Celebratio Mathematica

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

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’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.

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.

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.”

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.

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.

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.

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.”

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.

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.

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.

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.

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.

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.

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.”

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.”

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.

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!”

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.

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.

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.

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.

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

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.

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.