Celebratio Mathematica

I first learned of John Mor­gan through read­ing his joint book with Phil­lip Grif­fiths Ra­tion­al Ho­mo­topy The­ory and Dif­fer­en­tial Forms1 when I was a gradu­ate stu­dent at the Uni­versity of Cali­for­nia at San Diego. Grif­fiths taught a one-semester course on al­geb­ra­ic curves at Pek­ing Uni­versity in the spring of 1982. I was a first year gradu­ate stu­dent at Pek­ing Uni­versity then and was deeply at­trac­ted by Grif­fiths’ won­der­ful lec­tures. That was the first course since my un­der­gradu­ate stud­ies where I nev­er missed a single class. After I en­rolled at UC­SD, be­cause I had ac­cess to much more lit­er­at­ure, I star­ted to read many more pa­pers and books of Grif­fiths and came across John and Phil­lip’s book. This is a well writ­ten book from which I learned a lot, though I did not think that I fully un­der­stood its con­tents.

In the aca­dem­ic year 1987–1988, I trans­ferred from UC­SD to Har­vard. Dur­ing my stay there, I had a chance to at­tend a lec­ture by John. He talked about his work on the to­po­logy of com­plex sur­faces through gauge the­ory. It was a very clear and in­form­at­ive lec­ture which I en­joyed very much, and I even asked a ques­tion, which was rare for me at that time. Be­cause of my ques­tion, we had some ex­changes af­ter­wards. I think this was the first time I met him in per­son. After that, I heard many talks from him in vari­ous set­tings. I found that his lec­tures were al­ways clear and in­struct­ive. After my gradu­ation, I vis­ited Columbia Uni­versity sev­er­al times and dis­cussed math­em­at­ics with John, es­pe­cially in the spring of 1991, when I went to Columbia for a job in­ter­view, and in the spring of 2001, when I gave the Ei­len­berg lec­tures.

My closer con­tacts with John began in 2003, shortly after G. Perel­man pos­ted three pre­prints between Novem­ber 2002 and Ju­ly 2003, in which he presen­ted proofs of the Poin­caré con­jec­ture and Thur­ston’s geo­met­riz­a­tion con­jec­ture. His proofs were based on the Ricci flow in­tro­duced by R. Hamilton, along with many power­ful geo­met­ric tools. However, Perel­man was ex­tremely con­cise, leav­ing many es­sen­tial de­tails un­stated. As a res­ult, sub­stan­tial ef­fort was needed to veri­fy his proofs and make them more ac­cess­ible to the math­em­at­ic­al com­munity.

Dur­ing the 2003–2004 aca­dem­ic year, I taught a course on Perel­man’s work. In the fall of 2003, I fo­cused on his first pa­per, which in­tro­duced key break­throughs, such as the non­col­lapsing the­or­em for the Ricci flow. In the spring of 2004, I turned to his second pa­per and the proof of the Poin­caré con­jec­ture. John traveled from New York al­most every week to at­tend my lec­tures throughout both semesters. As John later said in his pub­lic lec­ture at ICM 2006, the Poin­caré con­jec­ture, which was posed in 1904, has been linked in one way or an­oth­er with most of the pro­gress in to­po­logy in the last 100 years. Many math­em­aticians had at­temp­ted its proof without suc­cess. Perel­man’s solu­tion, though ground­break­ing, was tech­nic­ally de­mand­ing even for ex­perts. At­tend­ance in my course gradu­ally dwindled, un­til only John and a few of my stu­dents re­mained. This gave us the op­por­tun­ity to have ex­tens­ive dis­cus­sions, dur­ing and after class, about Perel­man’s ar­gu­ments. John’s thought­ful ques­tions and sug­ges­tions deepened our un­der­stand­ing.

By late 2003, John and I had de­cided to write a book on Perel­man’s solu­tion to the Poin­caré con­jec­ture. For the next two years, we met reg­u­larly, either in Prin­ceton or at John’s home on Am­s­ter­dam Av­en­ue in New York City, to dis­cuss the ar­gu­ments and re­lated is­sues. We com­pleted the first draft in mid-May 2006 and sent the manuscript on May 19 to John Ball, then Pres­id­ent of the IMU, and Jim Carlson, then Dir­ect­or of the Clay Math­em­at­ics In­sti­tute. In his emails to them, John ex­pressed our be­lief that the proof of the Poin­caré con­jec­ture had been com­pleted. Our book Ricci Flow and the Poin­caré Con­jec­ture2 was fi­nal­ized in the fall of 2006 and pub­lished jointly by the Clay Math­em­at­ics In­sti­tute and the Amer­ic­an Math­em­at­ic­al So­ci­ety in 2007. In the book, we provided a com­plete, self-con­tained proof of the Poin­caré con­jec­ture. While it fol­lowed Perel­man’s ap­proach in his third pre­print, which was only sev­en pages long and omit­ted many de­tails, we worked hard to fill in the de­tails. Our proof ul­ti­mately spanned more than 80 pages. John played a cru­cial role in this pro­ject, and I felt for­tu­nate to col­lab­or­ate with him on a book of such sig­ni­fic­ance. I was ex­tremely im­pressed by his cour­age and tal­ent to ven­ture in­to a com­pletely new area of math­em­at­ics, trans­form­ing him­self from an out­sider to an ex­pert with­in just two years.

The sum­mer of 2006 was event­ful. The In­ter­na­tion­al Con­gress of Math­em­aticians (ICM) was held in Au­gust, amid wide­spread con­cern over wheth­er the Poin­caré con­jec­ture had truly been solved. John was in­vited to give a pub­lic lec­ture on Au­gust 24. We felt it im­port­ant to provide a clear an­swer. John did ex­actly that: In his open­ing re­marks, he de­clared, “It is a great pleas­ure for me to speak to you today about a stu­pendous achieve­ment in math­em­at­ics: Grigori Perel­man has solved the Poin­caré con­jec­ture.” It was John who first made an un­am­bigu­ous pub­lic con­firm­a­tion of the solu­tion. An­oth­er path to the Poin­caré con­jec­ture lay through the more gen­er­al geo­met­riz­a­tion con­jec­ture. Our first book did not cov­er this fully, though it es­tab­lished many found­a­tions to­ward a com­plete proof. So we began work on a se­quel. A pre­lim­in­ary draft was com­pleted in 2008, but fi­nal­iz­ing it took years. Both of us had new re­spons­ib­il­it­ies: John be­came the found­ing dir­ect­or of the Si­mons Cen­ter for Geo­metry and Phys­ics (SCGP), and I be­came in­creas­ingly oc­cu­pied with my work at Pek­ing Uni­versity. Non­ethe­less, John fre­quently in­vited me to Stony Brook so that we could con­tin­ue col­lab­or­at­ing. In 2011, Richard Bamler found an is­sue in Perel­man’s second pre­print con­cern­ing the long-time be­ha­vi­or of the Ricci flow — something we and oth­ers had over­looked. The is­sue had to be re­solved to com­plete the proof of the geo­met­riz­a­tion con­jec­ture. For­tu­nately, Bamler him­self re­solved it us­ing Perel­man’s tech­niques. With his help, we cor­rec­ted our manuscript, which was sub­mit­ted in 2012 and pub­lished by the Clay Math­em­at­ics In­sti­tute and the Amer­ic­an Math­em­at­ic­al So­ci­ety in 2014.3

After these pro­jects, I con­tin­ued to vis­it John and dis­cuss pos­sible fu­ture col­lab­or­a­tions. At one point, John even con­sidered of­fer­ing me a po­s­i­tion at SCGP, though it did not come to fruition, and I re­turned to work in China. My last meet­ing with him was in the fall of 2017, when he joined the vis­it­ing com­mit­tee in­vited by Pek­ing Uni­versity to eval­u­ate the Beijing In­ter­na­tion­al Cen­ter for Math­em­at­ic­al Re­search (BICMR). He gave us high praise and many valu­able sug­ges­tions, which my col­leagues and I deeply ap­pre­ci­ated.

I had the priv­ilege of work­ing with John on ma­jor pro­jects for nearly a dec­ade. I en­joyed the col­lab­or­a­tion im­mensely and learned a great deal from him. I ad­mired his cour­age to ex­plore new fron­ti­ers of math­em­at­ics, and his ded­ic­a­tion to serving the math­em­at­ic­al com­munity — wheth­er by con­firm­ing the proof of the Poin­caré con­jec­ture, lead­ing SCGP, or con­trib­ut­ing to re­search more broadly. Of course, I also treas­ure the many hours we spent talk­ing about math­em­at­ics in cof­fee shops and in his beau­ti­ful New York apart­ment. Time has flown — it has now been al­most eight years since I last saw John at Pek­ing Uni­versity. I hope to see him again soon, per­haps over a good meal in New York City.

Gang Tian is a lead­ing math­em­atician in geo­met­ric ana­lys­is, com­plex and sym­plect­ic geo­metry. He is cur­rently a Chair Pro­fess­or at Pek­ing Uni­versity and Dir­ect­or of the Beijing In­ter­na­tion­al Cen­ter for Math­em­at­ic­al Re­search. A re­cip­i­ent of the Alan T. Wa­ter­man Award and the Veblen Prize, he is a mem­ber of the Chinese Academy of Sci­ences and the Amer­ic­an Academy of Arts and Sci­ences.