsky was dif­fi­cult to fol­low and needed to be up­dated us­ing the new to­po­lo­gic­al meth­ods of al­geb­ra­ic geo­metry. Bott and I were in­tro­duced to this prob­lem by Lars Gård­ing, an au­thor­ity on hy­per­bol­ic equa­tions, and he es­sen­tially “com­mis­sioned” us to take on the pro­ject dur­ing a vis­it to Ox­ford in 1965.

An area in which Bott’s care­ful ap­proach and geo­met­ric in­sight paid di­vidends was his dis­cov­ery that when a man­i­fold is fo­li­ated, the bundle of tan­gent vec­tors to the leaves of the fo­li­ation can­not be an ar­bit­rary sub-bundle of the tan­gent bundle, but must sat­is­fy some glob­al to­po­lo­gic­al con­di­tions. This fairly simple ob­ser­va­tion gen­er­ated a sub­stan­tial fol­low-up, lead­ing to a whole new the­ory.

In a some­what re­lated area Bott col­lab­or­ated with Graeme Segal to study [13] the co­homo­logy of the Lie al­gebra of vec­tor fields on a man­i­fold, which had been in­tro­duced by Is­rael Gel­fand and D. B. Fuks. The res­ult was to ex­press the vec­tor-field co­homo­logy as that of a space nat­ur­ally as­so­ci­ated to the man­i­fold and its tan­gent bundle. In a sense, the out­come was dis­ap­point­ing, in that the Gel­fand–Fuks in­vari­ants gave noth­ing new — they could be iden­ti­fied as ho­mo­topy in­vari­ants.

A very dif­fer­ent col­lab­or­a­tion much earli­er was that between Bott and Shing-Shen Chern [8], in which they set out to gen­er­al­ize the clas­sic­al Nevan­linna the­ory of mero­morph­ic func­tions to high­er di­men­sions. A byproduct of their in­vest­ig­a­tion was the Bott–Chern class, which is a com­plex re­fine­ment of the dif­fer­en­tial form of the Chern class. This has proved ex­tremely use­ful in many sub­sequent de­vel­op­ments.

In one of Bott’s last sig­ni­fic­ant pa­pers [18], he and Clif­ford Taubes gave an el­eg­ant math­em­at­ic­al proof of Wit­ten’s ri­gid­ity the­or­em [e4]. This was the fruit of the ex­cit­ing in­ter­ac­tion dur­ing pre­ced­ing years between geo­met­ers and phys­i­cists — an ef­fort in which Bott took an act­ive part.

Bott’s in­flu­ence on the­or­et­ic­al phys­i­cists has been well de­scribed by Ed­ward Wit­ten:

I came to Har­vard, where Raoul Bott was a pro­fess­or, in the fall of 1976. This turned out to be just the peri­od when phys­i­cists were start­ing to ap­pre­ci­ate that a lot of mod­ern math­em­at­ic­al ideas that we didn’t know much about were rel­ev­ant to un­der­stand­ing quantum gauge the­or­ies. Raoul did a lot to edu­cate me and my con­tem­por­ar­ies. He loved ex­plain­ing things and had a knack for pick­ing out the key point that would make a dif­fi­cult sub­ject clear. Later on, I and many oth­er phys­i­cists learned much of our dif­fer­en­tial to­po­logy from the 1982 book [15] by Bott and Lor­ing Tu.

In 1979, Raoul was in­vited to the sum­mer school on particle phys­ics at Car­gèse, in Cor­sica. He began his lec­ture [14] by say­ing that he was go­ing to tell us about a fa­vor­ite sub­ject of his, which might be use­ful to us some day. The sub­ject was Morse the­ory. Quite pos­sibly, none of the phys­i­cists in at­tend­ance had ever heard of Morse the­ory, and cer­tainly I hadn’t. However, sev­er­al years later, in study­ing a phe­nomen­on known as spon­tan­eous break­ing of su­per­sym­metry, I ran in­to some puzz­ling phe­nom­ena. At some point, a dim re­col­lec­tion of Bott’s lec­ture sprang to mind and it be­came clear that the phe­nomen­on in ques­tion was closely re­lated to the fact that, in Morse the­ory, a crit­ic­al point of a func­tion can ex­ist for a good to­po­lo­gic­al reas­on. This led to my work [e2] re­lat­ing su­per­sym­metry to Morse the­ory.

In my pa­per on that sub­ject, I tried to de­scribe in terms of dif­fer­en­tial forms a second, su­per­fi­cially sim­il­ar, su­per­sym­met­ric con­struc­tion. Un­for­tu­nately, the math­em­at­ic­al set­ting for this second con­struc­tion was not clear. In hind­sight, things might have been clear­er if Bott had giv­en an­oth­er lec­ture in Car­gèse! As it was, this con­struc­tion was later put in its prop­er set­ting by Bott and Atiyah [17]. Bott went on to tu­tor me, and I be­lieve oth­er phys­i­cists, on the ba­sics of equivari­ant co­homo­logy, which has turned out to have nu­mer­ous ap­plic­a­tions in gauge the­ory and to­po­lo­gic­al field the­ory.


Bott at­trac­ted a large num­ber of tal­en­ted stu­dents dur­ing his many years at Har­vard. They grav­it­ated to­ward him be­cause of his friendly in­form­al man­ner as well, as his ob­vi­ous pas­sion for beau­ti­ful math­em­at­ics. Two of his stu­dents, Steph­en Smale and Daniel Quil­len, went on to win Fields medals, and many oth­ers had dis­tin­guished aca­dem­ic ca­reers. Just be­fore Bott died he con­fided to me, with mod­est pride, that he had re­ceived a let­ter from a re­cent No­bel laur­eate who told him how in­spired he had been by Bott’s teach­ing. This was George Aker­lof (of the Uni­versity of Cali­for­nia, Berke­ley), who shared the No­bel Prize in Eco­nom­ics for 2001. He wrote:

You prob­ably do not re­mem­ber but many years ago (39 to be ex­act) an eco­nom­ics stu­dent from MIT took your course in al­geb­ra­ic to­po­logy at Har­vard. I was that stu­dent. I worked very hard in it and learned a great deal. You were not just a great teach­er, but a fab­ulous teach­er. What I learned in your course was the found­a­tion of my whole ca­reer.

You did not just teach the tech­nic­al field of ho­mo­topy the­ory, but showed stu­dents how to de­com­pose prob­lems in­to their es­sen­tials and their tech­nic­al de­tails. This, of course, was the same skill that [Robert] So­low’s pa­pers demon­strated, and that I was learn­ing sep­ar­ately in eco­nom­ics at MIT.

This year I was named co-re­cip­i­ent of the No­bel Prize in Eco­nom­ics. I merely ap­plied to eco­nom­ics the com­mon sense about math­em­at­ics that I had learned from you.

I know that you are a great math­em­atician. I also want to thank you for be­ing such a fine and caring teach­er.

Raoul Bott lecturing at the University of Bonn in 1969
Photograph by Wolfgang Vollrath

Har­vard has “houses” modeled on the Ox­ford and Cam­bridge col­leges, and it was no sur­prise to Bott’s friends when they heard that he had been ap­poin­ted Mas­ter of Dun­ster House. He and his wife Phyl­lis moved in­to the mas­ter’s res­id­ence, over­look­ing the Charles River, and took an en­thu­si­ast­ic part in stu­dent life. Some years earli­er he had been ap­proached about the mas­ter­ship of St. Cath­er­ine’s Col­lege, Ox­ford, where he had spent a sab­bat­ic­al year and made a great im­pres­sion. He de­clined the some­what un­real­ist­ic of­fer, but sub­sequently (1985) be­came an hon­or­ary fel­low. In the same year, he was Hardy lec­turer of the Lon­don Math­em­at­ic­al So­ci­ety.

Bott re­ceived many oth­er hon­ors and awards, in­clud­ing the Wolf Prize in Math­em­at­ics (2000) and the U.S. Na­tion­al Medal of Sci­ence (1987). He had hon­or­ary doc­tor­ates from Notre Dame, Mc­Gill, Carne­gie Mel­lon, and Leicester Uni­versit­ies. He was a mem­ber of the Na­tion­al Academy of Sci­ences of the U.S.A. (elec­ted in 1964) and a for­eign as­so­ci­ate of the French Academy of Sci­ences. Be­cause he was too ill to come to Lon­don, ar­range­ments were made so that I could ad­mit him in Cali­for­nia, shortly be­fore he died, as a for­eign mem­ber of the Roy­al So­ci­ety.


I can think of no bet­ter way to con­vey something of Bott’s per­son­al­ity than to re­pro­duce here the text of the ad­dress I gave at the Har­vard Me­mori­al Church on Janu­ary 29, 2006, soon after his death:

“I first met Raoul over 50 years ago at the In­sti­tute for Ad­vanced Study in Prin­ceton, and it was an in­dic­a­tion of the im­port­ant part that Prin­ceton played in his life that, des­pite his ill­ness, he came back there last March and we met again at the 75th an­niversary of the in­sti­tute.

“Over those 50 years we be­came lifelong friends and worked to­geth­er in many oth­er places, in­clud­ing Har­vard, Ox­ford, and Bonn, where we joined Fritz Hirzebruch’s an­nu­al jam­bor­ee. We traveled the world to­geth­er to con­fer­ences in exot­ic places — In­dia, Mex­ico, China, Hun­gary. I re­call an event in Bud­apest when our bus was held up by a total traffic snarl. When time passed and the dead­lock con­tin­ued, Raoul took charge. He stood in the middle of the road and with great au­thor­ity ac­ted as a po­lice­man, skill­fully dir­ect­ing the traffic and un­lock­ing the jam — which showed he was a real Hun­gari­an!

“Even as a young man, Raoul ex­uded charm and made an im­me­di­ate im­pres­sion on all he met. I re­mem­ber when he was in­ter­viewed in 1964 for a vis­it­ing fel­low­ship at St. Cath­er­ine’s, my Ox­ford col­lege; the mas­ter, Alan Bul­lock, was so at­trac­ted to him that he felt sure Raoul would turn down the of­fer, so he was de­lighted when Raoul ac­cep­ted.

“In our early years to­geth­er, des­pite the fact that he was six years older than me, we were col­leagues on the same plane. Iron­ic­ally, as we grew older the re­la­tion­ship subtly changed and he be­came more of an avun­cu­lar, or fath­er, fig­ure. I think it may have been the beard, but in fact he just grew in­to his nat­ur­al role as a ‘pa­ter­fa­mili­as.’ He had in­deed a large lov­ing fam­ily of chil­dren and, even­tu­ally, grand­chil­dren, and he had a par­al­lel fam­ily of stu­dents and grand-stu­dents. With his large tower­ing frame and his wide em­brace he was really in his ele­ment as the head of these large and ex­ten­ded fam­il­ies. I be­came part of this fam­ily circle, of which Phyl­lis was, of course, a cent­ral fig­ure, shar­ing nearly 60 years of mar­ried life with Raoul and keep­ing him un­der friendly con­trol with her quiet hu­mor. It is very ap­pro­pri­ate that a joint por­trait of Raoul and Phyl­lis, as co-mas­ters, now hangs in Dun­ster House.

“It was im­possible to work with Raoul without be­com­ing en­tranced by his per­son­al­ity. Work be­came a joy to be shared rather than a bur­den to bear. His­tor­i­ans and bio­graph­ers fre­quently try to make a sharp dis­tinc­tion between the life and work of the cre­at­ive artist. No such sep­ar­a­tion makes sense for Raoul — his per­son­al­ity over­flowed in­to his work, in­to his re­la­tions with col­lab­or­at­ors and stu­dents, in­to his lec­tur­ing style, and in­to his writ­ing. Man and math­em­atician were hap­pily fused.

“This is not the place to de­scribe Raoul’s math­em­at­ics, but I should say something about the way he worked — his style. He loved to dis­cuss math­em­at­ics, and we would spend happy hours to­geth­er in front of a black­board toss­ing ideas about and, at Raoul’s in­sist­ence, do­ing cal­cu­la­tions. While he liked to see the big pic­ture, he was nev­er hap­pi­er than when he found a good ex­ample to work on in de­tail. He was sus­pi­cious of hand wav­ing or airy-fairy spec­u­la­tion. To him math­em­at­ics was a craft, where the ar­tis­an lov­ingly carved his handi­work in beau­ti­ful de­tail.

“Ex­pound­ing ideas simply was su­premely im­port­ant to him. He was a born teach­er who knew how to en­gage his audi­ence, get­ting them in­volved so that they could really un­der­stand. It is no ac­ci­dent that Raoul at­trac­ted so many tal­en­ted stu­dents who went on in­to suc­cess­ful ca­reers. Un­like some great math­em­aticians, he did not try to in­tim­id­ate his stu­dents by ex­pos­ing their ig­nor­ance. On the con­trary, he would des­cend to their level and provide en­cour­age­ment and ad­vice to suit the in­di­vidu­al stu­dent. When at­tend­ing a sem­in­ar he would fre­quently ask an ele­ment­ary ques­tion, even when he him­self knew the an­swer, in or­der to help the more in­hib­ited stu­dents in the audi­ence.

“He had great sens­it­iv­ity to people and situ­ations. I re­mem­ber one oc­ca­sion when I wrote the draft in­tro­duc­tion to a joint pa­per in which I re­ferred to the ‘mod­est con­tri­bu­tion’ that each of us had made in earli­er pa­pers. He told me to re­move the word ‘mod­est,’ say­ing it was false mod­esty. He was of course quite right: genu­ine mod­esty does not ad­vert­ise it­self.

“Hu­mor and laughter was an im­port­ant part of Raoul’s char­ac­ter. He en­joyed re­count­ing amus­ing epis­odes of the past, such as the time Steph­en Smale got them trapped between the rising tide and a sheer cliff, or the time when he ar­rived in In­dia without a visa but was giv­en red-car­pet treat­ment — while on our re­turn jour­ney I was in­car­cer­ated in quar­ant­ine at Cairo air­port!

“In any group he was al­ways the cen­ter of at­trac­tion — like the sun, he ra­di­ated warmth, and we plan­ets cir­cu­lated around. But be­neath the jol­lity and hu­mor there was a deeply ser­i­ous side. On oc­ca­sion Raoul would turn on you a pens­ive pen­et­rat­ing look that seemed to see in­to your soul. He could see through pre­ten­sions or poses. He was anchored to the hard core of his be­liefs, even though they rarely came to the sur­face. In Shakespearean terms, he was part Fal­staff and part Ham­let but without the ex­tremes of either, so that they hap­pily co­ex­is­ted.

“His love of beauty in math­em­at­ics was re­flec­ted in his deep love of mu­sic. His en­joy­ment of life found its coun­ter­part in the sparkle of Moz­art, while his more ser­i­ous side found its solace in the spir­itu­al­ity of Bach.

“All of us who knew Raoul un­der­stood what a mar­velous per­son he was, and any­thing we say is in­ad­equate. But let me give the last word to my son Robin who, as a young teen­ager said, after meet­ing Raoul, ‘Now I know what is meant by cha­risma’.”


Much of the ma­ter­i­al on Bott’s early life came from his own re­min­is­cences in [19]. Ad­di­tion­al ma­ter­i­al can also be found in [20].

I am also in­debted to George Aker­lof, Friedrich Hirzebruch, Graeme Segal, Jocelyn Scott, Lor­ing Tu, and Ed­ward Wit­ten for their help.


[1]R. J. Duffin and R. Bott: “Im­ped­ance syn­thes­is without use of trans­formers,” J. Ap­pl. Phys. 20 : 8 (1949), pp. 816. MR 0037753 article

[2]R. Bott: “The stable ho­mo­topy of the clas­sic­al groups,” Proc. Natl. Acad. Sci. U.S.A. 43 : 10 (1957), pp. 933–​935. See also Ann. Math. 70:2 (1959). MR 0102802 Zbl 0093.​03401 article

[3]R. Bott: “Quelques re­marques sur les théorèmes de péri­od­icité” [Some re­marks on peri­od­icity the­or­ems], Bull. Soc. Math. Fr. 87 (1959), pp. 293–​310. In French. Also pub­lished in To­po­lo­gie al­gébrique et géométrie différen­ti­elle (1960). MR 0126281 Zbl 0124.​38203 article

[4]R. Bott and H. Samel­son: “Ap­plic­a­tions of the the­ory of Morse to sym­met­ric spaces,” Am. J. Math. 80 : 4 (October 1961), pp. 964–​1029. Ded­ic­ated to Mar­ston Morse on his 65th birth­day. A cor­rec­tion was pub­lished in Am. J. Math. 83:1 (1961). MR 0105694 Zbl 0101.​39702 article

[5]M. F. Atiyah, R. Bott, and A. Sha­piro: “Clif­ford mod­ules,” To­po­logy 3 : Supplement 1 (July 1964), pp. 3–​38. MR 0167985 Zbl 0146.​19001 article

[6]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

[7]M. Atiyah and R. Bott: “On the peri­od­icity the­or­em for com­plex vec­tor bundles,” Acta Math. 112 : 1 (1964), pp. 229–​247. MR 0178470 Zbl 0131.​38201 article

[8]R. Bott and S. S. Chern: “Her­mitian vec­tor bundles and the equidistri­bu­tion of the zer­oes of their holo­morph­ic sec­tions,” Acta Math. 114 : 1 (1965), pp. 71–​112. A Rus­si­an trans­la­tion was pub­lished in Matem­atika 14:2 (1970). MR 0185607 Zbl 0148.​31906 article

[9]R. Bott: “The in­dex the­or­em for ho­mo­gen­eous dif­fer­en­tial op­er­at­ors,” pp. 167–​186 in Dif­fer­en­tial and com­bin­at­or­i­al to­po­logy: A sym­posi­um in hon­or of Mar­ston Morse (Prin­ceton, NJ, 1964). Edi­ted by S. S. Cairns. Prin­ceton Math­em­at­ic­al Series 27. Prin­ceton Uni­versity Press, 1965. MR 0182022 Zbl 0173.​26001 incollection

[10]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

[11]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

[12]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

[13]R. Bott and G. Segal: “The co­homo­logy of the vec­tor fields on a man­i­fold,” To­po­logy 16 : 4 (1977), pp. 285–​298. MR 0645730 Zbl 0387.​57012 article

[14]R. Bott: “Morse the­or­et­ic as­pects of Yang–Mills the­ory,” pp. 7–​28 in Re­cent de­vel­op­ments in gauge the­or­ies. Edi­ted by G. ’t Hooft, C. Itzyk­son, A. Jaffe, H. Lehmann, P. Mit­ter, I. Sing­er, and R. Stora. NATO Ad­vanced Study In­sti­tutes Series 59. Plen­um Press (New York), 1980. incollection

[15]R. Bott and L. W. Tu: Dif­fer­en­tial forms in al­geb­ra­ic to­po­logy. Gradu­ate Texts in Math­em­at­ics 82. Spring­er (New York), 1982. A Rus­si­an trans­la­tion was pub­lished as Dif­fer­ent­si­al’nye formy v al­geb­raicheskoj to­po­lo­gii (1989). MR 658304 Zbl 0496.​55001 book

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

[17]M. F. Atiyah and R. Bott: “The mo­ment map and equivari­ant co­homo­logy,” To­po­logy 23 : 1 (1984), pp. 1–​28. MR 721448 Zbl 0521.​58025 article

[18]R. Bott and C. Taubes: “On the ri­gid­ity the­or­ems of Wit­ten,” J. Am. Math. Soc. 2 : 1 (1989), pp. 137–​186. MR 954493 Zbl 0667.​57009 article

[19]R. Bott: Col­lec­ted pa­pers, vol. 1: To­po­logy and Lie groups. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston, MA), 1994. MR 1280032 Zbl 0820.​01026 book

[20] The founders of in­dex the­ory: Re­min­is­cences of Atiyah, Bott, Hirzebruch, and Sing­er. Edi­ted by S.-T. Yau. In­ter­na­tion­al Press (Somerville, MA), 2003. A 2nd edi­tion was pub­lished in 2009. MR 2136846 Zbl 1072.​01021 book