Celebratio Mathematica

Raoul H. Bott

Remembering Raoul Bott (1923–2005)

by Loring W. Tu

Lor­ing W. Tu coau­thored Dif­fer­en­tial Forms in Al­geb­ra­ic To­po­logy with Raoul Bott. A second volume, Ele­ments of Equivari­ant Co­homo­logy, in the works long be­fore Bott’s passing, is due to ap­pear in 2014.1

Making a problem your own
Figure 20. Raoul Bott, John Tate, and Jean-Pierre Serre at a reception in honor of O. Zariski and L. Ahlfors, Library of Harvard mathematics department, winter 1981–82.
Photo: Carol Tate.

The first time I met Raoul was at an ori­ent­a­tion lunch for in­com­ing gradu­ate stu­dents in math­em­at­ics at the Har­vard Fac­ulty Club. Raoul gave us some ad­vice on how to write a Ph.D. thes­is. He said it was like do­ing a home­work prob­lem, but a harder prob­lem. He ended by say­ing, “Make the prob­lem your own.” It puzzled me what it meant to “make a prob­lem my own,” but I was too in­tim­id­ated to ask. I thought it was one of those things, like the taste of a cer­tain fruit, that is im­possible to ex­plain ex­cept to those who have ex­per­i­enced it them­selves.

A few years later, when I was an as­sist­ant pro­fess­or at the Uni­versity of Michigan, my Ph.D. thes­is ad­visor, Phil Grif­fiths, came to vis­it. I picked him up from the air­port and drove him to a res­taur­ant. While in the car, we star­ted talk­ing about a math­em­at­ic­al prob­lem. I be­came so en­grossed that I lost all sense of time, place, and ori­ent­a­tion. The next thing I knew, a po­lice­man was hand­ing me a tick­et for driv­ing the wrong way on a one-way street.

Grif­fiths ad­vised me help­fully, “Go tell the judge that you were think­ing about math­em­at­ics.” So I showed up in court to dis­pute the charge, and I did as Grif­fiths told me. The judge took a look at my driver’s li­cense and said, “You live only one block away from this street. You have no ex­cuse!” He up­held the fine of sev­enty-five dol­lars. At that mo­ment, it dawned on me what Raoul had meant by “mak­ing a prob­lem your own.” I think it meant to be so ab­sorbed by the prob­lem that you for­get everything else — to be pos­sessed, so to speak.

It has happened to me a few more times, miss­ing a sub­way stop on my way to the air­port or jump­ing out of bed at night with a solu­tion. Each time I feel that I have fi­nally made a math­em­at­ic­al prob­lem my own.

Bott as a lecturer
Figure 21. Left to right: Joan Glashow, Dorothy Haag, Rudolf Haag, Sheldon Glashow, Arthur Jaffe, Barbara Dauschke, Raoul Bott, Phyllis Bott, Klaus Hepp, Konrad Osterwalder, Walter Kaufmann-Bühler at a dinner for Rudolf Haag, founding editor of Communications in Mathematical Physics, at Harvest Restaurant, Cambridge, MA, September 1982.
Photo courtesy of Arthur Jaffe.

Bott’s lec­tures were le­gendary. He had a knack for ex­plain­ing ideas in simple, eas­ily un­der­stood terms, no mat­ter how ab­struse, com­plic­ated, or ab­stract the top­ic. His lec­tures were al­ways clear and ex­cit­ing. They were ma­gic­al in that they gave you the feel­ing you had un­der­stood something, some­times even when you had not. Not sur­pris­ingly, his lec­tures were pop­u­lar and his courses heav­ily en­rolled. His courses had im­pact bey­ond math­em­at­ics stu­dents at Har­vard, for they were at­ten­ded also by stu­dents and fac­ulty from oth­er de­part­ments and oth­er uni­versit­ies. The phys­i­cist Cum­run Vafa cited Bott’s courses for chan­ging his per­cep­tion of mod­ern math­em­at­ics and pro­foundly in­flu­en­cing his later stud­ies ([e5], p. 277). Like­wise, Ed­ward Wit­ten cred­ited Bott’s lec­tures with teach­ing him tech­niques of geo­metry and to­po­logy, such as Morse the­ory and equivari­ant co­homo­logy, which have proven pivotal in his work on su­per­sym­metry.

Bott al­ways seemed glad to be in the classroom. His courses were a lot of fun. In every lec­ture there were spon­tan­eous mo­ments of laughter. This came about not through pre­par­a­tion and canned jokes but be­cause of his in­nate sense of hu­mor, unique per­spect­ive, col­or­ful phrases, and su­perb de­liv­ery. In his hands, the con­struc­tion of a spec­tral se­quence could be­come en­ter­tain­ing. He al­ways fo­cused on the cent­ral idea and simple but il­lu­min­at­ing ex­amples.


One year Bott taught the second semester of com­plex ana­lys­is, and the text­book he chose was Lars Ahlfors’s Com­plex Ana­lys­is. At some point he de­par­ted from the book and gave a dif­fer­ent defin­i­tion. Now stu­dents of­ten revere the text­book as the ul­ti­mate au­thor­ity, so a hand shot up and a stu­dent blur­ted out, “But Ahlfors says this, not that!” Bott replied calmly, “Yes, but Bott says that.” As usu­al, Bott un­der­stood things his own way and was not about to faith­fully fol­low any book. In fact, in to­po­logy courses he did not even fol­low his own books, be­cause usu­ally his un­der­stand­ing of the sub­ject had evolved since the book ap­peared.

A conscripted lecture
Figure 22. The poster for a conference in honor of Raoul Bott in 1984, with an ink painting by the artist and topologist Anatoly T. Fomenko depicting the Bott periodicity theorem.

One day in the early 1980s a poster ap­peared on the bul­let­in board of the Har­vard math­em­at­ics de­part­ment on the third floor of the Sci­ence Cen­ter. It looked just like any oth­er an­nounce­ment, but with a twist. On the top it said, “By pop­u­lar de­mand, Pro­fess­or Raoul Bott will give a lec­ture on ‘The Atiyah–Sing­er In­dex The­or­em: What It Really Means’ ”. The date, time, and place of the lec­ture were all clearly spelled out. What was un­usu­al about this poster was the pres­ence of an as­ter­isk next to Raoul Bott’s name and a foot­note at the bot­tom: “*Please in­form the speak­er.”

A few minutes be­fore the sched­uled time on the ap­poin­ted day, the room was packed. No one had the temer­ity to in­form the speak­er about the lec­ture, so we were all won­der­ing if Raoul Bott was go­ing to show up. At the ap­poin­ted time, he showed up, made a few jokes, and then pro­ceeded to de­liv­er a won­der­ful lec­ture on the Atiyah–Bott fixed point the­or­em and the Atiyah–Sing­er in­dex the­or­em, all in the al­lot­ted hour.

Finder’s fee

Nowhere were Bott’s powers of per­sua­sion more evid­ent than at the sev­enty-fifth an­niversary of the In­sti­tute for Ad­vanced Study at Prin­ceton in March 2005. On that oc­ca­sion he gave a talk re­min­is­cing about how the in­sti­tute in the fifties changed his life and launched his ca­reer (Fig­ure 38). A few days after the con­fer­ence, Bob MacPh­er­son, a pro­fess­or at the in­sti­tute, called him to say that a couple in the audi­ence that day were so moved by Bott’s talk that they donated two mil­lion dol­lars to the in­sti­tute. Bott re­coun­ted the story to me and ad­ded, “I should have asked for a find­er’s fee.”

Figure 23. Receiving the National Medal of Science from President Reagan in 1987.
Bott Family Collection.

Com­ing from a fam­ily of tee­totalers, I knew noth­ing about al­co­hol as a gradu­ate stu­dent. At one point I thought it would be good to re­pair my ig­nor­ance in this do­main. Raoul had the look of a bon vivant who might be know­ledge­able about such things. Just as some stu­dents might ask him for good ref­er­ences in to­po­logy, when I ran in­to him in the el­ev­at­or one day I asked him, “Pro­fess­or Bott, can you re­com­mend some li­quors to me?” He gave me a sly look side­ways, and said, “Candy is dandy, but li­quor is quick­er!” be­fore men­tion­ing a few brands. To this day I re­mem­ber the aph­or­ism but not the brands of li­quor he re­com­men­ded.

Joint books

When I first star­ted work­ing on the book Dif­fer­en­tial Forms in Al­geb­ra­ic To­po­logy with Raoul, I was a gradu­ate stu­dent. He thought that we made a great pair work­ing to­geth­er, be­cause as a gradu­ate stu­dent I would know first-hand the dif­fi­culties a stu­dent would en­counter in learn­ing the sub­ject. I think Raoul did not an­ti­cip­ate that it would end up tak­ing up so much of my time. In the end I was glad to have writ­ten the book with him. For me it was a form of ap­pren­tice­ship, and I felt that I had learned a tre­mend­ous amount of math­em­at­ics from a mas­ter.

Raoul was pleased with the res­ult­ing book. Once in a lec­ture I at­ten­ded, he men­tioned some facts — I for­get about what, maybe de Rham co­homo­logy or spec­tral se­quences — and told the audi­ence that they could find them all in the “Bible”. There was a mo­ment­ary per­plex­ity among the audi­ence, and then it tran­spired that Bott was re­fer­ring to our joint book. For a de­vout Cath­ol­ic like Bott to com­pare our book to the Bible must have been the highest form of com­pli­ment.

Figure 24. At the Harvard Science Center, with Lars Ahlfors in the background, 1988.
Photo: Arthur Jaffe.

Al­though we had pro­jec­ted a second volume, Raoul did not men­tion it after the com­ple­tion of the first, pos­sibly be­cause he did not want to put me through the ex­per­i­ence again. It was many years later that I brought it up. The book would be called Ele­ments of Equivari­ant Co­homo­logy. We worked on it for many years. My chief re­gret is that we did not fin­ish it while he was alive, but I have hope that it will soon see the light of day.

While work­ing on the books, Raoul of­ten told me to be “gen­er­ous with cred­it to oth­ers.” Hu­man nature be­ing what it is, we prob­ably all have the tend­ency to over­es­tim­ate our own con­tri­bu­tion and, con­versely, to un­der­es­tim­ate that of oth­ers. These days, whenev­er my baser nature threatens to come to the fore, I re­mem­ber this les­son from Raoul.

One reas­on we got along so well I think is that with my strict Con­fucian up­bring­ing, in which every edict is ser­i­ous, I found Raoul’s wit and ir­rev­er­ence re­fresh­ing. As for Raoul, he said that as he got older, he liked more and more the Con­fucian rev­er­ence for the aged.

Personal happiness
Figure 25. Phyllis and Raoul on Martha’s Vineyard in the 1980s.
Bott Family Collection.

Raoul had a play­ful streak that per­sisted throughout his life. He liked to tease every­one: his wife, chil­dren, friends, col­leagues, and even stu­dents. His in­ter­ac­tion with me was no ex­cep­tion.

His con­cern for me ex­ten­ded to my per­son­al hap­pi­ness. My time as a gradu­ate stu­dent at Har­vard over­lapped with that of Nancy Hing­ston, a good friend of mine and a stu­dent of his of whom he thought highly. I re­mem­ber at a con­fer­ence, Raoul once put his arm around her shoulder and ex­claimed to the pub­lic, “My finest stu­dent!” On the day that Nancy got mar­ried, Raoul said to me, “Lor­ing, you missed your chance.”

Dust bunnies
Figure 26. “Raoul, Raoul, Raoul your Bott.” Squiboncket Pond, Martha’s Vineyard, 1989.
Bott Family Collection.

In my first year as an as­sist­ant pro­fess­or at Michigan, I worked long dis­tance with Raoul on the book Dif­fer­en­tial Forms in Al­geb­ra­ic To­po­logy. That sum­mer I re­turned to Har­vard to fa­cil­it­ate our col­lab­or­a­tion. At the time Raoul and his wife, Phyl­lis, were co­mas­ters of Dun­ster House, a Har­vard un­der­gradu­ate house with three hun­dred un­der­gradu­ates. Too cheap to rent a place of my own, I asked Raoul if he had a guestroom for me in the Dun­ster House mas­ter’s res­id­ence. Bott read­ily agreed.

The guestroom was a room at­tached to the mas­ter’s res­id­ence but with a sep­ar­ate en­trance. This way I had my pri­vacy, but I could go in­to the mas­ter’s res­id­ence to use the kit­chen and din­ing room. To af­ford Raoul and Phyl­lis their pri­vacy, I nor­mally did not do that ex­cept when they were away. The Botts by then had a house on Martha’s Vine­yard and would of­ten spend a large part of the sum­mer there. I worked with Raoul on oc­ca­sion­al trips to the Vine­yard or when he re­turned to Cam­bridge from time to time.

As co­mas­ters of Dun­ster House, Raoul and Phyl­lis of­ten had to en­ter­tain on a large scale, hold­ing re­cep­tions for stu­dents and par­ents, for ex­ample, and so Har­vard provided them with live-in help, who were usu­ally gradu­ate stu­dents in fields oth­er than math­em­at­ics. The live-in help lived up­stairs from the Botts, so that sum­mer I found my­self liv­ing in the Dun­ster House mas­ter’s res­id­ence with three young wo­men, the live-in help of the year.

The first time Raoul came back in the sum­mer, he got very mad at the four of us; ap­par­ently we had been liv­ing in squal­or (though not in sin). Point­ing to dust bun­nies every­where, he said, “Look at this!” The three young wo­men were not used to clean­ing the house, be­cause dur­ing the school year there was a clean­ing staff from Har­vard. As for me, at that point of my life I was ob­li­vi­ous to dust bun­nies; they were simply in­vis­ible to me. It was strange that as in math­em­at­ics, where, after Raoul showed me his fixed-point the­or­ems, I began to see fixed-point phe­nom­ena every­where, in the same way, after Raoul poin­ted out those dust bun­nies, I began to no­tice dust bun­nies every­where. After that, each time just be­fore Raoul was to re­turn to Cam­bridge, my three house­mates and I would clean the mas­ter’s res­id­ence from top to bot­tom.

Book contract
Figure 27. Raoul steering a boat (not his own).
Bott Family Collection.

The dust-ball in­cid­ent was one of only two times that I saw Raoul get mad. The oth­er time had to do with the con­tract for our book. While work­ing on the book, we cir­cu­lated the manuscript to some col­leagues and stu­dents for feed­back. Pos­sibly be­cause of Raoul’s fame, the book was heav­ily cour­ted by pub­lish­ers. Both Wal­ter Kaufmann-Bühler, the math­em­at­ics ed­it­or at Spring­er, and Klaus Peters, the ed­it­or at Birkhäuser, at the time an in­de­pend­ent pub­lish­er,2 came to Har­vard to lobby us for their book series. We chose Spring­er, not only be­cause of its long his­tory and ex­cel­lent repu­ta­tion for qual­ity but in part be­cause of the bet­ter roy­alty Spring­er offered.

After the book was pub­lished, Kaufmann-Bühler was quite happy, be­cause as he told me, “The book was selling like hot­cakes.” He passed away a few years later and was re­placed by a suc­ces­sion of ed­it­ors at Spring­er. At one point, one of the new ed­it­ors sent me a let­ter, plead­ing dif­fi­cult fin­an­cial cir­cum­stances at Spring­er and ask­ing Raoul and me to sign a new con­tract with a lower roy­alty rate.

For Raoul, I think the roy­alty was not an is­sue at all, but for me, a low-paid as­sist­ant pro­fess­or at the time, it was much more sig­ni­fic­ant. With the let­ter in hand, I walked in­to Raoul’s of­fice, look­ing frantic. When Raoul saw me and read the let­ter, he got quite mad. He said, “They signed a con­tract. Tough luck.” He then picked up the phone and called the ed­it­or. In his usu­al au­thor­it­at­ive voice, he told the ed­it­or firmly that we had no in­ten­tion of rene­go­ti­at­ing the con­tract. That was the end of it. Spring­er backed off and seems to have flour­ished.


At a con­fer­ence in Montreal in 2008, Mi­chael Atiyah said that someday his­tor­i­ans of math­em­at­ics may want to de­cipher joint pa­pers to fig­ure out who wrote what. In some cases this may be quite easy. Raoul was a con­sum­mate styl­ist. His writ­ings were pithy. He had a col­or­ful, in­im­it­able way of ex­press­ing him­self. People have of­ten come up to me to tell me how much they like our book. Some­times, as if to prove that they have read it, they cite spe­cif­ic pas­sages that they like best. Much to my chag­rin, these are usu­ally not the ones I wrote.

Sleeping in another woman’s bed
Figure 28. Raoul Bott in 1991.
Bott Family Collection.

Jane Kister was a young lo­gi­cian at the Uni­versity of Ox­ford in the sev­en­ties. In the fall of 1978, just after mar­ry­ing the to­po­lo­gist Jim Kister, Jane spent a sab­bat­ic­al semester at MIT. At a re­cep­tion at Har­vard, Raoul put his arm around her and an­nounced, “I’ve slept in this wo­man’s bed.” Jane’s face turned beet red. What happened was that Jane was also on sab­bat­ic­al in the spring of 1977 and had ren­ted her house in Ox­ford to the Botts. It was in­deed true that Raoul had slept in Jane’s bed, though not sim­ul­tan­eously with her.

While vis­it­ing Eng­land in the early eighties, Raoul thought that he had also slept in Queen Eliza­beth’s bed, but of course without the queen in it. In his Col­lec­ted Pa­pers he cred­ited this ex­per­i­ence with his sud­den joint in­sight with Mi­chael Atiyah in­to the re­la­tion between equivari­ant co­homo­logy and the mo­ment map ([4], p. xiii): “Pos­sibly the night I had spent in the erstwhile bed of Queen Eliza­beth had something to do with it!” Ac­cord­ing to a re­cent mes­sage from Atiyah, the queen was Vic­tor­ia, not Eliza­beth. Raoul had stayed with the Atiyahs in the Mas­ter’s Lodge at Trin­ity Col­lege, Cam­bridge, where Atiyah was then the mas­ter. In her time, Queen Vic­tor­ia and her con­sort, Prince Al­bert, did in fact stay as guests at Trin­ity Col­lege, and the four-poster bed that they used be­came a guest bed.

Lecture preparation

One year when I was at the Uni­versity of Michigan, Raoul was in­vited to give a lec­ture in a pres­ti­gi­ous series. Dur­ing his vis­it to Ann Ar­bor, Raoul stayed with me in my one-bed­room apart­ment. The morn­ing of the lec­ture, he was writ­ing his lec­ture notes. After writ­ing sev­en pages, he said, “That’s enough. I will not be able to cov­er more than five pages in an hour.” I have found this to be a use­ful rule of thumb: five to sev­en pages of hand­writ­ten notes are about right for an hour lec­ture on the black­board. I learned more from Raoul’s leis­urely but well-timed pace of five hand­writ­ten pages in an hour than from oth­er people’s fifty slides, each densely packed with in­form­a­tion.

Another narrow escape
Figure 29. Phyllis and Raoul on Martha’s Vineyard in the 1990s.
Bott Family Collection.

Raoul’s life seemed to be blessed. He left his nat­ive Hun­gary/Slov­akia be­fore the Nazi in­va­sion, sur­vived near-drown­ing in an ex­ped­i­tion or­gan­ized by Steph­en Smale, and vis­ited In­dia without a visa at a time when visas were re­quired. In Ann Ar­bor he also had a nar­row es­cape.

At the end of his vis­it to Ann Ar­bor, I drove him to the De­troit In­ter­na­tion­al Air­port, twenty miles away, in my Ford Mav­er­ick. It was a used car that I had bought from a de­part­ing postdoc at the Uni­versity of Michigan. Soon after I pur­chased the car, I no­ticed that it was leak­ing trans­mis­sion flu­id, but the rate of the leak was so slow — just one or two drops a day — that it did not seem worth­while to re­place the en­tire trans­mis­sion. On the high­way as we were head­ing to­wards the air­port, the car star­ted smoking un­der the hood. We were alarmed, but Raoul had a plane to catch and the air­port was not so far away, so I con­tin­ued driv­ing at full speed.

Just as we ar­rived at the air­port, dense white smoke bil­lowed from un­der the hood and the car went dead. It looked like it could ex­plode. Raoul hur­riedly ran to his flight, and I jumped out of the car. After his re­turn to Bo­ston he called me to make sure that I was still alive.

The Toaster Incident at Dunster House
Figure 30. Raoul Bott, George Mackey, and Arthur Jaffe at Arthur Jaffe’s wedding, Lime Rock, Connecticut, September 12, 1992.
Photo: Arthur Jaffe.

Raoul nav­ig­ated the per­ils of aca­dem­ic polit­ics with con­sum­mate skill. He and Phyl­lis were co­mas­ters of Dun­ster House for six years. After they stepped down, an­oth­er pro­fess­or was ap­poin­ted as the mas­ter. To dis­tin­guish him from Raoul, I will call him the new mas­ter. The new mas­ter was a very nice man, but his term was marked by con­tro­versy. I will give one ex­ample. It stemmed from a toast­er oven.

Some Jew­ish stu­dents did not want to eat the food in the din­ing hall for reas­ons of keep­ing kosh­er. They asked the new mas­ter for a toast­er oven so that they could heat up their own kosh­er food. The new mas­ter bought a toast­er oven for them. One of the tu­tors (aca­dem­ic ad­visors) at Dun­ster House, an act­iv­ist with strong prin­ciples, wrote a let­ter to the stu­dent pa­per, the Har­vard Crim­son, cri­ti­ciz­ing the use of house funds to buy the toast­er oven, be­cause in his view this was an act of fa­vor­it­ism to­wards one par­tic­u­lar re­li­gion, akin to a vi­ol­a­tion of the sep­ar­a­tion of church and state, a found­ing prin­ciple of our re­pub­lic.

The new mas­ter fired this tu­tor. More let­ters fol­lowed in the Crim­son. It was no longer about the toast­er oven, but about the new mas­ter’s lead­er­ship. Oth­er tu­tors wrote let­ters, ac­cus­ing the mas­ter of auto­cracy and par­ti­al­ity, of fa­vor­ing some tu­tors over oth­ers. There were calls for the mas­ter’s ouster. Stu­dents or­gan­ized demon­stra­tions in Har­vard Yard sup­port­ing the fired tu­tor. Pro­fess­or Ed­mund Lin, a former chair of the De­part­ment of Mo­lecu­lar Ge­net­ics at Har­vard Med­ic­al School and a mem­ber of the Seni­or Com­mon Room of Dun­ster House, wrote a let­ter to Pres­id­ent Ruden­stein of Har­vard, call­ing for the mas­ter’s resig­na­tion. Only at Har­vard could there be a ra­ging de­bate about con­sti­tu­tion­al prin­ciples arising from a toast­er oven. This was when the new mas­ter’s five-year term was up for re­new­al. Pres­id­ent Ruden­stein asked to meet with Raoul, evid­ently be­cause he val­ued Raoul’s judg­ment. Know­ing that I was a close friend of Ed­mund Lin, Raoul asked me if I knew what was go­ing on. I did, not only be­cause of my friend­ship with Ed­mund Lin but also be­cause I read the Crim­son every day. Raoul did not read the Crim­son.

Figure 31. Lecturing at Harvard in the 1990s.
Bott Family Collection.

When I ex­plained the in­cid­ent to Raoul, his im­me­di­ate re­ac­tion was “An act­iv­ist trouble­maker? You should nev­er fire someone like that. If you do, there is no end to the trouble. You should give him ten­ure!” Raoul had a very good nose for stay­ing out of trouble. Of course, this did not mean that he would give every act­iv­ist ten­ure. It just meant that in this case the stakes were not high enough to fire the tu­tor. Raoul then said pens­ively, “Ed Lin was al­ways so quiet when I was the mas­ter. He must have thought that I was do­ing a good job.”

It so happened that the new mas­ter was an eth­nic Chinese from In­done­sia, a res­id­ent tu­tor whom he par­tic­u­larly liked and was ac­cused of be­ing par­tial to was a Chinese-Amer­ic­an, and the pro­fess­or call­ing for the mas­ter’s ouster was a Chinese from China. Raoul turned to me and asked, “Is this one of those Chinese battles so in­scrut­able to us West­ern­ers?”

I do not know what he said to Pres­id­ent Ruden­stein. Ruden­stein re­newed the con­tract of the new mas­ter. The con­tro­versy died down after the stu­dents gradu­ated. Ed­mund Lin told me af­ter­wards, “I am sure it was Raoul who saved the new mas­ter’s skin.”

Foreign languages

Raoul had a won­der­ful self-de­prec­at­ing sense of hu­mor. He was a tal­en­ted lin­guist. He spoke Ger­man, Hun­gari­an, and Slov­ak flu­ently, not to men­tion Eng­lish, of which he was a mas­ter. But there is a lim­it to the num­ber of lan­guages one can learn or need to learn. I like his ex­per­i­ence with Itali­an. Be­fore a con­fer­ence in Italy, he bought a cas­sette course on Itali­an. Re­peat­ing the sen­tences on the cas­sette tape, he stud­ied Itali­an for two weeks. When he got to Italy, he found that he had for­got­ten all the sen­tences ex­cept for one. He told me that the one sen­tence he could say in Itali­an was “Ascolti e ri­peta,” which means “Listen and re­peat.”

Nonmathematical activities
Figure 32. Raoul Bott, Isadore M. Singer, Friedrich Hirzebruch, and Michael Atiyah at a Journal of Differential Geometry reunion dinner, Cambridge, MA, 1999. The four founders of index theory are holding paintings by Milen Poenaru depicting their work.
Bott Family Collection.

In spite of his prodi­gious out­put in math­em­at­ics, Raoul found time to do oth­er things. As co­mas­ters of Dun­ster House, Raoul and Phyl­lis act­ively par­ti­cip­ated in the life of the un­der­gradu­ates, shar­ing meals with them, meet­ing with their par­ents, and or­gan­iz­ing and at­tend­ing cul­tur­al activ­it­ies in the house. Raoul played the pi­ano well enough to give pub­lic per­form­ances. Ever the good sport, he took part in an un­der­gradu­ate theat­er pro­duc­tion, play­ing a Hun­gari­an lin­guist in My Fair Lady. At one Hal­loween party, Raoul and Phyl­lis dressed up as a pir­ate king and a young maid­en, but two stu­dents up­staged them by dress­ing up as Raoul and Phyl­lis Bott! The male stu­dent spor­ted a big beard and was chock-full of gray hair, and to top it off, he was car­ry­ing Raoul’s sig­na­ture briefcase (Fig­ure 33).

An avid swim­mer and a reg­u­lar on the cloth­ing-op­tion­al beach of Martha’s Vine­yard, Raoul earned him­self the sobri­quet “The May­or of Lucy Vin­cent Beach”. He played ten­nis and bi­cycled to work. Once when I vis­ited his home, he showed me with great pride some kit­chen renov­a­tion, say­ing that he did it all with a router.

Material enjoyment
Figure 33. Raoul Bott as a pirate king, Phyllis Bott as a maiden, and two students impersonating Phyllis and Raoul at a Dunster House Halloween party, late 1970s.
Bott Family Collection.

From Raoul, I learned that a lifelong ded­ic­a­tion to in­tel­lec­tu­al pur­suits is not in­com­pat­ible with en­joy­ment of ma­ter­i­al things.

Raoul bought a beau­ti­ful house on Martha’s Vine­yard. Al­though the house was not right on the wa­ter, it was sur­roun­ded by an ex­panse of wild ve­get­a­tion and had an un­ob­struc­ted view of the ocean. There was even a brook on the prop­erty. Since most of the houses there were hid­den in dense fo­liage, Raoul’s house had a view of nature with no oth­er sign of hu­man hab­it­a­tion. One day an­oth­er house rose up, tower­ing above the can­opy of trees in full view from Raoul’s win­dow, the only house vis­ible in oth­er­wise pristine nature. Raoul said it stuck out like a sore thumb, but he was philo­soph­ic­al about it. After all, his own house might be a sore thumb to the oth­er own­er.

While we were work­ing on the book Dif­fer­en­tial Forms in Al­geb­ra­ic To­po­logy, he teased me about the enorm­ous amount of time I was spend­ing on it, ask­ing me if I thought that with the ex­pec­ted roy­alty it would come out to min­im­um wage. Then he said, “I want to buy a boat with it.” I thought he was jok­ing, but years later he did buy a boat.

Raoul had a fas­cin­a­tion with cars, and on one vis­it he proudly showed me his col­lec­tion, a single 2-inch ex­act rep­lica of a Jag­uar that he said a stu­dent of his gave him. Fi­nally, at the age of sev­enty-four, he bought a BMW, ex­em­pli­fy­ing an­oth­er piece of ad­vice he gave me: “Live it up!”

Mineral collection

One of the pleas­ures of talk­ing to Raoul was the un­ex­pec­ted in­sight that he of­ten offered. Some­time in the early nineties, Raoul re­ceived in the mail a cal­en­dar of Steve and Clara Smale’s price­less col­lec­tion of nat­ur­al crys­tals, lov­ingly and beau­ti­fully pho­to­graphed by Steve Smale him­self. Raoul showed me the cal­en­dar in his of­fice, and while ad­mir­ing the breath­tak­ing beauty of the min­er­als, he said, “What a way to avoid in­her­it­ance tax! You just have to slip a few of these to your chil­dren.” Of course, he did not mean it as an es­tate-plan­ning tip; be­sides, I had neither a for­tune nor chil­dren to be­ne­fit from this ad­vice, but it was so char­ac­ter­ist­ic of Raoul to have a unique per­spect­ive on everything.

Practical advice
Figure 34. Jean-Pierre Serre and Raoul Bott, c. 2000.
Bott Family Collection.

Fresh out of gradu­ate school, I once vis­ited Raoul on Martha’s Vine­yard to work on our joint book. Sit­ting on a bench sur­vey­ing his beau­ti­ful es­tate, he said to me, “Lor­ing, buy land.” At the time I was too poor to buy any­thing, but time has borne out the wis­dom of his ad­vice, es­pe­cially when the land is in a well-chosen loc­a­tion like Martha’s Vine­yard.

One of Raoul’s ob­ser­va­tions on life has played a cru­cial role in my men­tal equi­lib­ri­um. When he was at the In­sti­tute for Ad­vanced Study at Prin­ceton in 1949–51, he once had a con­ver­sa­tion with John von Neu­mann, a fel­low Hun­gari­an who was at the time a pro­fess­or at the in­sti­tute. Von Neu­mann told Raoul that he had known only one great math­em­atician, Dav­id Hil­bert, and that hav­ing been a prodigy in his youth, he nev­er felt that he had lived up to his prom­ise. Raoul wrote in ([4], p. 270), “So you see, it is not dif­fi­cult to be found want­ing — one just needs an ap­pro­pri­ate meas­ur­ing rod.” If even von Neu­mann felt in­ad­equate in his achieve­ment in com­par­is­on with Hil­bert’s, what chance for pro­fes­sion­al sat­is­fac­tion do we or­din­ary mor­tals have? After Raoul re­coun­ted this in­cid­ent to me, I re­solved nev­er to com­pare my­self with any­one else, es­pe­cially not with my friends and class­mates who have achieved great­ness.

I was for­tu­nate to be in the job mar­ket dur­ing a brief win­dow of op­por­tun­ity when there were many jobs avail­able, and so I ac­tu­ally had a few choices. Tufts had a fine repu­ta­tion and ex­cel­lent col­leagues, but what clinched the deal was what Raoul said to me, “It will be nice to have you in the back­yard.” The phys­ic­al prox­im­ity made col­lab­or­a­tion easi­er, and after mov­ing to Tufts, I worked on a few more joint pro­jects with him and had the pleas­ure of at­tend­ing more of his courses.

Favorite theorems
Figure 35. Receiving the Wolf Prize from President Ezer Weizman of Israel in 2000.
Bott Family Collection.
When I was writ­ing “The life and works of Raoul Bott” in 2001, I in­ter­viewed Raoul and asked him to list three of his own the­or­ems that he liked the best. He had trouble do­ing it, say­ing that it was like ask­ing him which of his chil­dren he liked best. Even­tu­ally he came up with a list of the top five. The Atiyah–Bott fixed point the­or­em for el­lipt­ic com­plexes was not one of them.

After the me­mori­al ser­vice for Raoul in Janu­ary 2006, Mi­chael Atiyah gave a com­pel­ling lec­ture on why the Atiyah–Bott fixed point the­or­em should have been one of Raoul’s top five fa­vor­ite the­or­ems. I think Raoul would have agreed. The list of five was a rather ar­ti­fi­cial frame­work and should prob­ably not be taken too lit­er­ally. It was what came to Raoul’s mind on the spur of the mo­ment, but he simply could not fit all of his fa­vor­ite the­or­ems in there. In the end, my art­icle in­cluded an­oth­er thir­teen in ad­di­tion to the top-five list.

The Wolf Prize
Figure 36. Visiting his childhood home in Dioszeg, Slovakia, in 2002.
Bott Family Collection.

Raoul used to say that there were two kinds of math­em­aticians, smart ones and dumb ones. The smart ones were people like Mi­chael Atiyah and Jean-Pierre Serre, who un­der­stood new ideas quickly. He clas­si­fied him­self as a dumb math­em­atician, be­cause un­der­stand­ing came to him slowly. This may be so, but his un­der­stand­ing was pro­found, as his cor­pus of many beau­ti­ful and deep the­or­ems at­tests. If he did not un­der­stand something, he had no hes­it­a­tion in say­ing so. When he was awar­ded the Wolf Prize, he told me that he was in very good com­pany, be­cause he was shar­ing the prize with Serre.

One of them had to give a speech in the Knes­set, the Is­raeli par­lia­ment. Ac­cord­ing to Raoul, Serre wanted him to give the speech, be­cause Serre thought that Raoul “had a bet­ter stage pres­ence” and that Raoul “looked more like a math­em­atician.” But how to ex­plain to the Is­raeli law­makers the re­search for which they were be­ing awar­ded the prize? This is the usu­al conun­drum of pure math­em­aticians called upon to ex­plain their work. Serre came up with a gem that Bott in­cor­por­ated in­to his speech:

Mr. Pres­id­ent of the State, Mr. Speak­er of the Knes­set, Mr. Min­is­ter of Edu­ca­tion, Mem­bers of the Dip­lo­mat­ic Corps, Dear Col­leagues and Guests:

It is a great hon­or for me to rise in this beau­ti­ful cham­ber and in so dis­tin­guished a com­pany to ac­cept the Wolf Prize in Math­em­at­ics on be­half of Jean-Pierre Serre and my­self.

Thank you.

In our field alone the pre­vi­ous win­ners of this Prize in­clude both her­oes of our youth and cher­ished friends. And if we look bey­ond, well, who would not be de­lighted — as well as humbled — to join a list that, so to speak, starts with Marc Chagall!

My first words of thanks here are in trib­ute to Ri­cardo and Fran­cis­ca Wolf for set­ting up a found­a­tion so much in tune with the most es­sen­tial need of our ever-shrink­ing plan­et. The uni­ver­sal­ity of their pur­pose speaks for it­self:

“To pro­mote sci­ence and art for the be­ne­fit of man­kind.”

And how in­spired of them to see the com­mon­al­ity of art and sci­ence, and to in­clude math­em­at­ics, where these two spheres of en­deavor are well nigh in­dis­tin­guish­able, in their gen­er­ous be­quest.

But we feel doubly honored that a small and re­l­at­ively new coun­try, with so many press­ing and highly non­trivi­al — as we say in our math­em­at­ic­al jar­gon — prob­lems on its agenda, nev­er­the­less finds time to be­stow this award at its highest level. This act alone is a mov­ing trib­ute to the life of the spir­it in a world mostly con­cerned with more mundane things.

Un­for­tu­nately, the very term “Math­em­at­ics” strikes ter­ror in most mor­tal hearts, and so it is pos­sibly ap­pro­pri­ate here to put our sub­ject in­to some sort of per­spect­ive. And I can think of no bet­ter way of do­ing this than to di­vulge to you just how my ju­ni­or, but much wiser, col­league Jean-Pierre Serre ca­joled me in­to be­ing the one to de­liv­er this ac­cept­ance speech. “For if I were to give the speech,” he ar­gued, “then all I would say is that while the oth­er sci­ences search for the rules that God has chosen for this Uni­verse, we math­em­aticians search for the rules that even God has to obey.” And I cer­tainly couldn’t let him get away with that!

But, after this little tongue in cheek, my time is def­in­itely up!

Still, please per­mit me two more words of thanks. The first is to the com­mit­tee that had a long enough memory to settle on us from amongst so large an ar­ray of worthy and young­er can­did­ates. And our fi­nal thank you is to our fam­il­ies and es­pe­cially our wives, who for a life­time have put up with our ab­sent-minded ways and have been our an­chors in the real world.

Final years
Figure 37. Loring Tu with Phyllis and Raoul Bott, Boston, 2004.
Photo: Mary Moise. Courtesy of Loring Tu.
After Phyl­lis be­came par­tially dis­abled fol­low­ing an op­er­a­tion, the Botts moved to Cali­for­nia in the fall of 2004, where the year-round good weath­er per­mit­ted Phyl­lis more op­por­tun­it­ies for out­door mo­bil­ity in a wheel­chair. In [e3] I men­tioned some of the co­in­cid­ences in Raoul’s life and my own in terms of the places where we ended up — Mc­Gill, Prin­ceton, Har­vard, Michigan — wherever he went, I fol­lowed a few dec­ades later, if only in the vi­cin­ity some­times. The fi­nal co­in­cid­ence was that the town the Botts moved to, Carls­bad, Cali­for­nia, was only twenty-five miles from my par­ents’ house! So it was easy for me to con­tin­ue to vis­it the Botts.

Soon after their move, Raoul was dia­gnosed with lung can­cer. In spite of the poor pro­gnos­is, he was his usu­al cheer­ful self. He ex­plained the prin­ciple of chemo­ther­apy to me this way: “It tries to kill the can­cer faster than it kills you.” He faced the pro­spect of death with equan­im­ity. When I asked him if he would be re­turn­ing to Mas­sachu­setts at some point, he poin­ted to the ground and said, “I am go­ing in here.”3

Figure 38. Speaking at the 75th Anniversary of the Institute at Princeton, March 2005.
Photo: Cliff Moore.

It has of­ten been said that math­em­at­ics is a young per­son’s game. Raoul’s life is a par­tic­u­larly in­spir­ing counter­example. I saw him three weeks be­fore he passed away. I had been work­ing on a prob­lem with him on the volume of a sym­plect­ic quo­tient. He was in top form men­tally. He ex­plained to me a new way of look­ing at the prob­lem that greatly sim­pli­fied it. I cried, “This is so simple!” He said, “That’s the way I like it.”

At the age of eighty-two, bat­tling can­cer, he was still try­ing to un­der­stand in­teg­ra­tion on a sym­plect­ic quo­tient. There was a pa­per of Vic­tor Guille­min and Jaap Kalk­man on the sub­ject, but he wanted to un­der­stand it in his own way. Clearly, his mo­tiv­a­tion was not any ex­tern­al re­ward, like an NSF grant or more hon­ors. He simply wanted to un­der­stand. He was a true math­em­atician.

His life showed us what is hu­manly pos­sible. He con­tin­ued to make beau­ti­ful dis­cov­er­ies and pub­lish im­port­ant pa­pers to the very end.

Royal Society
Figure 39. Induction into the Royal Society, ceremony at UC Santa Barbara, October 2005. (Photo courtesy of the Kavli Institute for Theoretical Physics, UC Santa Barbara.)

In the fi­nal year of his life, Bott was in­duc­ted in­to the Roy­al So­ci­ety. The Roy­al So­ci­ety dates back to 1660 and is a roster of lu­minar­ies in the his­tory of sci­ence. Each new fel­low signs in a book that has the sig­na­tures of all former and cur­rent fel­lows. For health reas­ons, Bott was not able to travel to Lon­don for the sign­ing, but Mi­chael Atiyah, a former pres­id­ent of the Roy­al So­ci­ety, brought to Cali­for­nia the ac­tu­al page from the book Bott was to sign. For good meas­ure, Atiyah also brought Raoul a scanned and bound copy of the pre­ced­ing pages. An in­duc­tion ce­re­mony was held at the Kavli In­sti­tute for The­or­et­ic­al Phys­ics, Uni­versity of Cali­for­nia, Santa Bar­bara, in Oc­to­ber 2005.

When I vis­ited Raoul in Cali­for­nia a month later, he ex­citedly showed me pages from his copy of the Roy­al So­ci­ety book, ex­claim­ing, “Look at this! Chris­toph­er Wren! Isaac New­ton! George Stokes! Lord Kelvin!” For a man of sci­ence, this may be the ul­ti­mate good com­pany.


[1]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

[2]R. Bott: Col­lec­ted pa­pers, vol. 2: Dif­fer­en­tial op­er­at­ors. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston, MA), 1994. MR 1290361 Zbl 0807.​01033 book

[3]R. Bott: Col­lec­ted pa­pers, vol. 3: Fo­li­ations. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston, MA), 1995. MR 1321886 book

[4]R. Bott: Col­lec­ted pa­pers, vol. 4: Math­em­at­ics re­lated to phys­ics. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston, MA), 1995. MR 1321890 Zbl 0823.​01011 book

[5] R. Bott: Col­lec­ted pa­pers, vol. 5. Edi­ted by E. by L. W. Tu. Con­tem­por­ary Math­em­aticians. Spring­er/Birkhäuser, Cham (Ber­lin), 2017. MR 3848672 book