Celebratio Mathematica

This is a brief mem­oir (writ­ten at the re­quest of R. MacPh­er­son) which ap­peared in volume 3 of Bott’s col­lec­ted works.

While a gradu­ate stu­dent at Prin­ceton (some­where in the early 1960s), I went to a lec­ture by Bott. He spoke on the Atiyah–Sing­er in­dex the­or­em, which was quite new at the time. Here was a friendly and some­what over­sized man en­thu­si­ast­ic­ally de­scrib­ing this mar­velous in­dex for­mula. He ended with a burst of great in­tens­ity, say­ing, “Well, I just wanted to tell you about this the­ory, which I find so beau­ti­ful.”

This was my first in-per­son con­tact with Bott. Earli­er (as a be­gin­ning gradu­ate stu­dent un­der the pa­tient guid­ance of Nor­man Steen­rod), I had stud­ied his pa­pers ap­ply­ing Morse the­ory to the to­po­logy of Lie groups. I had greatly ad­mired these pa­pers, but I was even more de­lighted with the au­thor him­self. His talk on the Atiyah–Sing­er in­dex the­or­em made a deep im­pres­sion on me. I was fas­cin­ated by the mix­ture of to­po­logy and ana­lys­is, and I was also in­trigued by the love of math­em­at­ics that came through so strongly in his talk.

About four years later, I was in a very dis­cour­aged mood. I was now an as­sist­ant pro­fess­or at Prin­ceton and I had the feel­ing I was march­ing back­wards. In work that every­body, in­clud­ing me, be­lieved I had done, gaps and dif­fi­culties were ap­pear­ing. In­stead of mov­ing ahead, I was wast­ing time and en­ergy try­ing to re­pair ar­gu­ments that I thought I had suc­cess­fully fin­ished one or two years be­fore.

In­to the midst of this gloom came Raoul. He gave a talk at IDA about his residue for­mula for holo­morph­ic vec­tor fields. Again, I was keenly in­ter­ested in the math­em­at­ics he presen­ted. I de­cided right then and there dur­ing his talk to simply aban­don the prob­lems I had been think­ing about and start try­ing to fur­ther de­vel­op Bott’s point of view on vec­tor fields.

So Jeff Chee­ger and I got to work to see what the residue for­mula was for a holo­morph­ic vec­tor field with nonde­gen­er­ate sub­man­i­folds of zer­oes. It took us some months to crack this — i.e., to see a clear, simple for­mula com­ing out of Bott’s meth­od. Once we had it, I wrote to Bott telling him our ex­pli­cit an­swer for these residues. He wrote back in­form­ing me that he had already done this, but sug­gest­ing that Jeff and I write up the case of Killing vec­tor fields. We did this, and the pa­per ap­peared in To­po­logy. Even­tu­ally these ideas emerged in a some­what changed form in work of Chee­ger and Gro­mov.

Next, in­spired by a com­ment of G. Wash­nitzer, I de­cided to see how Bott’s the­or­em for holo­morph­ic vec­tor fields might ex­tend to mero­morph­ic vec­tor fields. On a com­pact, com­plex-ana­lyt­ic man­i­fold holo­morph­ic vec­tor fields are really quite rare. But mero­morph­ic vec­tor fields are plen­ti­ful, so it seemed worth­while to gen­er­al­ize Bott’s the­or­em to the mero­morph­ic case. By this time Chee­ger had moved on to oth­er things, so I found my­self work­ing alone on the mero­morph­ic vec­tor field prob­lem. In the late spring of 1968, I felt I had cor­rectly gen­er­al­ized Bott’s meth­od to the mero­morph­ic situ­ation. I wrote to Bott about this and re­ceived a let­ter from him say­ing, “Dear Paul — We cer­tainly are work­ing in \( \| \)!” We were think­ing about ex­actly the same points. Shortly after this, we met in Berke­ley where we were both at­tend­ing the meet­ing on glob­al ana­lys­is. I out­lined to Bott the for­mula and proof that I had in mind. To my as­ton­ish­ment, I found that I was ac­tu­ally slightly ahead of him. When he real­ized that my proof worked, he spread his arms wide and cheer­fully shouted, “It works like a charm!” The tech­nic­al trick on which the proof hinged he re­ferred to as “a slightly mys­ter­i­ous Kun­st­griff”.

After I ar­rived at Brown in Septem­ber 1968, there fol­lowed a peri­od of about two years when I would sporad­ic­ally vis­it Bott at Har­vard, and we worked on a residue for­mula for sin­gu­lar­it­ies of holo­morph­ic fo­li­ations. Dur­ing this time I came to ap­pre­ci­ate an­oth­er of Bott’s qual­it­ies — his hu­mor. Bott can be very funny. We had many good laughs to­geth­er.

Strangely, al­most every time I vis­ited Bott at Har­vard it would rain. When I was leav­ing, Bott would say, “Good-bye, Paul. See you on the next rainy day.” We made some pro­gress on the residue ques­tion. Our work here is closely re­lated to the dif­fer­en­tial char­ac­ter the­ory of Chee­ger–Chern–Si­mons. We proved a ri­gid­ity the­or­em and we for­mu­lated a ra­tion­al­ity con­jec­ture that ap­pears to be a deep and dif­fi­cult prob­lem. At the Stan­ford dif­fer­en­tial geo­metry meet­ing, I learned that Si­mons had made the op­pos­ite con­jec­ture. He was con­vinced that the residues in ques­tion are not al­ways ra­tion­al. The prob­lem re­mains un­solved.

One rainy day I was with Raoul in his of­fice in the Geo­graph­ic In­sti­tute dis­cuss­ing these mat­ters when a tall, thin, shy gradu­ate stu­dent came in and said, “Pro­fess­or Bott, where should I ap­ply for a job?” I said, “Ap­ply to Brown.” The stu­dent was Bob MacPh­er­son. His com­ing to Brown was one of the main events in the met­eor­ic rise of the Brown Math De­part­ment.

To­wards the end of my two-year col­lab­or­a­tion with Bott, he was work­ing on un­der­stand­ing the Hae­fli­ger clas­si­fy­ing space for fo­li­ations and I had star­ted work­ing on Riemann–Roch with Bill Fulton and Bob MacPh­er­son. I had a num­ber of very in­ter­est­ing con­ver­sa­tions with Raoul about the Hae­fli­ger clas­si­fy­ing space. Ap­prox­im­ately twenty years later, while work­ing with Alain Connes, I re­called these con­ver­sa­tions and they helped guide Alain and me to­wards the clas­si­fy­ing space for prop­er ac­tions that we need in our con­jec­ture on the K-the­ory of group C\( ^{*} \) al­geb­ras.

What did I learn from Bott? I learned — per­haps I should say re­learned — from Bott things that I already knew: that there is great beauty and won­der and fun in math­em­at­ics; that the best math­em­at­ics is not ne­ces­sar­ily the most dif­fi­cult; that there is a main­stream to our sub­ject that tal­en­ted math­em­aticians like Bott have an in­stinct­ive feel for; that ideas should be presen­ted clearly and without non­sense so that oth­er math­em­aticians can un­der­stand and ap­pre­ci­ate them.

I con­sider my­self for­tu­nate to have be­nefited from the clar­ity and beauty of Bott’s math­em­at­ics, and from the warmth and joy of his per­son­al­ity.