Celebratio Mathematica

André Haefliger

Interview with André Haefliger

by Allyn Jackson

André Haefliger in Nyon, Switzerland in 2018.
Photo courtesy of Allyn Jackson.

Every­one likes to talk to An­dré Hae­fli­ger. His math­em­at­ic­al in­sight and vivid en­thu­si­asm have for dec­ades made him a be­loved in­ter­locutor among math­em­aticians. But talk to him about any­thing — mu­sic, cheese, books, people, the state of the world — and you will find the con­ver­sa­tion buoyed along by his sin­gu­lar in­tel­lect as well as his joy­ous laugh. Likely tak­ing part in the con­ver­sa­tion is An­dré’s wife Her­mina (née Po­lakovitch), uni­ver­sally known by her nick­name Min­ouche, whose in­tel­li­gence, warmth, and charm in­crease the en­joy­ment.

In his 1958 PhD thes­is, writ­ten un­der the dir­ec­tion of Charles Ehresmann, An­dré Hae­fli­ger ob­tained ground-break­ing res­ults in fo­li­ation the­ory. After mak­ing sig­ni­fic­ant con­tri­bu­tions to em­bed­ding and im­mer­sion the­ory, he re­turned to fo­li­ations to again pro­duce fun­da­ment­al ad­vances, in­clud­ing what is now known as the Hae­fli­ger clas­si­fy­ing space. His work in geo­met­ric group the­ory set the stage for many ad­vances in this now-bur­geon­ing sub­ject.

Hae­fli­ger spent two postdoc­tor­al years as the as­sist­ant of Hassler Whit­ney at the In­sti­tute for Ad­vanced Study, where his two main du­ties were to do math­em­at­ics and to play cham­ber mu­sic with Whit­ney. In 1961 he ac­cep­ted a pro­fess­or­ship at the Uni­versity of Geneva and re­mained there, apart from fre­quent vis­its abroad, un­til his re­tire­ment in 1995.

The fol­low­ing art­icle is based on a series of in­ter­views with Hae­fli­ger, con­duc­ted by Allyn Jack­son. The in­ter­views took place in May 2018, at the Hae­fli­ger home near Nyon, Switzer­land. Min­ouche was present dur­ing por­tions of the in­ter­view, as was D. Kotschick of the Uni­versity of Mu­nich.

Early life

An­dré Hae­fli­ger was born in 1929 in Nyon, Switzer­land, and since the 1960s has lived in a small vil­lage less than a mile away. Nyon sits on the north­ern shore of Lake Geneva about 15 miles east of Geneva and about 25 miles west of Lausanne. His fath­er worked in a bank and was good at catch­ing arith­met­ic mis­takes; he once found that the date had been ad­ded to an ac­count bal­ance. An­dré’s moth­er, who was about 20 years young­er than her hus­band, also worked at the bank, al­though she stopped work­ing when her chil­dren were born. The couple already had two chil­dren, a boy and a girl, when An­dré and his twin broth­er were born on May 22, 1929. At that time their fath­er was 60 years old. An­dré’s older broth­er, now de­ceased, stud­ied law and at the end of his life worked in Lausanne as the head of the Land Re­gis­trar of the Can­ton of Vaud. An­dré’s sis­ter, who was the one who taught An­dré to read, loved chil­dren and be­came a teach­er. She still lives in Nyon near An­dré and Min­ouche. They also re­main close to one of his cous­ins, who as a girl was ad­op­ted by An­dré’s moth­er. Now in her 90s, this cous­in lives in Florence.

Hae­fli­ger: My twin broth­er had psy­chi­at­ric prob­lems. He had schizo­phrenia. It was very dif­fi­cult for the whole fam­ily. Now he lives in an apart­ment close to us and is do­ing much bet­ter. This is thanks to my old­est son, Ivan, who is a med­ic­al doc­tor.1 He takes care of my broth­er and does everything for him, pay­ing bills and so on. He is very de­voted to help­ing people. It is in his nature!

Jack­son: That’s good. So your twin broth­er could not do any stud­ies?

Hae­fli­ger: He did some stud­ies, but he had dif­fi­culties. But he is very in­tel­li­gent. He used to read po­etry, for ex­ample Baudelaire. He could re­mem­ber many poems by heart. He also had a tal­ent for im­it­at­ing people. He would go to the front of the little church in our neigh­bor­hood and im­it­ate the pas­tor! He used to play the flute and was very in­ter­ested in mu­sic. He still is.

Jack­son: He wasn’t in­ter­ested in math­em­at­ics?

Hae­fli­ger: No, not at all.

Jack­son: Did you have some early in­flu­ences in­spir­ing you to study math­em­at­ics?

Hae­fli­ger: Yes, I had a very good teach­er when I was 16. He was very en­thu­si­ast­ic and com­pet­ent. At that time I liked to go to the pub­lic lib­rary. I used to bor­row the Dis­quisi­tiones Arith­met­icae!

Jack­son: Did you read it?

Hae­fli­ger: I tried! It was in Lat­in! Do you know Jean-Luc God­ard?

Jack­son: The film­maker?

Hae­fli­ger: Yes. He also lived in Nyon. We were very good friends. My best friend mar­ried his sis­ter. God­ard made a doc­u­ment­ary on René Thom.

Jack­son: Did you sug­gest that to God­ard?

Hae­fli­ger: No, I did not sug­gest it, but I used to see him very of­ten in Par­is, and we would talk to­geth­er. Thom had great suc­cess with cata­strophe the­ory. God­ard was in­ter­ested in mak­ing doc­u­ment­ar­ies of people. You can listen to this in­ter­view on You­Tube. It’s ter­rible! Ter­rible for Thom! Thom is try­ing to ex­plain with great pa­tience some ba­sic cata­strophe the­ory, and God­ard is mak­ing com­ments. It was a big chore for Thom!

Jack­son: You went to a clas­sic­al sec­ond­ary school, with Lat­in and an­cient Greek.

Hae­fli­ger: Yes, in Nyon. We had very good teach­ers. For in­stance one was a spe­cial­ist in Sanskrit, but he taught in Nyon be­cause he had no po­s­i­tion in the uni­versity. Af­ter­ward I went to the Collège Calv­in in Geneva.

Jack­son: Was that a re­li­gious school?

Hae­fli­ger: No. At that time there was only one Gym­nas­i­um, and it was in the Collège Calv­in, a pres­ti­gi­ous school that still ex­ists.

When I ar­rived in Geneva from Nyon, the math­em­at­ics teach­er told me, “Don’t push, don’t come too early in­to math­em­at­ics.” The reas­on is that there was in Geneva a geni­us in math­em­at­ics, who was pushed that way and got his doc­tor­ate in math­em­at­ics one year after his ma­tur­ité.2 He had writ­ten a thes­is that was in­ter­est­ing but not first-class. Then he was sent to Chica­go to An­dré Weil. It was ter­rible for him. An­dré Weil was very tough!

Jack­son: So your teach­er gave you good ad­vice.

Hae­fli­ger: Yes — “Don’t do like this guy!”

Jack­son: Did you start study­ing vi­ol­in very early?

Hae­fli­ger: Yes, as a child. My moth­er lost her fath­er when she was 10 years old. She had be­gun pi­ano, but could not con­tin­ue. Her moth­er had no pen­sion and there­fore had to work very hard. My moth­er loved mu­sic, so she was in­sist­ent that all the mem­bers of our fam­ily learn an in­stru­ment.

Jack­son: And your fath­er, did he also play mu­sic?

Hae­fli­ger: Yes, he played in the Nyon or­ches­tra. But I don’t re­mem­ber him play­ing, he was too old at that time.

Jack­son: You fin­ished your stud­ies at the Collège Calv­in, and then you went to the Uni­versity of Lausanne. Why did you not go to the Uni­versity of Geneva?

Hae­fli­ger: Be­cause the only pos­sib­il­ity I thought I could have was to be­come a teach­er in a sec­ond­ary school. The can­ton of Vaud is much big­ger than the can­ton of Geneva, so there would be more pos­sib­il­it­ies to get a job. This was be­fore the ex­plo­sion of uni­versit­ies, and I didn’t have the idea I could have an aca­dem­ic ca­reer.

Jack­son: When you entered uni­versity, you had de­cided to study math­em­at­ics?

Hae­fli­ger: Sure. There was no ques­tion.

Jack­son: But you also got a cer­ti­fic­ate from the con­ser­vat­ory in vi­ol­in.

Hae­fli­ger: Yes, dur­ing my stud­ies in Lausanne, I got a cer­ti­fic­ate.

Jack­son: Did you con­sider be­com­ing a mu­si­cian?

Hae­fli­ger: No. I real­ized math­em­at­ics was much easi­er!

Jack­son: Too much prac­ti­cing in mu­sic?

Hae­fli­ger: Yes, and it’s hard to find a job. I nev­er con­sidered it. But I played with many people, in­clud­ing my teach­ers in Lausanne. Later on I played with [Henri] Cartan, [Wil­li­am] Browder, [Mi­chael] Artin, [Hassler] Whit­ney, [Eu­genio] Calabi, Dusa Mc­Duff, as well as the wives of some of the math­em­aticians. Mu­sic is really a way of meet­ing people. Min­ouche also took part, as she plays pi­ano, vi­ol­in, and vi­ola. She and I have played cham­ber mu­sic to­geth­er ever since we first met.

Jack­son: That’s won­der­ful to be able to share that. What did Cartan play?

Hae­fli­ger: Cartan played pi­ano. He was a very good pi­an­ist. He had a broth­er who was a com­poser and had con­tact with many mu­si­cians in Par­is.

Jack­son: You took many courses with Georges de Rham in Lausanne. What were your ini­tial im­pres­sions of him?

Hae­fli­ger: He was a very im­press­ive man. He had a sem­in­ar, and we had to pre­pare sub­jects to present. He was very de­mand­ing, very pre­cise, but still very kind. Later he be­came a friend. Un­for­tu­nately in his last years he com­pletely lost his memory. We in­vited him re­gard­less, very of­ten. Once he for­got to come. One of our col­leagues was here and went to pick him up. De Rham had already eaten, but he ate again with us! His memory loss was dif­fi­cult, be­cause he real­ized it. So it was very sad. He was a great alpin­ist. He loved to read Mal­larmé.

Jack­son: You had to do army ser­vice around this time. What did you have to do?

Hae­fli­ger: I was in aer­i­al de­fense. After one month, the cap­tain asked us to write our opin­ion about something. So I wrote, “I want to be in the Com­mun­ist Party.” After that I was con­sidered a ter­rible danger! So they put me in the kit­chen! This was not long after the war. Switzer­land was really afraid of com­mun­ism.

Jack­son: But you were not afraid of say­ing this?

Hae­fli­ger: No. At the end of this army ser­vice, you got a grade. My grade was “surnuméraire”[re­dund­ant]! Also dur­ing this army ser­vice, I was in­jured when a can­non got pushed onto my feet, so I had to go to the hos­pit­al. This was mar­velous, be­cause I did math­em­at­ics! I don’t know how I got books, but I re­mem­ber read­ing Fe­lix Klein and oth­er clas­sic books.

Jack­son: Go­ing back to your uni­versity work, how did you de­cide to go on to get a doc­tor­ate?

Hae­fli­ger: At first I thought I would do sec­ond­ary teach­ing. It was Jean de Siebenth­al who told me, “No, you should get an aca­dem­ic de­gree.” So I asked for a sti­pen­di­um, and de Rham helped me with that. I did not want to do a doc­tor­ate with de Rham. I wanted to be more free. He re­com­men­ded to me not to go to Par­is. He said, “You will be lost there, it’s bet­ter for you to go to Stras­bourg.” He was a very good friend of Ehresmann, who was at Stras­bourg then. In fact I was also very much in­ter­ested in the pa­pers of Ehresmann. I stud­ied them in a sem­in­ar with de Siebenth­al.

Jack­son: So Stras­bourg was a good choice for you.

Hae­fli­ger: Yes. It was fant­ast­ic. At that time [1954], I was the only gradu­ate stu­dent. Wu Wentsun and Georges Reeb had been stu­dents in Stras­bourg, but they had left by then. It is strange that I was the only stu­dent! There were very good people there: [Jean-Louis] Koszul, [Georges] Cerf (the fath­er of Jean Cerf), and René Thom, of course. And Ehresmann in­vited many, many people.

At this time I got in very good con­tact with Thom. Later Ehresmann was nom­in­ated as a pro­fess­or in Par­is, and I fol­lowed him, but I used to go very of­ten to Stras­bourg to dis­cuss with Thom. Our dif­fer­ence in age was not so big. I was born in 1929, and Thom was born in 1923. Six years, it’s not so much. But when I met him, I knew al­most noth­ing about math­em­at­ics! I was really a be­gin­ner. I didn’t know what dir­ec­tion to go in, how to ori­ent my­self.

Jack­son: What were your im­pres­sions of Thom?

Hae­fli­ger: I ad­mire his way of un­der­stand­ing math­em­at­ics. He would be listen­ing to you and then would make re­marks which were ex­tremely help­ful. And he was very kind. I can show you let­ters he sent me, many let­ters. I be­came like a con­fid­ant for him — I don’t know why! It was a time when he was very much in­ter­ested in sin­gu­lar­it­ies. A con­crete part of my thes­is used the the­ory of sin­gu­lar­it­ies of dif­fer­en­ti­able map­pings.

We were friends with him and his fam­ily, un­til the end. We spent hours and hours with him when he was very ill, and we some­times stayed overnight to help the fam­ily. He had a leg am­pu­tated and was in a wheel­chair. And he lost his memory. It was very hard for his wife, who had a very clear mind.3

Reeb’s thesis an inspiration

“It was René Thom who ex­plained to me Ehresmann’s ideas about holonomy dur­ing the winter of 1954–55 (Ehresmann was of­ten trav­el­ing). I real­ized pretty quickly that, in the dif­fer­en­ti­able case, the holonomy of a prop­er leaf \( F \) (that is, a leaf such that the leaf to­po­logy co­in­cides with the in­duced to­po­logy) per­mit­ted one to re­con­struct the germ of the fo­li­ation along \( F \) and to char­ac­ter­ize it. The fol­low­ing year I showed the non-ex­ist­ence of ana­lyt­ic fo­li­ations of codi­men­sion 1 on com­pact simply con­nec­ted man­i­folds, in par­tic­u­lar re­spond­ing, in the neg­at­ive, to the ques­tion Reeb had posed con­cern­ing the ex­ist­ence of an ana­lyt­ic fo­li­ation on the 3-sphere.”

 — An­dré Hae­fli­ger, “Nais­sance des Feuil­letages, d’Ehresmann–Reeb à Novikov” (trans­lated from the French).4

Jack­son: Your thes­is work was closely re­lated to the work of Georges Reeb. Did you get to know Reeb?

Hae­fli­ger: Yes, yes.

Jack­son: He was already in Gren­oble when you were in Stras­bourg. Did he come back to Stras­bourg some­times?

Hae­fli­ger: Yes. He was very kind and his wife also. He was truly Alsa­tian. He didn’t go to the École Nor­male, so he was really an out­sider! Of course we saw him very of­ten, in Brazil, in many places. He had a whole school around him. In the end, he was in­ter­ested in non­stand­ard ana­lys­is. It was like a re­li­gion! He wanted to per­suade people they should know about it.5

Jack­son: Did you learn about fo­li­ations from Reeb?

Hae­fli­ger: Yes, by read­ing his thes­is. I re­mem­ber in 1954 I went to the In­ter­na­tion­al Con­gress in Am­s­ter­dam. I didn’t know the thes­is of Reeb be­fore that. I saw it on a table, and I opened it, and I found it fas­cin­at­ing. He posed sev­er­al prob­lems in his thes­is.

I was also very much in­flu­enced [in my thes­is] by Ehresmann’s way of think­ing about math­em­at­ics. And people dis­liked that very much.

Jack­son: Why?

Hae­fli­ger: Why? You should read the re­port of [Richard] Pal­ais in Math Re­views!

Jack­son: Oh yes, he wrote things like, “An ex­treme Bourbaki-like gen­er­al­iz­a­tion…sev­er­al hun­dred pre­lim­in­ary defin­i­tions…”

Hae­fli­ger: Yes, but he mixed up a few things! And my thes­is was very long. When I ar­rived in Prin­ceton, every­body was jok­ing about this! Later Pal­ais and I talked about this, and we had a very good re­la­tion­ship after that. He was sorry to have writ­ten that, be­cause it’s clear that there was a con­fu­sion in his mind between \( H^0 \) and \( H^1 \)!

At this time I had a ter­rible con­flict with de Rham. He wanted my thes­is to be pub­lished in the Com­ment­arii. When he heard that Reeb wanted to pub­lish it in the An­nales de l’In­sti­tut Four­i­er, he wrote me a ter­rible let­ter, say­ing that I was not thank­ful for the help the Swiss Na­tion­al Fund gave to me. This let­ter was so ter­rible that I des­troyed it!

In the end, de Rham pub­lished my thes­is in the Com­ment­arii. He made a great ef­fort to be very quick. At that time you could not get your de­gree be­fore your thes­is was prin­ted. I had to send to the Sor­bonne more than 100 cop­ies of my thes­is. And of course they are lost! It was the cus­tom at that time.

Jack­son: So your thes­is did not ap­pear in the An­nales de l’In­sti­tut Four­i­er. What did Reeb think of that? Was he angry?

Hae­fli­ger: No, no — he was not that kind of man. He was very kind. I prob­ably told him about the let­ter from de Rham, and he un­der­stood. But de Rham was right in a sense. He had helped me very much.

Jack­son: What was Ehresmann like?

Hae­fli­ger: He was a very open and gen­er­ous man. His of­fice was open, any­body could come. He liked to work in res­taur­ants in Par­is. He had prob­lems with his wife, who stayed in Stras­bourg — she was a strong per­son­al­ity. So when he was in Par­is he was alone. He in­vited me very of­ten to very good res­taur­ants, be­cause he liked to eat good food, and we would dis­cuss math­em­at­ics. We had a meal in a res­taur­ant in Par­is after my thes­is de­fense.

Haefliger celebrates his PhD defense in Paris (1958). Charles Ehresmann is at left facing the camera (in what looks like a tweed jacket and a dark tie); at his left is Wu Wentsun and George Reeb. On the right, closest to the camera is Shi Weishu, and to his right, Haefliger himself and Minouche.

When I dis­cussed my thes­is work with Ehresmann, he said, “If you could prove in every fo­li­ation on the 3-sphere there is a tor­us, then it is enough for your thes­is.” I didn’t prove that!6 Reeb was a stu­dent of Ehresmann, and it was Ehresmann who in­tro­duced Reeb to the the­ory of fo­li­ations. When I was a stu­dent in Lausanne, de Siebenth­al, [Georges] Vin­cent, and I or­gan­ized a sem­in­ar on the the­ory of fiber bundles, very clas­sic­al, and Ehresmann was men­tioned there. I was at­trac­ted by his way of think­ing. He was very geo­met­ric­al, like Reeb.

Later Ehresmann de­veloped a very ab­stract way of look­ing at math­em­at­ics — cat­egor­ies and ba­sic struc­tures and so on. It was not my way of look­ing at math­em­at­ics. So he had changed com­pletely his way of un­der­stand­ing math­em­at­ics.

Jack­son: Why did he change so much?

Hae­fli­ger: I don’t know.

Jack­son: This oc­curred after he moved to Par­is?

Hae­fli­ger: Some­how, yes. Maybe it was the in­flu­ence of Bourbaki. He was a mem­ber of Bourbaki.

Jack­son: Were you ever in Bourbaki?

Hae­fli­ger: No.

Jack­son: Were you asked?

Hae­fli­ger: No. But when I was a stu­dent at Geneva, I used to go by bike to France to buy the fas­cicule of Bourbaki. It was not a long bike ride!

Jack­son: What are your thoughts on the in­flu­ence of Bourbaki?

Hae­fli­ger: It was an im­port­ant en­ter­prise. But they wanted to make something that would re­main forever, and math­em­at­ics moves so quickly that the first pub­lic­a­tions were already not valu­able any­more. So that is the bad thing. Bourbaki pub­lished some use­ful volumes, but dif­fer­en­tial to­po­logy — it was too loose. They were not able to do that, be­cause they did not take ac­count of the evol­u­tion. In the States, so many people were work­ing in dif­fer­en­tial geo­metry, it was pro­gress­ing so much, and Bourbaki was too ri­gid. The best volumes of Bourbaki are those ded­ic­ated to Lie groups. [Ar­mand] Borel made a big con­tri­bu­tion to that, and [Jacques] Tits too.

Strasbourg and Paris

Hae­fli­ger: Min­ouche and I got mar­ried the first of March in 1958, and we went im­me­di­ately to Par­is. I had a sti­pen­di­um, a very small one, but it was very cheap at that time. We could not stay in the same hotel for more than two weeks. That was in­ter­est­ing! We used to pre­pare some of our meals in the hotel. It was not well ac­cep­ted, but we did it!

Jack­son: You had gone from Stras­bourg to Par­is. What dif­fer­ences in the math­em­at­ic­al cli­mate did you see?

Hae­fli­ger: Oh, it was com­pletely dif­fer­ent. In Par­is there was the Cartan sem­in­ar. This was very im­port­ant, with a lot of people. At that time there was the École Nor­male Supérieure and the In­sti­tut Henri Poin­caré. The École Poly­tech­nique was also in the cen­ter of Par­is, but there were no oth­er uni­versit­ies. Or­say didn’t ex­ist then. So people would meet each oth­er very of­ten.

[Jean-Pierre] Serre and [Ro­ger] Go­de­ment were in Par­is, and [Jean] Leray. The dir­ect­or of the thes­is of Borel was Leray. So Borel had a very good know­ledge of sheaf the­ory, and he ex­plained the the­ory of Leray to people in Zurich be­fore they could un­der­stand it!7 In fact Serre said that Borel ex­plained all that stuff to him.

Jack­son: Did you speak in the Cartan sem­in­ar?

Hae­fli­ger: Yes, I spoke about sin­gu­lar­it­ies. [Claude] Che­val­ley and An­dré Weil were in the audi­ence. It was hard for me.

Jack­son: Why is that?

Hae­fli­ger: They wanted to un­der­stand the de­tails. I gave maybe three talks, but I could not give all the de­tails.

Thom gave sem­inars on sin­gu­lar­it­ies, and Serre could not un­der­stand this.

Jack­son: Why could Serre not un­der­stand Thom?

Hae­fli­ger: Be­cause Thom was not pre­cise enough for him, es­pe­cially his talks — and later on, his pub­lic­a­tions!

Jack­son: They were very dif­fer­ent as math­em­aticians.

Hae­fli­ger: Yes, very dif­fer­ent ways of un­der­stand­ing math­em­at­ics.

Jack­son: And Cartan? Could he un­der­stand both Thom and Serre?

Hae­fli­ger: Oh, yes. He had great es­teem for Thom.

Jack­son: Did Serre have es­teem for Thom even though he did not un­der­stand him?

Hae­fli­ger: At first, yes. I can show you let­ters he wrote to Thom. They are very friendly.

Jack­son: And later?

Hae­fli­ger: No, not at all. Thom wrote let­ters to Serre, and Serre des­troyed most of them. Serre, and also Borel, did not real­ize the im­pact of the non-math­em­at­ic­al works of Thom in many fields. They were not in­ter­ested. But Borel was very in­ter­ested in things out­side math­em­at­ics, more than Serre. Borel was fas­cin­ated by In­di­an mu­sic and jazz. He wrote many art­icles about the his­tory of math­em­at­ics. He had a more pro­found view of the world. And Borel was very crit­ic­al of what Bourbaki had done.

Jack­son: You got to know Borel very well.

Hae­fli­ger: Yes. Min­ouche met him be­fore me. She stud­ied math­em­at­ics in Geneva and was in Borel’s al­gebra course. When he was at Prin­ceton he some­times spent his sum­mers in La Con­ver­sion. The moth­er of his wife Gaby had a house there, which still ex­ists, with a beau­ti­ful view of the lake. Some­times we would cross Lake Geneva by sail­boat and stop in Lut­ry, near La Con­ver­sion. We would meet there and have a very good time!

As I said, I first went to Stras­bourg on the re­com­mend­a­tion of de Rham. There were a lot of vis­it­ors, for in­stance, Robert Her­mann, an Amer­ic­an who was in­ter­ested in phys­ics and math­em­at­ics. Like an Amer­ic­an, he sat in a chair with feet on the table. Koszul did not like that!

I also re­mem­ber meet­ing [Re­in­hold] Rem­mert and go­ing with him to the Stras­bourg cathed­ral, to the top of the tower. He got me in­ter­ested in com­plex ana­lys­is. My second thes­is was on this sub­ject.

Jack­son: What is the second thes­is?

Hae­fli­ger: At that time you wrote a second thes­is on a top­ic not too close to the sub­ject of your thes­is, to be sure that you learned something in an­oth­er do­main. Pierre Lel­ong was my ex­am­iner for the second thes­is. It was not pub­lished be­cause there was noth­ing so in­ter­est­ing in it. But Serre and Borel had second theses that were pub­lished and were im­port­ant pa­pers!

Reeb stability

Around the time of Hae­fli­ger’s thes­is, he and Reeb dis­covered a way to gen­er­al­ize the sta­bil­ity the­or­em for fo­li­ations that Reeb had proved in his own thes­is. They nev­er pub­lished their pa­per, but an un­dated draft is de­pos­ited with Hae­fli­ger’s pa­pers at the Uni­versity of Geneva. Much later their res­ult was su­per­seded in a pa­per of Wil­li­am Thur­ston [e7]. Then Reeb, to­geth­er with Paul Sch­weitzer, used non­stand­ard ana­lys­is to pro­duce a one-page proof of Thur­ston’s res­ult [e9]. An ad­dendum to the Reeb–Sch­weitzer pa­per, writ­ten by Wal­ter Schach­er­may­er, uses two pages to present a trans­la­tion in­to stand­ard ana­lys­is [e10].

Hae­fli­ger: Reeb and I were sup­posed to write a pa­per to­geth­er, a very good pa­per, but as I went to the States, I did not have time to take care of it. The top­ic was a sta­bil­ity res­ult that Thur­ston de­veloped later on. I changed sub­jects when I ar­rived in Prin­ceton in the fall of 1959. I didn’t do any­thing about fo­li­ations. It was not fash­ion­able!

In Prin­ceton I met Smale and dis­cussed a lot with him. At this time he was do­ing im­mer­sions, handle­bod­ies, clas­si­fic­a­tion of man­i­folds, things like that. He was sup­posed to stay in Prin­ceton an­oth­er term or so, but he left early to go to Brazil and got in­ter­ested in dy­nam­ic­al sys­tems.

Jack­son: In Prin­ceton you got in­ter­ested in em­bed­dings and im­mer­sions and so on. Did Whit­ney have any­thing to do with that?

Hae­fli­ger: No. He was in­ter­ested in many fas­cin­at­ing sub­jects, like real al­geb­ra­ic man­i­folds. Whit­ney wrote mar­velous pa­pers in many dif­fer­ent sub­jects. It’s really im­press­ive.

Be­fore work­ing on im­mer­sions and em­bed­dings, I wrote a joint pa­per with Borel [2], which is in fact my first pub­lic­a­tion after my ar­rival at Prin­ceton. I will de­scribe the gen­es­is of that pa­per.

I gave a talk in the frame­work of the sem­in­ar or­gan­ized by An­dré Weil, called Cur­rent Lit­er­at­ure, at Prin­ceton Uni­versity. The top­ic of my talk was a re­port on the work of Thom on sin­gu­lar­it­ies of dif­fer­en­ti­able map­pings that I gave with [Ant­oni] Kos­in­ski in the Cartan sem­in­ar in 1957 [1], as men­tioned be­fore. A few days af­ter­ward I met Borel at the tra­di­tion­al tea at the In­sti­tute [for Ad­vanced Study] and men­tioned to him Thom’s claim that a real al­geb­ra­ic ana­lyt­ic set car­ries a fun­da­ment­al class mod­ulo 2, a fact ad­mit­ted without proof. Borel real­ized im­me­di­ately that the so-called Borel–Moore co­homo­logy was the nat­ur­al tool to prove it. Borel left Prin­ceton in Decem­ber 1959 to be­gin a long trip to In­dia, so most of our col­lab­or­a­tion was made by postal mail ex­changes.

Min­ouche: While we were in Prin­ceton we went to Bal­timore for a con­fer­ence at the in­sti­tute of [So­lomon] Lef­schetz.

Hae­fli­ger: When I saw him for the first time, Lef­schetz said, “Where do you come from?” I said, “Geneva.” And he said, “I hate Geneva!”

Jack­son: How did it come about that Lef­schetz in­vited you?

Hae­fli­ger: I was look­ing for of­fers, and he gave me an of­fer from his in­sti­tute, though later he wrote to tell me it would not be pos­sible. Dur­ing the vis­it, I was on duty to ex­plain to him the work of some Rus­si­ans, or to find a mis­take in it.

In 1958, So­lomon Lef­schetz came out of re­tire­ment to es­tab­lish the math­em­at­ics di­vi­sion of the Re­search In­sti­tute for Ad­vanced Stud­ies, foun­ded by the Glen L. Mar­tin Com­pany of Bal­timore, which is now Mar­tin Mari­etta. He later moved the group to Brown Uni­versity, and it con­tin­ues to this day as the Lef­schetz Cen­ter for Dy­nam­ic­al Sys­tems. In 1960, Lef­schetz in­vited Hae­fli­ger to vis­it RI­AS and gave him a pa­per by two Rus­si­an math­em­aticians.8 The pa­per claimed a solu­tion to the second part of Hil­bert’s 16th Prob­lem, which con­cerns the num­ber and loc­a­tion of lim­it cycles for a planar poly­no­mi­al vec­tor field. Hae­fli­ger found the mis­take in the pa­per, as did Yu. Ily­ashen­ko and S. P. Novikov around the same time in Rus­sia (see [e16]). In the early 1980s, when Shi Song Ling vis­ited Hae­fli­ger in Geneva, he found an ex­pli­cit counter­example [e12].

Professorship in Geneva

By 1961, at the end of his time in Prin­ceton, Hae­fli­ger had sev­er­al of­fers from in­sti­tu­tions in the United States, in­clud­ing the Uni­versity of Cali­for­nia at Berke­ley and Stan­ford Uni­versity. Georges de Rham worked quickly to en­tice Hae­fli­ger back to Switzer­land with an of­fer from the Uni­versity of Geneva, which Hae­fli­ger ac­cep­ted. Dur­ing 1961–62, the Cartan sem­in­ar in Par­is was de­voted to the work of Smale, in par­tic­u­lar his pa­per “On the struc­ture of man­i­folds [e3] and was or­gan­ized jointly with the de Rham sem­in­ar in Geneva and Lausanne. Hae­fli­ger, who had lived next door to Smale in the IAS hous­ing and had had many dis­cus­sions with him, co­ordin­ated the sem­in­ar activ­it­ies through many phone calls and let­ters with Cartan and also presen­ted sev­er­al of the lec­tures.

Hae­fli­ger: When I star­ted at Geneva, pro­fess­ors had no of­fices. When de Rham and I were to phone the Min­is­ter of Edu­ca­tion in Geneva, we had to go to a pub­lic phone be­cause there was no phone avail­able at the uni­versity! But the phys­i­cists already had phones. They were well es­tab­lished, be­cause of the prox­im­ity of CERN.

One of the first math­em­at­ics pro­fess­ors we hired was [Al­fred] Frölich­er. We also hired Michel Ker­vaire and Raghavan Narasim­han. We were very am­bi­tious. We wanted to have Borel, we wanted to have [Pierre] Carti­er, [Jacques] Tits. We were very pre­ten­tious!

In 1970, when Ker­vaire came to Geneva, de Rham was be­gin­ning to re­tire. De Rham was the ed­it­or of Com­ment­arii, and we had a dis­cus­sion with Ker­vaire about the journ­al. De Rham said, “Maybe it’s Hae­fli­ger who has to take care of Com­ment­arii, and Ker­vaire L’En­sei­gne­ment Mathématique.” So I be­came an ed­it­or of Com­ment­arii. It was fant­ast­ic! I could have the work of Thur­ston pub­lished and oth­er in­ter­est­ing pa­pers. I was an ed­it­or for a long time.

Jack­son: Were you the ed­it­or in chief?

Hae­fli­ger: No, nev­er, be­cause if you are the ed­it­or in chief, you have to be an ad­min­is­trat­or — you re­ceive a lot of pa­pers, you must re­fuse some of them, you must find ref­er­ees, and so on. It’s a huge job. [Jo­hann Jakob] Burck­hardt at the Uni­versity of Zurich did this for a long time [1950–1982].

Jack­son: What is the his­tory of the journ­al L’En­sei­gne­ment Mathématique?

Hae­fli­ger: It was foun­ded a long time ago [1899], by Henri Fehr from Geneva and Charles-Ange Lais­ant from Par­is. It was in col­lab­or­a­tion with the French Math­em­at­ic­al So­ci­ety and the In­ter­na­tion­al Com­mis­sion on Math­em­at­ic­al In­struc­tion. All de­part­ments of math­em­at­ics in France would re­ceive L’En­sei­gne­ment Mathématique and would con­trib­ute to it. It does not pub­lish pa­pers in math­em­at­ics edu­ca­tion, but the pa­pers should be un­der­stand­able to gradu­ate stu­dents. They should not be spe­cial­ized.

One thing that is very im­port­ant for our lib­rary is that L’En­sei­gne­ment Mathématique would ad­vert­ise new books that came out. We got the books free, in ex­change for a sub­scrip­tion to the journ­al. So the math­em­at­ic­al lib­rary at Geneva got es­sen­tially all books be­cause of that.

In 1966, Borel, Hae­fli­ger, and de Rham or­gan­ized a sym­posi­um to be held in the Swiss Ro­mande, just be­fore the In­ter­na­tion­al Con­gress of Math­em­aticians was to take place in Mo­scow that year (at this time de Rham was pres­id­ent of the In­ter­na­tion­al Math­em­at­ic­al Uni­on). Among the par­ti­cipants in the Swiss sym­posi­um were Wal­ter Baily, Wil­li­am Browder, Mor­ris Hirsch, Nic­olas Kuiper, John Mil­nor, An­thony Phil­lips, Steph­en Smale, Den­nis Sul­li­van, René Thom, Jack Wag­on­er, and C. T. C. Wall. Phil­lips grew up in Par­is, and as it happened the ed­it­or of the loc­al news­pa­per Tribune de Genève was an old fam­ily friend. In­trigued by the high-level in­ter­na­tion­al con­fer­ence tak­ing place in the city, the ed­it­or asked Phil­lips to write an art­icle about it. Phil­lips com­plied, but be­fore sub­mit­ting the piece, he read it out loud to an­oth­er sym­posi­um par­ti­cipant, the blind to­po­lo­gist Bern­ard Mor­in  — who then did a com­plete re­write. The art­icle ap­peared in the Tribune de Genève on Au­gust 2, 1966. It dis­cussed the hope of the sym­posi­um or­gan­izers to cre­ate an In­sti­tut Ro­mande de Mathématique — a math­em­at­ics re­search in­sti­tute in the Swiss Ro­mande, sim­il­ar to ven­tures like the In­sti­tute for Ad­vanced Study in Prin­ceton (where Borel was on the fac­ulty) and the Forschungsin­sti­tut für Math­em­atik at the ETH in Zurich (foun­ded by Beno Eck­mann in 1964).

Hae­fli­ger: This in­sti­tute in the Suisse Ro­mande was the dream of Borel.

Jack­son: What be­came of it?

Hae­fli­ger: Noth­ing. [laughter]

You know, the name of Borel is nev­er evoked in Lausanne. He is really a for­eign­er for them. As a boy he was a stu­dent in a pub­lic sec­ond­ary school in Geneva and later on a stu­dent for one year in a private school close to Nyon. You can­not enter uni­versity without a cer­ti­fic­ate de ma­tur­ité, and he did that es­sen­tially by him­self. The ex­am­in­a­tion is very hard, and the grades of the year be­fore don’t count at all. He was a uni­versity stu­dent at the ETH. Later, in the 1980s, he joined the ETH fac­ulty, and then there was huge ten­sion in Zurich between Borel and Eck­mann and Moser.

Min­ouche: And we were friends with all of them!

Hae­fli­ger: One per­son from the Uni­versity of Zurich, be­fore Borel would enter a lec­ture room, would write on the black­board, “Borel is not wel­come.” It was ter­rible! Borel had a strong char­ac­ter. I re­mem­ber when he died, there were people on the in­ter­net say­ing, “It is time that the earth is get­ting rid of him!”

Min­ouche: He was not a dip­lo­mat! We were very good friends with him, but maybe we have not such strong char­ac­ters!

Jack­son: Was he really in­ter­ested in com­ing back to Switzer­land?

Hae­fli­ger: Oh, yes. He liked Switzer­land a lot.

Connections with Asia

Start­ing early in his ca­reer, Hae­fli­ger had con­tacts with math­em­aticians from Asia. While a stu­dent in Stras­bourg, for ex­ample, he met Shoshi­chi Kobay­ashi and Yozo Mat­sushi­ma. Later on, his con­tacts with Itiro Tamura and Takashi Tsuboi helped to build a thriv­ing school for fo­li­ations in Ja­pan. In 1979, at the in­vit­a­tion of Wu Wentsun, An­dré and Min­ouche went for the first time to China, and on the same trip made a vis­it in Ja­pan.

Min­ouche: We ar­rived in China on the first of Oc­to­ber 1979, when there was a big meet­ing for the 30th an­niversary of Mao Zedong’s lead­er­ship of China and the Cul­tur­al Re­volu­tion. Wu had to be at this meet­ing, so he in­vited us to it. There was an in­cred­ible feast. The whole of China was rep­res­en­ted, with people in cos­tume.

Hae­fli­ger: There were people who had sur­vived the Long March with Mao. It was very im­press­ive. The meet­ing was held in the Great Hall of the People in Tianan­men Square. When we ar­rived in Beijing, Wu told me, “I have sad news to tell you. Ehresmann died.” I didn’t know that. He had also been a stu­dent of Ehresmann.

Jack­son: Was there a math­em­at­ic­al meet­ing in China that you went to?

Hae­fli­ger: No, but I gave a course.

Min­ouche: We were not al­lowed to go out by ourselves. We had a guide. So one day the guide came to pick me up. We went to the uni­versity, but An­dré had not yet fin­ished his lec­ture. The guide entered in the lec­ture room and said, “It’s time to leave!”

Jack­son: So your lec­ture was over! Why were you not al­lowed to go out by yourselves?

Hae­fli­ger: It was a dif­fi­cult time in China. It was just after the Cul­tur­al Re­volu­tion. Things were not yet stable.

Jack­son: But Wu Wentsun was still able to in­vite you to China.

Hae­fli­ger: Yes, be­cause he was very re­spec­ted by the gov­ern­ment. Dur­ing the Cul­tur­al Re­volu­tion, he was asked to do re­search on math­em­at­ics in China. He wrote about this time when he got the Shaw Prize.9

Our last trip to China was close to the time when Wu re­ceived a big prize from the Chinese gov­ern­ment. A spe­cial­ist in ag­ri­cul­ture re­ceived one of the prizes, and Wu Wentsun got the oth­er prize.10

In China we also met Li Bang He. He was a stu­dent of Wu Wentsun and later stayed two years in Geneva, with an­oth­er Chinese, Shi Song Ling.

Min­ouche: Li was already 40 when we met him. His stud­ies had been in­ter­rup­ted be­cause of the Cul­tur­al Re­volu­tion. To thank us, Li later in­vited us for one month in China, and Tsuboi in­vited us in Ja­pan for two or three weeks. Be­cause of the time they spent in Switzer­land, they could de­vel­op them­selves.

Jack­son: On your first trip to China in 1979, you also vis­ited Ja­pan, and that was when you first met Tsuboi and Tamura.

Hae­fli­ger: Yes.

Jack­son: When did the Ja­pan­ese start work­ing in fo­li­ations ser­i­ously?

Hae­fli­ger: I would say Tamura was the first, around 1970.

Jack­son: Did Tamura ever come to Europe?

Hae­fli­ger: I don’t think so. There were many Ja­pan­ese vis­it­ing Stras­bourg when I was a stu­dent. But they were not do­ing fo­li­ations. The sub­ject was com­pletely un­known to most people then.

Jack­son: Right after your first trip to Ja­pan, an act­ive ex­change in fo­li­ations star­ted.

Hae­fli­ger: Yes. You know, at that time, the Ja­pan­ese didn’t travel so much, un­less they already were big pro­fess­ors. Ja­pan­ese stu­dents didn’t travel. To say, “I want to go to Geneva to study” — it was out of the ques­tion. But Tsuoboi spent two years here while he was a gradu­ate stu­dent. I got a sti­pen­di­um for him from the Swiss Na­tion­al Fund. Then he got his PhD [in 1983] un­der Tamura.

Of course Tsuboi is a mar­velous per­son. He traveled all over Europe and was in­vited by many uni­versit­ies, mainly in France. He learned French in Fri­bourg, at a school for for­eign stu­dents. The school had a very strict and ef­fi­cient way of teach­ing the French lan­guage. Later Vaughan Jones and his wife learned French at the same place.

Jack­son: How long was this lan­guage course?

Hae­fli­ger: About three months.

Jack­son: That’s not a long time.

Hae­fli­ger: Yes, but they worked very hard!

From classifying space to group theory

“An­dré Hae­fli­ger is a geo­met­er. In the worlds of form that math­em­aticians sound in or­der to dis­cov­er the deep struc­ture with­in, he was able to per­ceive forms of which no one be­fore him was aware. He thus showed the many ways in which the simplest sur­faces wrap them­selves up in­side spaces of high­er di­men­sion. The tech­nic­al terms ad­op­ted by spe­cial­ists of­ten car­ried his name: these are the knot­ted Hae­fli­ger spheres, or the Hae­fli­ger clas­si­fy­ing space in the the­ory of fo­li­ations. This clas­si­fy­ing space form­al­izes one of his most fer­tile ideas, which al­lows one to un­der­stand in what sense it is pos­sible to clas­si­fy all the ways in which mul­tiple ob­jects of small di­men­sion fold them­selves in­side struc­tures of large di­men­sion — like the lay­ers in puff pastry or in a block of mica.”

 — from the cita­tion for the Prix de la ville de Genève, awar­ded to Hae­fli­ger in 2003 (trans­lated from the French).

Kotschick: Den­nis Sul­li­van once said that there are two schools of dif­fer­en­tial to­po­logy in the United States. There is the Prin­ceton school, where they do na­ked man­i­folds without any ad­di­tion­al struc­ture.

Hae­fli­ger: Not sin­gu­lar spaces.

Kotschick: Right, just na­ked man­i­folds. Then there is the Berke­ley school, where they do man­i­folds with some ad­di­tion­al struc­ture — a dy­nam­ic­al sys­tem, or a fo­li­ation. Of course, most people are not com­pletely in one school or the oth­er. But you star­ted with man­i­folds with ex­tra struc­ture, with a fo­li­ation, and then you went to na­ked man­i­folds after that.

Hae­fli­ger: Yes, be­cause it was a sub­ject that in­ter­ested all the people!

Kotschick: That was when you met Smale, and he was also do­ing na­ked man­i­folds. But af­ter­wards he star­ted adding on dy­nam­ic­al sys­tems. You did em­bed­dings and im­mer­sion the­ory for a while, but then you re­turned to fo­li­ations.

Hae­fli­ger: I re­turned to fo­li­ations in 1970, fol­low­ing dis­cus­sions with Tony Phil­lips.

Kotschick: Who had also been act­ive in im­mer­sion the­ory!

Hae­fli­ger: But the main in­flu­ence was the the­or­em of Bott about the an­ni­hil­a­tion of char­ac­ter­ist­ic classes — the Bott van­ish­ing the­or­em.

Kotschick: When did your in­tro­duc­tion of the clas­si­fy­ing space hap­pen?

Hae­fli­ger: It was in 1969, when I gave a talk in Mont-Aig­ou­al, which is a con­fer­ence cen­ter close to Mont­pel­li­er. I re­mem­ber Gil­bert Hec­tor — he was a stu­dent of Reeb — said, “It’s fo­li­ation the­ory, but you don’t see the leaves! Where are the leaves?” I had in­tro­duced the no­tion of \( \Gamma \)-struc­ture already in my thes­is. But I had not yet had the idea of clas­si­fy­ing space for that.

Kotschick: With the clas­si­fy­ing space, you in­tro­duced a lot more al­geb­ra­ic to­po­logy to the study of fo­li­ations. Be­fore that, it was more geo­met­ric.

Hae­fli­ger: Yes, be­cause, how do you com­pute this huge clas­si­fy­ing space? When I met Gel­fand for the first time, he was col­lab­or­at­ing with Fuks, who is a gi­ant. And Gel­fand said to me, “But you are so small!” — he had the idea of such a big clas­si­fy­ing space!

Kotschick: It seems the Bott van­ish­ing the­or­em and the dis­cov­ery of God­bil­lon and Vey were two in­de­pend­ent events. But they im­me­di­ately fed in­to the de­vel­op­ment of char­ac­ter­ist­ic classes for fo­li­ations. Then there was a burst of activ­ity.

Hae­fli­ger: God­bil­lon once lis­ted all pa­pers pub­lished on fo­li­ations in the 1960s and 1970s — it’s in­cred­ible.11

Kotschick: Dur­ing this burst of activ­ity, there was an an­nounce­ment by you and [Raoul] Bott [5], but it seems the full pa­per was nev­er pub­lished. What happened there?

Hae­fli­ger: Bott was work­ing with many oth­er people, like Graeme Segal. And I am lazy to write up things! There was also a mis­take in the an­nounce­ment, which maybe was re­moved just be­fore pub­lic­a­tion.

We spent days and days talk­ing with Bott in his house, with his fam­ily, with his daugh­ters. He had very good con­tact with Min­ouche, be­cause he was Hun­gari­an and Min­ouche’s fath­er had lived in Bud­apest. Bott worked a lot with [Mi­chael] Atiyah, who was very quick to write pa­pers. Many ideas came from Bott, but Bott by him­self was not so quick to write things.

Kotschick: How did you first get in­ter­ested in group the­ory?

Hae­fli­ger: It was fash­ion­able! There was this book of Serre, Arbres, Am­al­games, et \( \mathrm{SL}_2 \) [e8]. Serre looks at an ori­ented graph, but one can also look at just a graph. Then the lan­guage some­how is sim­pler. This is the ba­sic idea of the book Brid­son and I wrote on CAT(0) [6].

Kotschick: How did the col­lab­or­a­tion with Brid­son hap­pen?

Hae­fli­ger: Brid­son was a PhD stu­dent of Kar­en Vo­gt­mann, at Cor­nell. I got a grant from the Swiss Na­tion­al Fund to bring Mar­tin to Geneva for a couple of years. We spent the sum­mer in the moun­tains with him. We ren­ted a chalet.

Min­ouche: One sum­mer, and they thought they would fin­ish the book in the au­tumn. It las­ted for years!

Hae­fli­ger: Ten years! From 1990 un­til 1999. The book is quite long. The struc­ture of the book be­came clear only five years after we began, be­cause there was a lot of com­pet­i­tion. There were sev­er­al books pub­lished on CAT(0) space, for ex­ample the one by Ball­mann, Gro­mov, and Schroeder [e13].

So we de­cided to write a book for be­gin­ners, so that we ex­plain everything and give every defin­i­tion, to get a dif­fer­ent fla­vor from those books that were more ad­vanced.

Jack­son: Your book with Brid­son has be­come a stand­ard ref­er­ence for this area.

Hae­fli­ger: It’s com­pletely out of fash­ion now, but it is good for be­gin­ners.

Kotschick: Many of the people in that area, like some of the ones you men­tioned, Ball­mann, Schroeder, and so on — came from Rieman­ni­an geo­metry and dif­fer­en­tial geo­metry. Your in­terests and back­ground had been com­pletely dif­fer­ent.

Hae­fli­ger: No — I wrote many pa­pers on pseudogroups of iso­met­ries. But the sub­ject of Rieman­ni­an fo­li­ations is huge now.

A pair of pants, a pair of brilliant students

In 1976, Paul Sch­weitzer or­gan­ized a con­fer­ence at his home in­sti­tu­tion, the Pon­ti­fic­al Uni­versity in Rio de Janeiro. The con­fer­ence at­trac­ted many of the lead­ing lights in geo­metry and to­po­logy, in­clud­ing Hae­fli­ger and Raoul Bott. In auto­bi­o­graph­ic­al re­marks in his Col­lec­ted Pa­pers, Bott writes that many par­ti­cipants at this con­fer­ence wound up as rob­bery vic­tims. Dur­ing a beach out­ing, a rob­ber stole Hae­fli­ger’s pants, which he had bought at a store in Rio just a short while be­fore go­ing to the beach.

Hae­fli­ger: But the pants were so cheap! They cost something like 10 Swiss francs.

At this meet­ing I spoke about the work of my stu­dent [Au­gustin] Ban­yaga. He came from Rwanda, which was a Bel­gian colony. His fath­er had a ba­nana plant­a­tion and nev­er learned to read. It was a fam­ily of many chil­dren. A teach­er no­ticed that Ban­yaga was very tal­en­ted, so he could go to school. He got a schol­ar­ship to come to Switzer­land [in 1967] to study min­ing en­gin­eer­ing, be­cause there were a lot of mines in Rwanda. He knew French very well. He had to take a course in math­em­at­ics, and then for him it was clear that he would like to do math­em­at­ics.

Jack­son: Was he in a math course you were teach­ing?

Hae­fli­ger: Yes. He at­ten­ded many lec­tures of mine, and I talked a lot with him. He used to go to the Troisième Cycle in Lausanne.12 I re­mem­ber ex­plain­ing many things to him about Thur­ston’s work. He wrote a diplôme [an ori­gin­al work writ­ten at the end of un­der­gradu­ate stud­ies], a very in­ter­est­ing work, in­spired by a pa­per of Jürgen Moser. I sent it to Lich­ner­ow­icz for the Comptes Ren­dus, and it was pub­lished there [e6]. Af­ter­ward Ban­yaga wrote a PhD thes­is [e11] in­spired by the work of Thur­ston.

Jack­son: When Ban­yaga came to Geneva, was it very dif­fi­cult for him to ad­just?

Hae­fli­ger: No, be­cause the uni­versity had a pro­gram for stu­dents from for­eign coun­tries. They would get lodging and a French course if they needed it — but he already knew French very well. They also or­gan­ized some sight­see­ing. Ban­yaga was like a ray­on de soleil! He came to vis­it us many, many times.

Jack­son: When he first came to Geneva, was he mar­ried?

Hae­fli­ger: No, but he was en­gaged. When the war began in Rwanda between Tut­si and Hutu, he was quickly able to get his fiancée to come to Geneva.

Min­ouche: They mar­ried and had their first child here. At one point he was asked to be a min­is­ter in the gov­ern­ment in Rwanda, but he re­fused.

Hae­fli­ger: He did not want that, be­cause if you be­come a min­is­ter, you have to do polit­ics, and you can’t do math­em­at­ics any­more. He star­ted a PhD pro­gram in Rwanda, but it died after a while.

Min­ouche: At some point he was afraid to go back to Rwanda and not to be able to get out.

Hae­fli­ger: Sev­er­al years ago, there was a con­fer­ence at the In­sti­tut Henri Poin­caré, for Marc Chap­er­on. Ban­yaga gave a talk at the meet­ing, and he asked for a minute of si­lence, be­cause this was at the time of an earth­quake and tsunami in Haiti.

Jack­son: We are talk­ing about one of your out­stand­ing stu­dents, Ban­yaga. Of course you had an­oth­er out­stand­ing stu­dent, Vaughan Jones, who also came from very far away, from New Zea­l­and. Can you tell the story of how Vaughan Jones ended up in Geneva?

Hae­fli­ger: He wanted to work with a phys­i­cist, and it happened that the phys­i­cist died when Vaughan ar­rived. Vaughan is also like a ray­on de soleil, and people loved him.

Min­ouche: As this pro­fess­or had died, Vaughan went one day and asked An­dré if he can give him a job.

Hae­fli­ger: And I said, “You can be­gin today!” He had already fol­lowed my lec­tures.

Jack­son: Was his tal­ent ap­par­ent when he came to you?

Hae­fli­ger: Of course! I think I feel im­me­di­ately who has tal­ent.

Jack­son: His work was very far from what you were do­ing.

Hae­fli­ger: Yes. Pierre de la Harpe was really the only one in Geneva who was close to his sub­ject. In the French parts of Switzer­land, many people knew that sub­ject, so Vaughan was in­vited to give talks. I was form­ally his thes­is ad­visor, and Alain Connes was on his thes­is com­mit­tee. It was clear that, for a geni­us like Vaughan, one had to ask for an ex­pert of the same level.

He got his PhD in 1979, so he was here for 4 or 5 years. When he got the Fields Medal at the ICM in Kyoto in 1990, he wanted very much that I come to Kyoto. But I thought they would present me as his thes­is dir­ect­or, and I had noth­ing to do with it!

Min­ouche: Vaughan al­ways claimed he learned a lot from An­dré, but An­dré al­ways says he didn’t give him any­thing.

Hae­fli­ger: He knew a lot of math­em­at­ics, more than me!

Spreading the word about Gromov’s work

In 1969, Mikhail Gro­mov upen­ded the world of dif­fer­en­tial to­po­logy with his doc­tor­al thes­is, writ­ten un­der V.A. Rokh­lin at Len­in­grad Uni­versity. His work con­tained far-reach­ing ap­prox­im­a­tion the­or­ems gen­er­al­iz­ing the known em­bed­ding and im­mer­sion the­or­ems that had been de­veloped throughout the 1950s and 1960s.

The first West­ern­er to hear about this was prob­ably An­thony Phil­lips: “I was in the USSR in Spring 1969, on the inter-academy ex­change pro­gram. My ment­or Sergei Novikov sent me to Len­in­grad to meet Gro­mov, who had just com­pleted his thes­is. Misha gave me a copy to take back to the hotel and I im­me­di­ately real­ized what he had ac­com­plished (since it ex­ten­ded my own work on sub­mer­sions, I knew ex­actly where to look).”13

Phil­lips pro­ceeded to War­wick, where he par­ti­cip­ated in a sum­mer-long sym­posi­um and spoke about Gro­mov’s work. In Oc­to­ber 1969 Hae­fli­ger wrote to Phil­lips: “I hear that you spoke at War­wick about a very gen­er­al in­ven­tion of Gro­mov that ax­io­mat­izes the­or­ems of this type. Can you give me any leads on this ques­tion?” In an­oth­er let­ter two months later, Hae­fli­ger thanks Phil­lips for sup­ply­ing a copy of Gro­mov’s thes­is. Hav­ing also worked in im­mer­sion the­ory for the pre­vi­ous ten years, Hae­fli­ger im­me­di­ately grasped the im­port of Gro­mov’s work and in 1970 lec­tured about it at the École Poly­tech­nique in Par­is14 and at the Uni­versity of Par­is Or­say.15

An­oth­er six­teen years would pass be­fore Gro­mov’s book Par­tial Dif­fer­en­tial Re­la­tions ap­peared in the Ergeb­n­isse series of Spring­er Ver­lag. The the­ory laid out in this book, and in works of Yakov Eli­ash­berg and oth­ers, pop­ular­ized the con­cepts em­an­at­ing from Gro­mov’s thes­is, now known as the “h-prin­ciple”.


[1] A. Hae­fli­ger and A. Kosiński: “Un théorème de Thom sur les sin­gu­lar­ités des ap­plic­a­tions différen­ti­ables” [A the­or­em of Thom on the sin­gu­lar­it­ies of dif­fer­en­ti­able ap­plic­a­tions] in Quelques ques­tions de to­po­lo­gie [Some ques­tions of to­po­logy]. Secrétari­at Mathématique (Par­is), 1958. Ex­posé no. 8 from Henri Cartan Sem­in­ar, 9th year (1956/57). MR 124063 Zbl 0178.​26604 incollection

[2] A. Borel and A. Hae­fli­ger: “La classe d’ho­mo­lo­gie fon­da­mentale d’un es­pace ana­lytique” [The fun­da­ment­al ho­mo­logy class of an ana­lyt­ic space], Bull. Soc. Math. France 89 (1961), pp. 461–​513. MR 149503 Zbl 0102.​38502 article

[3]A. Hae­fli­ger: Lec­tures on the work of Gro­mov, at the École Poly­tech­nique and Paris Or­say, 1970. Lec­ture notes by Alain Chen­cin­er. misc

[4] A. Hae­fli­ger: “Lec­tures on the the­or­em of Gro­mov,” pp. 128–​141 in Pro­ceed­ings of Liv­er­pool Sin­gu­lar­it­ies Sym­posi­um II (Liv­er­pool, Septem­ber 1969–Au­gust 1970). Edi­ted by C. T. C. Wall. Lec­ture Notes in Math­em­at­ics 209. Spring­er (Ber­lin), 1971. MR 334241 Zbl 0222.​57020 incollection

[5] R. Bott and A. Hae­fli­ger: “On char­ac­ter­ist­ic classes of \( \Gamma \)-fo­li­ations,” Bull. Am. Math. Soc. 78 : 6 (November 1972), pp. 1039–​1044. MR 307250 Zbl 0262.​57010 article

[6] M. R. Brid­son and A. Hae­fli­ger: Met­ric spaces of non-pos­it­ive curvature. Grundlehren der Math­em­at­ischen Wis­senschaften 319. Spring­er (Ber­lin), 1999. MR 1744486 Zbl 0988.​53001 book

[7] A. Hae­fli­ger: “Nais­sance des Feuil­letages, d’Ehresmann–Reeb à Novikov,” pp. 335–​354 in Géométrie au XXe siècle, 1930–2000: His­toire et ho­ri­zons. Edi­ted by J. Koun­ei­h­er, D. Fla­ment, P. Na­bon­nand, and J. J. Szcze­cini­arz. Her­mann (Par­is), 2014. incollection