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Celebratio Mathematica

Raoul H. Bott

Shing-Tung Yau

Shing-Tung Yau re­flects on Raoul Bott’s in­flu­ence on him — math­em­at­ic­al, per­son­al, and pro­fes­sion­al.1

I first met Raoul Bott about forty years ago when he briefly vis­ited Shi­ing-Shen Chern and the Berke­ley math­em­at­ics de­part­ment. Bott was a great and fam­ous math­em­atician then, while I was merely a gradu­ate stu­dent. My teach­er, Chern, was in­ter­ested in his pa­per on the loc­al­iz­a­tion of Chern num­bers for Kähler man­i­folds and went through it sev­er­al times dur­ing a sem­in­ar. I was, of course, very im­pressed by Bott’s el­eg­ant the­ory. Little did I know that much later this the­ory would be de­veloped in­to an ex­tremely valu­able tool for com­pu­ta­tions in geo­metry. I used this the­ory my­self, along with my coau­thors, Bong Li­an and Ke­feng Liu, in solv­ing the mir­ror sym­metry con­jec­ture (in­de­pend­ently solved by Givent­al), which was part of the broad­er the­ory of Calabi–Yau man­i­folds.

Figure 18. Foreground, left to right: Susan Bombieri, Enrico Bombieri, Phyllis and Raoul Bott, Shing-Tung Yau, Michael Atiyah. Background: Lars Gårding in the back on far right. Forbidden City, Beijing, 1980.
Photo: Shiu-Yuen Cheng.

In 1971 there was a spe­cial pro­gram on fo­li­ations led by Bott at the In­sti­tute for Ad­vanced Study in Prin­ceton. Since I was gradu­at­ing from Berke­ley, the IAS was an at­tract­ive place for me to go. When I ap­plied to sev­er­al uni­versit­ies, I got a few good of­fers. Al­though I could have got­ten a high­er salary else­where, Chern urged me to spend some time at the IAS, partly be­cause of Bott’s pro­gram. So I went and en­joyed my year tre­mend­ously. At the IAS, I be­came in­ter­ested in ways of con­struct­ing met­rics with spe­cial curvature prop­er­ties and ap­ply­ing them to solve ques­tions in to­po­logy. For ex­ample, I thought about con­struct­ing met­rics with pos­it­ive scal­ar curvature to cre­ate ob­struc­tions for a nona­beli­an group ac­tion on a man­i­fold. (I later wrote a pa­per with Lawson based on this idea.) By study­ing the wedge product of dif­fer­en­tial forms un­der a circle ac­tion, I found ob­struc­tions to the ex­ist­ence of a to­po­lo­gic­al circle ac­tion on a man­i­fold; these ob­struc­tions ex­ist in the co­homo­logy ring of the man­i­fold. I showed my work on group ac­tions to Raoul. He was pleased, and his en­cour­age­ment was really im­port­ant to a young man like me.

Later on, Raoul had many more in­ter­ac­tions with me. After I proved the Calabi con­jec­ture and the pos­it­ive mass con­jec­ture, the lat­ter with Richard Schoen, he tried hard to con­vince me to come to Har­vard, which of­fer I did not ac­cept at first. Dur­ing that pro­cess he in­vited my wife and me to his home for din­ner sev­er­al times. At the time, he was the mas­ter of Dun­ster House at Har­vard. It was in­spir­ing to see how much time and en­ergy he in­ves­ted in col­lege un­der­gradu­ates. I was truly grate­ful for his hos­pit­al­ity dur­ing my vis­its to Har­vard. In re­turn, I tried to en­ter­tain him well when he vis­ited Beijing at the in­vit­a­tion of Chern in 1980. Dur­ing that vis­it I pro­posed the Chinese name Bo Le to him. Bo Le was a fam­ous per­son­age in Chinese his­tory re­puted to have the abil­ity to re­cog­nize ex­cel­lent horses, those that can run a thou­sand miles. Apart from the apt­ness of its mean­ing, the name was ap­pro­pri­ate phon­et­ic­ally also: “Bo” is the Chinese sur­name closest to “Bott,” and “Le” is about as close to “Raoul” as a Chinese char­ac­ter can sound. Raoul told me he liked this Chinese name.

Figure 19. Left to right: Tsai-Han Kiang, Shiing-Shen Chern, Hsio-Fu Tuan, Shan-Tao Liao, Raoul Bott, Shing-Tung Yau, Guang-Lei Wu (in white shirt), Mrs. Yau (in red dress) in Beijing, 1980.
Photo: Shiu-Yuen Cheng.

The pivotal mo­ment of my life was the time when I was hav­ing some trouble in the math­em­at­ics de­part­ment at the Uni­versity of Cali­for­nia in San Diego. I needed help with a de­cision. Raoul was vis­it­ing Berke­ley, and I flew to Oak­land to have din­ner with him. After din­ner we had a long dis­cus­sion about my fu­ture. A true states­man, he laid out the pros and cons of what I should do. I felt greatly re­lieved after talk­ing with him and made the most im­port­ant de­cision of my ca­reer, which was to come to Har­vard, a de­cision that I have nev­er re­gret­ted.

Of course, I learned much more from Raoul dur­ing my years at Har­vard, no less in states­man­ship than in math­em­at­ics — he was ex­traordin­ar­ily skilled in hand­ling de­part­ment­al af­fairs. I felt truly sad when he passed away. I gave a talk on his life’s work at a Journ­al of Dif­fer­en­tial Geo­metry con­fer­ence. In pre­par­a­tion for the talk, I re­searched his con­tri­bu­tions to math­em­at­ics. I was amazed to learn how much he had ac­com­plished and how much he had done that I did not know about.

Raoul cer­tainly ranks among the most in­flu­en­tial math­em­aticians of the last cen­tury. His work was deep, his vis­ion far reach­ing, and his im­pact dur­able. May his spir­it al­ways be with us!