Celebratio Mathematica

Augustin Banyaga

Interview with Augustin Banyaga

by Allyn Jackson

Augustin Banyaga.
Photo: Judith Mukaruziga.

Born in 1947, Au­gustin Ban­yaga is the first Rwandan to earn a PhD in math­em­at­ics. He had a happy child­hood grow­ing up on his fam­ily’s ba­nana plant­a­tion not far from the Rwandan cap­it­al, Kigali. In sec­ond­ary school, he stood out as the top stu­dent. He ob­tained a schol­ar­ship to study min­ing en­gin­eer­ing at the Uni­versity of Geneva but soon switched to math­em­at­ics — and there too be­came the top stu­dent. Un­der the guid­ance of An­dré Hae­fli­ger, Ban­yaga earned three de­grees from Geneva: Li­cen­ce ès Sci­en­ces Ma­thé­ma­tiques (1971), Di­plôme de Ma­thé­ma­ti­cien (1972), and Doc­teur ès Sci­en­ces Ma­thé­ma­tiques (1976). With Hae­fli­ger as a mag­net for re­search in geo­metry and to­po­logy, Ban­yaga found in Geneva the op­por­tun­ity to meet many of the out­stand­ing math­em­aticians of the day.

After it be­came clear he could not ob­tain a po­s­i­tion in Rwanda, Ban­yaga spent a year (1977–1978) as a postdoc at the In­sti­tute for Ad­vanced Study in Prin­ceton and then four years at Har­vard Uni­versity as a Ben­jamin Peirce In­struct­or. He taught at Bo­ston Uni­versity un­til 1984, when he joined the fac­ulty of Pennsylvania State Uni­versity. Today he is a pro­fess­or of math­em­at­ics and dis­tin­guished seni­or schol­ar at Penn State. He be­came a United States cit­izen in 1995.

Ban­yaga is a lead­ing fig­ure in geo­metry and to­po­logy, hav­ing made im­port­ant con­tri­bu­tions in sym­plect­ic and con­tact geo­metry/to­po­logy and in Morse the­ory. He has also de­voted a great deal of en­ergy to de­vel­op­ing math­em­at­ic­al tal­ent in Africa, where his stature in re­search, to­geth­er with his joy­ful en­thu­si­asm for math­em­at­ics, have had a ma­jor in­flu­ence. He is on the sci­entif­ic com­mit­tee for the In­sti­tut de Mathématiques et Sci­ences Physiques in Ben­in and has served on the se­lec­tion com­mit­tees for the Afric­an Math­em­at­ics Mil­len­ni­um Sci­ence Ini­ti­at­ive and the Si­mons Found­a­tion’s Africa Math­em­at­ics Pro­ject. He serves on the ed­it­or­i­al boards of Afrika Matem­atika and the Afric­an Di­a­spora Journ­al of Math­em­at­ics. He was elec­ted a Fel­low of the Afric­an Academy of Sci­ences in 2009.

What fol­lows is an ed­ited ver­sion of an in­ter­view with Au­gustin Ban­yaga, con­duc­ted in Decem­ber 2019.

Chasing pigs on the plantation

Jack­son: Can you tell me about your par­ents?

Ban­yaga: They were born in Rwanda, in a vil­lage called Gas­abo, near the cap­it­al Kigali. The vil­lage in which I was born is also near Kigali — it’s called Gisozi.

Jack­son: Did they also come from farm­ing fam­il­ies?

Ban­yaga: Every­body was a farm­er. In Rwanda at that time, there were very few jobs. My dad learned ma­sonry. Of course he didn’t go to school. There was al­most no school at the time he was grow­ing up.

Jack­son: And your moth­er also did not go to school?

Ban­yaga: No.

Jack­son: Did they learn to read?

Ban­yaga: No.

Jack­son: You are Hutu, is that right?

Ban­yaga: Yes — how do you know?

Jack­son: The Hae­fli­gers told me about you and your wife and said that one of you is Hutu and the oth­er is Tut­si. You come from a farm­ing back­ground, and I read that the Hu­tus were farm­ers and the Tut­si were cattle ranch­ers. So I guessed that you are Hutu and your wife is Tut­si.

Ban­yaga: Yes, but my wife’s par­ents had no cattle at all! Her fath­er was a car­penter. We are both from the same vil­lage. I knew her when she was very small — she is 6 years young­er than me.

Jack­son: Did your par­ents already have the ba­nana plant­a­tion when you were born?

Ban­yaga: Yes, they were already on the plant­a­tion, in the vil­lage Gisozi. Each kid in our fam­ily was sup­posed to have a piece of the prop­erty and make his or her own plant­a­tion. So I had a few ba­nana trees I planted my­self.

Jack­son: You had sib­lings?

Ban­yaga: Yes. We were 5 chil­dren. My two broth­ers are still in Rwanda, and my two sis­ters are with me in the US. They came as refugees.

Jack­son: When did they come to the US?

Ban­yaga: About 20 years ago. After the civil war, they were refugees in Congo, and we man­aged to get them to the US.

Jack­son: What was life like on the ba­nana plant­a­tion?

Ban­yaga: It was really great. I loved it. In the morn­ing I walked to school, and this was quite long, maybe a few miles. When I came back, I would go to my plant­a­tion. We kids used to play to­geth­er. I had a couple of pigs, and we would run and chase them. It was fun.

Jack­son: So you had a lot of free­dom to play and be out­doors.

Ban­yaga: Ab­so­lutely.

Jack­son: Did your fam­ily have meals to­geth­er every day?

Ban­yaga: Yes. There was no short­age of any kind. Mainly we ate sweet pota­toes, beans, cas­sava, and a bit of goat meat.

Jack­son: Were your par­ents very strict?

Ban­yaga: No. Ac­tu­ally, all the kids were so nice, the par­ents did not have too much to do. We were very cor­rect. But if some­body did something stu­pid, the par­ents would pun­ish them a little bit.

Jack­son: Where were the ba­na­nas sold?

Ban­yaga: The ba­na­nas we grew were sold in the loc­al mar­ket. But we also used them to eat them and to make ba­nana wine.

Jack­son: Does the wine taste like ba­na­nas?

Ban­yaga: No. It tastes al­most like white wine. When I was a stu­dent in Geneva, I used to tell my friends that they were drink­ing Swiss wine, but I had giv­en them ba­nana wine! It’s not very dif­fer­ent from Swiss wine.

Jack­son: Did you make the wine your­self?

Ban­yaga: No, but I knew how to ex­tract the juice. You first let the ba­na­nas fer­ment. They be­come yel­low and soft. Then you re­move the skin and ma­cer­ate them us­ing some herbs. Little by little, the juice comes out. It’s not easy.

Jack­son: What was your primary school like?

Ban­yaga: It was very nice. Half of the school was giv­en in French, so we had to learn a lot of French in the be­gin­ning. We used books from Bel­gi­um, be­cause we Rwandans didn’t have any books at all — or, we didn’t pro­duce our books. The math books would have prob­lems where a train would leave Brus­sels at this time and ar­rive in Lux­em­bourg at that time; how many hours did you spend on the train? This was the kind of prob­lem we were do­ing.

Jack­son: I read that there were no train lines in Rwanda.

Ban­yaga: No, there wer­en’t! They just trans­lated the European situ­ation to Rwanda. Some kids got lost be­cause of that. But if you have some ima­gin­a­tion, you can think about it and ab­stract it.

Jack­son: Where were the teach­ers from?

Ban­yaga: They were Rwandans who had stud­ied in the school for teach­ers. Some­times we had white priests as teach­ers, but that was mostly in the sec­ond­ary school, not in the primary school.

Jack­son: You had to learn French and math­em­at­ics. What oth­er sub­jects did you study?

Ban­yaga: Re­li­gion — re­li­gion was very im­port­ant. Also his­tory and geo­graphy.

Jack­son: The re­li­gion that you stud­ied was Cath­ol­ic re­li­gion?

Ban­yaga: Yes.

Jack­son: Were you raised as a Cath­ol­ic?

Ban­yaga: Yes. My mom used to go to the mass very of­ten.

Jack­son: And your fath­er?

Ban­yaga: Less! But he went too.

“The people who poison others”

Jack­son: What lan­guage did you speak at home — the Rwandan lan­guage?

Ban­yaga: Yes. Rwanda is one of the very rare coun­tries in Africa that has a unique lan­guage for the whole coun­try — just Rwanda, Bur­undi, and a few oth­er coun­tries. Kin­yar­wandan is a very pleas­ant lan­guage. When I was a kid, I liked to read and to write. Ac­tu­ally, I pub­lished a book in Rwandan lan­guage when I was 18. It was a short nov­el.

Jack­son: What was it about?

Ban­yaga: The title was Murin­gur­iza, which means “the people who pois­on oth­er people.” There were le­gends around in the coun­try that, each time some­body dies, it’s not by a nat­ur­al cause, but be­cause some­body poisoned the per­son. Even some­body who doesn’t know you, who doesn’t even hate you, might pois­on you, ac­cord­ing to the le­gend. The nov­el is about people who pois­on oth­er people for no reas­on at all.

Jack­son: This sounds scary and grim.

Ban­yaga: Yes. But the goal of my book was to tell people how this is stu­pid. For in­stance, when my moth­er died, people said she had been poisoned by our neigh­bor. I didn’t be­lieve it. She went in­to the hos­pit­al, so I asked the doc­tor about her, and the doc­tor told me she had had can­cer. So I didn’t be­lieve the le­gend. But every­body else be­lieved it.

People be­lieved that the people who pois­on did it ran­domly to any­body, with no reas­on. This is why I thought it was com­pletely ab­surd.

Jack­son: Did your sib­lings be­lieve she had been poisoned?

Ban­yaga: Yes.

Jack­son: And your fath­er also?

Ban­yaga: Yes. Every­body be­lieved it.

Jack­son: How old were you when she died?

Ban­yaga: Maybe about 15 years old.

Jack­son: That’s very sad. But at 15, you un­der­stood this was a su­per­sti­tion, even though every­one around you be­lieved she’d been poisoned. You were dif­fer­ent. What did your fam­ily think of your nov­el?

Ban­yaga: They liked it, but if you be­lieve the le­gend com­pletely, you are not go­ing to be­lieve what I am say­ing in the nov­el.

Jack­son: So they liked the nov­el, but it didn’t change their minds.

Ban­yaga: No, it didn’t.

Jack­son: How did you man­age to get a nov­el pub­lished when you were only 18 years old?

Ban­yaga: There were peri­od­ic­al journ­als that were eager to pub­lish in­ter­est­ing things. They pub­lished one chapter each week.

Jack­son: So you just sent a chapter to one of the journ­als, and they agreed to pub­lish it?

Ban­yaga: Yes.

Jack­son: Your fam­ily be­lieved the su­per­sti­tion when your moth­er died. But you listened to the doc­tor. Why were you dif­fer­ent?

Ban­yaga: I can’t tell you. I don’t know! I was dif­fer­ent. And I am still dif­fer­ent!

Jack­son: You’re skep­tic­al, be­cause you are a math­em­atician! Do you still have a copy of this nov­el?

Ban­yaga: Un­for­tu­nately everything dis­ap­peared dur­ing the civil war in Rwanda. I had it in my house there, and my house was van­dal­ized. I had also brought some nice math books there. Everything was lost.

The seminary as educational opportunity

Jack­son: Be­ing a skep­tic­al-minded kid, what did you think of the church and Cath­oli­cism when you were grow­ing up?

Ban­yaga: Ac­tu­ally, after primary school, I went in­to a sem­in­ary to be­come a priest. Of course, I didn’t be­come a priest! But this is how I went to a sem­in­ary and how I stud­ied Lat­in and Greek.

Jack­son: Did you de­cide to go to the sem­in­ary be­cause you wanted to get an edu­ca­tion?

Ban­yaga: Ex­actly — this was the one op­por­tun­ity I had. I am Hutu, and there were not many schools that were open to Hu­tus at that time, maybe just the sem­in­ary. The good, big, nice school was only for Tut­sis. I stud­ied at the sem­in­ary for three years. Then I went to an­oth­er school that was not a sem­in­ary, it was an or­din­ary high school in Kigali. There the em­phas­is was sci­ence and Lat­in.

Jack­son: Did your sib­lings also go on to sec­ond­ary school?

Ban­yaga: No. It was very hard to enter the sec­ond­ary school at that time.

Jack­son: When you stud­ied at the sem­in­ary, did you live there also?

Ban­yaga: Yes.

Jack­son: Was that hard for you, to be away from the plant­a­tion?

Ban­yaga: No, it was a good change. Ac­tu­ally the school was also in a rur­al area, so you didn’t see too much dif­fer­ence when you looked out­side.

Jack­son: How did it come about that you switched to the sec­ond­ary school in Kigali?

Ban­yaga: At some point I de­cided I did not want to be a priest any­more. I had seen that there were prob­lems between white priests and black Rwandan priests. There were con­flicts, and they were shout­ing at each oth­er. I didn’t find this very Chris­ti­an, to fight among them­selves. I told our boss, the bish­op, that the priests don’t live a good im­age of Chris­tian­ity, be­cause they hate each oth­er. I didn’t want to be a part of this sys­tem. It was ri­dicu­lous. So I left.

Jack­son: What was the con­flict about?

Ban­yaga: Ba­sic­ally it was about ra­cism. The priests from Bel­gi­um didn’t think that the Rwandans were good enough to be priests. And the Rwandans thought that the priests were sent by the col­on­izers to help col­on­iz­a­tion.

Jack­son: The Rwandans felt that the priests were not there only for re­li­gious reas­ons, but to help the col­on­izers.

Ban­yaga: Ex­actly.

Jack­son: When you told the bish­op about the con­flict between black and white priests, what did he say?

Ban­yaga: He was un­happy. He was a very good guy and helped me to find the oth­er school. Ac­tu­ally, he was Swiss.

Jack­son: But the bish­op could not help to re­solve the con­flict between the black and white priests.

Ban­yaga: No, it was too big.

Jack­son: Did you have any es­pe­cially good teach­ers, at the sem­in­ary or later at the school in Kigali?

Ban­yaga: I don’t re­mem­ber them, but I think they were okay.

Jack­son: And the math­em­at­ics in­struc­tion in these schools?

Ban­yaga: That was okay too, but I have no spe­cial re­col­lec­tions.

Jack­son: Are there ex­per­i­ences that you had as a young per­son that you now with hind­sight see were in­spir­a­tion­al in a math­em­at­ic­al sense?

Ban­yaga: I really loved sym­met­ries. This is why I col­lec­ted stones, quartz, flowers, etc. I think this was very im­port­ant and stayed in my mind for a long time. Ac­tu­ally, this is geo­metry. Geo­metry is nature, the sym­metry in nature.

Going to Switzerland by mistake

Jack­son: At your school in Kigali, you gradu­ated num­ber one in your class, in 1966. Hav­ing been first in your class en­abled you to get a schol­ar­ship to go to Switzer­land, is that right?

Ban­yaga: I didn’t get the schol­ar­ship im­me­di­ately, be­cause again there was this Hutu/Tut­si stuff be­hind the scenes. If you were Hutu, you had few­er op­por­tun­it­ies. There was an­oth­er prob­lem too, the prob­lem of re­gion­al­ism — if you are not from a cer­tain re­gion, you had few­er op­por­tun­it­ies.

Jack­son: You are a Hutu, so that re­duced your chances, but you were from the “wrong” re­gion also?

Ban­yaga: Yes! Some­how, I have been a vic­tim all my life, be­cause of in­equal­it­ies about what you are, where you are from, who you are. And now I am a black in Amer­ica! I man­aged al­ways to be in the “wrong” group! But it’s okay. Ac­tu­ally, I have been lucky.

You know, I went to Switzer­land by mis­take.

Jack­son: Really?

Ban­yaga: I was offered a schol­ar­ship to study min­ing in Bel­gi­um. The Swiss con­su­late offered me a schol­ar­ship, with more money, to study min­ing in Switzer­land. They told me that Switzer­land is a beau­ti­ful coun­try, which is true, and that they have an ex­cel­lent school of min­ing in Geneva, which is wrong! But I be­lieved it. The gen­er­al be­lief was that the Bel­gians had bet­ter schools for min­ing. So I went to Switzer­land and star­ted in the min­ing school.

I stud­ied math­em­at­ics, phys­ics, and geo­logy. The good thing is that my lin­ear al­gebra course was with An­dré Hae­fli­ger, and I star­ted con­nect­ing with math more than with geo­logy. So the mis­take of go­ing to Switzer­land be­came a very nice thing!

Jack­son: When you took for ex­ample Hae­fli­ger’s lin­ear al­gebra class, did you find the back­ground you had from school in Rwanda was suf­fi­cient?

Ban­yaga: Yes. Ac­tu­ally, I pro­gressed very quickly and even be­came the best in the class in Geneva.

Jack­son: You de­cided not to con­tin­ue in en­gin­eer­ing stud­ies, but could you still keep the schol­ar­ship?

Ban­yaga: They gave me one year to de­cide. If I de­cided to quit min­ing en­gin­eer­ing, then they would cut off the schol­ar­ship. But at the same time, the math de­part­ment saw that I was pro­gress­ing well, and it soon offered me an as­sist­ant­ship.

Jack­son: Was it a big cul­ture shock for you to go to Switzer­land?

Ban­yaga: Not really.

Jack­son: Was it the first time you had been out of Rwanda?

Ban­yaga: Yes. But people were so nice. I nev­er felt lost in Geneva. When I first ar­rived in Switzer­land, I didn’t go dir­ectly to Geneva; I went to Fri­bourg. A lady there came to greet and to help the group of schol­ar­ship stu­dents. I still re­mem­ber her. She is very old now. The last time we vis­ited Switzer­land, we met her again. She is won­der­ful.

Jack­son: A per­son like that makes it easi­er to ad­just to the new coun­try, new hemi­sphere — new everything! Did the Swiss weath­er and the cold both­er you much?

Ban­yaga: No. Some­times I went to places in the moun­tains where people skied. The cold didn’t both­er me.

Jack­son: You have called your lin­ear al­gebra course with Hae­fli­ger a turn­ing point in your life. Can you tell me about that?

Ban­yaga: I learned later that Hae­fli­ger was ac­tu­ally one of the greatest math­em­aticians in the world. But of course I couldn’t have known that! I just saw a guy who was smil­ing, who was caring and nice, and who spoke ex­tremely simply. He didn’t go too fast. He was de­voted to mak­ing sure that we stu­dents un­der­stood the small de­tails. Af­ter­ward, I took oth­er classes from oth­er people. I was su­per lucky to have a class with Georges de Rham. I tell my stu­dents now that I had de Rham as my teach­er, and they say, “Wow!”

There is a joke de Rham told us. He said that what is im­port­ant for math­em­aticians is the defin­i­tions. “I define old as any per­son older than me,” he said. “The­or­em: I will nev­er be old! Be­cause I will nev­er be older than my­self!” This was a class in ana­lys­is — a long time ago.

Jack­son: He was Hae­fli­ger’s teach­er also, in Lausanne.

Ban­yaga: Yes. Geneva had a lot of nice people. An­oth­er was Michel Ker­vaire. Ac­tu­ally, all the im­port­ant math­em­aticians I en­countered later, I had already met in Geneva. They used to come to vis­it Hae­fli­ger — [Wil­li­am] Thur­ston, [John] Math­er, [Raoul] Bott, even Jim Si­mons, who is now very rich and star­ted a found­a­tion. All the people I got to know later in Prin­ceton, I had met in Geneva, ex­cept prob­ably [John] Mil­nor. And Dusa Mc­Duff I met in Cov­entry, Eng­land.

Jack­son: Around this time you mar­ried your wife Ju­dith. Did you get mar­ried in Geneva?

Ban­yaga: No, in Rwanda. On a trip from Geneva back to Rwanda in the early 1970s, I met her again, and we were then both adults. I went to Rwanda again in 1973, and at that time, the coun­try was go­ing through some polit­ic­al troubles. Tut­sis were be­ing beaten and some­times killed. I wanted to res­cue Ju­dith, to save her. So I went there for about two weeks and we got mar­ried, and then I took her back to Geneva with me.

Jack­son: There was a mil­it­ary coup in Rwanda in 1973.

Ban­yaga: Yes, that was a few months after I left Rwanda with Ju­dith.

Jack­son: Your tim­ing was good! What was the situ­ation of her fam­ily, and what happened to them in the coup?

Ban­yaga: They were okay. Her fath­er was work­ing as a car­penter in the mil­it­ary. The moth­er, like most the moth­ers, stayed home and took care of the chil­dren.

Jack­son: Was it hard for Ju­dith to go to Geneva?

Ban­yaga: I hope not! I had already an apart­ment, I had my life there.

Jack­son: But to leave so quickly like that is hard. It wasn’t safe for her to stay in Rwanda?

Ban­yaga: It was com­pletely un­safe to stay in Rwanda. As I said, a few months later, there was a coup by the mil­it­ary guys. The coun­try was in a bad situ­ation.

No one understood it: an unpublished paper by Thurston

Jack­son: So this was 1973, and you had gone to Geneva in 1967, is that right?

Ban­yaga: Yes. I fin­ished my bach­el­or’s de­gree in 1971, my mas­ter’s in 1972, and my PhD in 1976. I was pretty fast.

Jack­son: Can you tell me about your PhD thes­is work? Were you very much in­flu­enced by the work of Thur­ston?

Ban­yaga: Ac­tu­ally, his work was the mo­tiv­a­tion. Thur­ston had writ­ten a pa­per,1 and nobody would pub­lish it, be­cause nobody could un­der­stand it! Hae­fli­ger, after he vis­ited Prin­ceton, came to me and said, “Au­gustin, there is this pa­per that nobody has un­der­stood. Try to un­der­stand it, and try to do something with it.” Ac­tu­ally, I un­der­stood it, which is un­be­liev­able. And I im­proved on it. I cre­ated a the­ory in which the res­ult of Thur­ston would be a small part. This is re­lated to the work of Math­er too.

Jack­son: How is it that you man­aged to un­der­stand Thur­ston’s pa­per that nobody else un­der­stood? Did you talk to Thur­ston about it?

Ban­yaga: No. I just stayed in my room for hours and hours and hours, for weeks and weeks. And fi­nally some light came.

Jack­son: Did you sud­denly have a flash of in­sight?

Ban­yaga: Yes. I think this is the way things hap­pen. You work a long time without any­thing, and very sud­denly, something comes. Ac­tu­ally, that work opened all the doors for me, so that I was able to get po­s­i­tions at the IAS in Prin­ceton and at Har­vard.

Jack­son: Was Thur­ston’s pa­per even­tu­ally pub­lished?

Ban­yaga: No. I don’t even have a copy any­more. My copy was prob­ably des­troyed in Rwanda. I don’t have the pa­per, but I know what’s in­side!

Jack­son: What was the sub­ject of Thur­ston’s pa­per?

Ban­yaga: The sub­ject was to study trans­form­a­tions that pre­serve a cer­tain struc­ture. The struc­ture that in­ter­ested me was more re­fined, more dif­fi­cult than the struc­ture of Thur­ston. This was re­lated to Hae­fli­ger’s the­ory of fo­li­ations. I have ac­tu­ally spent most of my life study­ing these groups of trans­form­a­tions, and the book I wrote ex­plained the main res­ults.2

A few years after my thes­is, I was able to prove a res­ult on Klein’s Er­lan­gen Pro­gram.3 Klein wanted to show that any geo­metry is com­pletely char­ac­ter­ized by its trans­form­a­tion groups. Now most of my math­em­at­ic­al work fo­cuses on prov­ing that sym­plect­ic geo­metry is char­ac­ter­ized by its group of auto­morph­isms. This is the biggest part of what I have done.

Jack­son: Hae­fli­ger said that you also wrote a very good mas­ter’s thes­is.4

Ban­yaga: This gen­er­al­ized ideas of an­oth­er Swiss math­em­atician, Jürgen Moser, from ETH Zurich.5 It doesn’t hap­pen very of­ten that a mas­ter’s thes­is is pub­lished.

Jack­son: Can you tell me about this work?

Ban­yaga: If you have two volume ele­ments on a man­i­fold, there are dif­feo­morph­isms that loc­ally trans­form one in­to the oth­er. Is there a glob­al dif­feo­morph­ism that does this? Moser proved that, for a man­i­fold without bound­ary, the an­swer is yes. What I did was to prove the same thing for a man­i­fold with bound­ary. This is a com­pletely dif­fer­ent situ­ation, be­cause you have to study what goes on near the bound­ary.

Jack­son: Where was your PhD thes­is pub­lished?

Ban­yaga: In Com­ment­arii Matem­atici Hel­vetici.6 This is a very im­port­ant Swiss journ­al.

Jack­son: While you were in Switzer­land, did you travel around much in Europe?

Ban­yaga: Yes. I went to Mont­pel­li­er and Par­is, and many times to Mar­seille. Michèle Aud­in in­vited me to Stras­bourg. I like her, she is won­der­ful. Michèle said that I am a mem­ber of the “Gang of Souriau.” Jean-Mar­ie Souriau is one of the founders of sym­plect­ic geo­metry. He was in Mar­seille his whole life. Un­for­tu­nately he is dead now. I worked with one of his stu­dents, Paul Donato, for many years.

Jack­son: When you fin­ished your PhD, you tried to get a po­s­i­tion at the Uni­versity of Rwanda. But there was no pos­sib­il­ity for you there?

Ban­yaga: The math­em­at­ics fac­ulty of the Uni­versity of Rwanda at that time were ex­pat­ri­ates who had at most mas­ter’s de­grees. They were afraid that if some­body who is more highly trained came around, it would be a danger for them. This is why they were not so thrilled with me com­ing there.

Jack­son: But you wanted to go back?

Ban­yaga: Yes. But since the Uni­versity of Rwanda did not want to hire me, I wrote to the In­sti­tute for Ad­vanced Study and asked to vis­it. They im­me­di­ately answered me, pos­it­ively. And after that I went to Har­vard Uni­versity.

Home follows you

Jack­son: I wanted to fol­low up on a couple of earli­er ques­tions. In school you were an out­stand­ing stu­dent. What was the re­ac­tion of your fam­ily?

Ban­yaga: They were proud. Ac­tu­ally, my broth­ers learned pro­fes­sions; one is a car­penter, the oth­er is a tail­or. So they are suc­cess­ful. They have a good life in Rwanda.

Jack­son: Did they have much trouble there dur­ing the civil war?

Ban­yaga: No, they kept quiet. That was not like me! I gave a tele­vi­sion in­ter­view and said that I didn’t like that people were at­tack­ing the coun­try. The people who were at­tack­ing are in power in the gov­ern­ment now.

Jack­son: What happened after this in­ter­view?

Ban­yaga: Noth­ing, but it means that I am not wel­come to go to Rwanda now. And I don’t in­tend to go there any­way.

Jack­son: Say­ing that a coun­try should not be at­tacked sounds like a reas­on­able mes­sage. How was what you said per­ceived?

Ban­yaga: It de­pends. If you say you are against the at­tack, the people who sup­por­ted the at­tack are against you. The people who are against the at­tack, they love you!

Jack­son: You were the first Rwandan to get a PhD in math­em­at­ics. Is that why you were in­ter­viewed on TV? Are you well known in Rwanda?

Ban­yaga: Yes. I know per­son­ally some people who have been in the gov­ern­ment. The present gov­ern­ment is Tut­si and is from a group that came with an army from Uganda and took over Rwanda. The Rwandan gen­o­cide star­ted at that time. The pre­vi­ous pres­id­ent [Juvénal Habyar­im­ana], who died in a plane crash in 1994, had asked me to be prime min­is­ter. Of course I re­fused.

Jack­son: Why did you re­fuse?

Ban­yaga: I told him that the coun­try had al­most com­pletely dis­in­teg­rated already. The ag­gres­sion was so high, the people were very di­vided. Ac­tu­ally, if I had ac­cep­ted, I would have been killed too.

Jack­son: Why did he ask you? You’re not a politi­cian. You are a math­em­atician!

Ban­yaga: Yes! I don’t know why. Maybe he thought that, if some­body is able to think cor­rectly and do math­em­at­ics, that per­son can save the coun­try.

Jack­son: He knew you were smart and ac­com­plished.

Ban­yaga: And I don’t have in my heart hatred of any­body. I don’t hate Tut­sis, I don’t hate Hu­tus. So prob­ably he thought I could make a good link­age between all the people.

Jack­son: Were you temp­ted at all to take that po­s­i­tion?

Ban­yaga: Not at all. I was writ­ing beau­ti­ful pa­pers. I was do­ing good math­em­at­ics.

Throughout my life, I have not been in­volved in polit­ics. Now I am 72. When I left Rwanda, I was 20. You see, I spent al­most all my life abroad. Usu­ally I nev­er talk about Hutu or Tut­si. But all these things are con­nec­ted with my life some­how.

Jack­son: They have fol­lowed you.

Ban­yaga: Ex­actly. I try not to be in­ter­ested in what is go­ing on in Rwanda, but I find my­self look­ing at news from Rwanda every day! I can’t help it.

Jack­son: I can un­der­stand that. You are in­ter­ested in your home coun­try even if in some sense you think it might be bet­ter not to be in­ter­ested! When were you last in Rwanda?

Ban­yaga: After 1990, I didn’t go to Rwanda at all.

Jack­son: Rwanda was col­on­ized by Ger­mans in the late 19th cen­tury and was part of Ger­man East Africa. Then it was trans­ferred to Bel­gian ad­min­is­tra­tion after the first World War. Did Ger­man cul­ture re­main in Rwanda?

Ban­yaga: Not at all. Nobody spoke any Ger­man. The Ger­man pres­ence was small and for a very short time. Gen­er­ally, the col­on­iz­a­tion did not have too much in­flu­ence on the reg­u­lar people. There was really no con­tact with the col­on­izers.

Jack­son: So you feel that a lot of the cul­ture of the Rwandan people was pre­served?

Ban­yaga: Yes.

Jack­son: And the lan­guage also.

Ban­yaga: Yes. Ac­tu­ally the people who had a big in­flu­ence are the Cath­ol­ic priests. They are the ones who were go­ing to vil­lages, talk­ing to people. They learned Kin­yar­wanda so they could talk to people.

Jack­son: Where were your chil­dren born?

Ban­yaga: Two were born in Geneva, and one was born in Bo­ston.

Jack­son: The ded­ic­a­tion in your book The Struc­ture of Clas­sic­al Dif­feo­morph­ism Groups lists your wife Ju­dith and the names of your chil­dren. One of the names is RK.

Ban­yaga: RK is my son. His name is Ru­gi­gana Kavama­hanga, which is too long! So we call him RK.

Jack­son: You chose Rwandan names for your kids.

Ban­yaga: Yes. In the past in Rwanda, you didn’t ne­ces­sar­ily take your par­ents’ name. For in­stance, my name is Ban­yaga, which by the way means “the con­quer­or.” My wife’s name is not Ban­yaga, it’s Mukaruziga. And each of our kids has a dif­fer­ent name. There is no fam­ily name. This is a spe­cial tra­di­tion in Rwanda. But now it has changed, and in Rwanda if you marry a girl, she takes your name.

Jack­son: Do you speak Rwandan lan­guage at home with your kids?

Ban­yaga: Yes. Un­for­tu­nately many Rwandans who stayed a long time abroad lost the habit of speak­ing Kin­yar­wandan. But we didn’t do that. We kept speak­ing Kin­yar­wandan.

Jack­son: Have your kids been to vis­it in Rwanda?

Ban­yaga: Yes. I think 1989 was the last time we all went to­geth­er.

The wonders of symplectic geometry

Jack­son: In 1970, Hae­fli­ger gave some lec­tures about Mikhail Gro­mov’s PhD thes­is, which Gro­mov had just fin­ished in St. Peters­burg. Did you meet Gro­mov in Geneva?

Ban­yaga: I didn’t see him in Geneva, but I saw him later many times. I re­mem­ber Hae­fli­ger’s lec­tures. Every­body loved them. Maybe they were the reas­on Gro­mov could enter Europe.

Jack­son: It was not long af­ter­ward that Gro­mov left the So­viet Uni­on. Hae­fli­ger spread the word about his work.

Ban­yaga: Ab­so­lutely. Hae­fli­ger has been a big in­flu­ence on many people in the math­em­at­ic­al world. And Gro­mov has had an in­flu­ence on every­body! I just fin­ished writ­ing the proof of a the­or­em re­lated to the Gro­mov–Eli­ash­berg the­or­em. It is in a pa­per I wrote with a stu­dent and sub­mit­ted last week.

Gro­mov is an etern­al in­flu­ence on geo­metry. He was the first to no­tice and use the fact that there is a con­nec­tion between sym­plect­ic geo­metry, Rieman­ni­an geo­metry, and com­plex ana­lys­is. He in­ven­ted the ter­min­o­logy “pseudo­holo­morph­ic curves” and wrote a fun­da­ment­al pa­per about them. Pierre Pansu, a very good French math­em­atician, wrote a huge thes­is just on that pa­per and has spent all his life study­ing this. The the­ory of pseudo­holo­morph­ic curves is at the cen­ter of everything now in sym­plect­ic geo­metry. Gro­mov also made up the term “soft and hard sym­plect­ic geo­metry” — “soft” mean­ing, everything that does not in­volve his new the­ory! And “hard” mean­ing, any­thing in­volving his the­ory!

Jack­son: He doesn’t mean “hard” as “not easy,” and “soft” as “easy”?

Ban­yaga: Not really. It’s po­et­ic­al ter­min­o­logy. Ac­tu­ally, soft can be harder than hard! And hard can be easi­er.

Jack­son: What kind of math­em­at­ics do you like? What kinds of prob­lems or ideas ap­peal to you?

Ban­yaga: I have been work­ing in sym­plect­ic geo­metry. Grosso modo, sym­plect­ic geo­metry is something that is at the base of math­em­at­ic­al phys­ics. It’s the basis of mech­an­ics. What is nice about it is that, from there, it goes in all math­em­at­ic­al dir­ec­tions, even in­to ap­plied math. I have been work­ing also on the gen­er­al­iz­a­tion of sym­plect­ic geo­metry. For the last ten years, I have been think­ing about a new idea of try­ing to un­der­stand what hap­pens if you don’t ask the struc­ture to be dif­fer­en­ti­able. Dif­fer­en­tial geo­metry, by defin­i­tion, is geo­metry in which you use ana­lys­is. Dif­fer­en­tial func­tions are nice to use, but in real­ity, you don’t see them much. Real­ity is less smooth and more con­tinu­ous.

There is a trend now to treat the theme, “What is con­tinu­ous sym­plect­ic geo­metry?” without us­ing dif­fer­en­ti­ab­il­ity. The main idea is what is called now the ri­gid­ity of Gro­mov and Eli­ash­berg. This was done by Gro­mov in 1986, 1987.

Jack­son: What does ri­gid­ity refer to?

Ban­yaga: Ri­gid­ity means that you won­der what hap­pens if you are al­lowed to de­form things con­tinu­ously. What doesn’t get killed when you push con­tinu­ously? Prop­er­ties that don’t get killed are called ri­gid. The big dif­fer­ence is between con­tinu­ity and dif­fer­en­ti­ab­il­ity.

Jack­son: You men­tioned that sym­plect­ic geo­metry comes up in ap­plied math­em­at­ics too. Apart from in phys­ics, are there real-world ap­plic­a­tions of sym­plect­ic geo­metry?

Ban­yaga: Yes. I once dir­ec­ted a thes­is7 on the ap­plic­a­tion of sym­plect­ic geo­metry to radar and son­ar, which are used in the mil­it­ary to de­tect fly­ing ob­jects. My stu­dent and I wrote a pa­per on radar and son­ar.

Jack­son: Was it pub­lished in a math journ­al?

Ban­yaga: Ac­tu­ally, it was writ­ten for the US Navy and was clas­si­fied. But I think it’s de­clas­si­fied now — we could pub­lish it if we want. My stu­dent hoped to get a job with the Navy after his thes­is, but he didn’t get the job. Our the­ory covered the situ­ation in which the radar works in an at­mo­sphere that is not clean and clear, or works un­der wa­ter. Our the­ory would be more use­ful than the reg­u­lar the­ory of radar. You know what they told us? They said, “The Cold War is over, so we don’t need it!” I told my stu­dent, you are the first cas­u­alty of peace!

Jack­son: How did you come upon this prob­lem of radar and son­ar?

Ban­yaga: My stu­dent came to tell me about it, be­cause he was already work­ing for the Army. He asked me what we can do to im­prove radar. The ques­tion is re­lated to group rep­res­ent­a­tions, and group rep­res­ent­a­tions is very close to sym­plect­ic geo­metry.

Jack­son: What is the stu­dent do­ing now?

Ban­yaga: He works for a uni­versity in Thai­l­and, in Bangkok. He is half Amer­ic­an, half Thail­andese.

Jack­son: Is that the only time in your ca­reer when you worked on such an ap­plied prob­lem?

Ban­yaga: Yes.

Jack­son: When you think about math­em­at­ics, what takes place in your mind? Do you visu­al­ize things? Can you char­ac­ter­ize your think­ing?

Ban­yaga: I think math­em­aticians, like all people, have a mis­sion. My mis­sion is to try to un­der­stand some in­triguing things in the world. There is a guy who I like very much, Le­onid Pol­ter­ovich. He talks about the “won­ders of sym­plect­ic geo­metry.” The Eli­ash­berg–Gro­mov ri­gid­ity is one of them, and there are a couple of oth­ers. I have been fas­cin­ated by these won­ders. My mis­sion is to try to un­der­stand them.

When you think about math­em­at­ics, there is al­ways a concept. You try to make it both real and ab­stract, at the same time. If your idea is just real and has no ab­strac­tion, you can­not make pro­gress.

For in­stance, there is a no­tion I like called dis­place­ment. You have a sub­set, and you look at all the mo­tions that dis­place the sub­set from it­self. Now you look at the size of the trans­form­a­tion that moves it from it­self. To do this, you have to put a norm on the trans­form­a­tions. One of the won­ders of sym­plect­ic geo­metry is the norm that was in­ven­ted by [Helmut] Hofer in 1990.8 The min­im­um of this norm is called the dis­place­ment en­ergy.

Hofer’s norm is one of the fam­ous res­ults of the 20th cen­tury. I have worked on this and have been able to gen­er­al­ize the norm to big­ger groups than the group on which he was work­ing. This is the kind of ob­ject I have been study­ing for the last five or ten years.

Jack­son: When you think of the dis­place­ment, are you visu­al­iz­ing something mov­ing?

Ban­yaga: Ex­actly. Mov­ing, dis­tort­ing it­self, and in the end it goes out­side of the place in which it was be­fore.

Jack­son: You said you are try­ing to make things real and at the same time ab­stract. Dis­pla­cing is very real.

Ban­yaga: And what is ab­stract is the no­tion of size, the size of the trans­form­a­tion. When you bring in this ab­strac­tion, you have a huge amount of ma­ter­i­al in math­em­at­ics that you can use.

Giving back to Africa

Jack­son: After your PhD you wanted to re­turn to Rwanda, but that did not work out. Later you star­ted a PhD pro­gram there.

Ban­yaga: Yes, later I found a way with fund­ing from UN­ESCO, which gave me the op­por­tun­ity to vis­it Rwanda oc­ca­sion­ally and start a pro­gram. This was in the 1980s.

Jack­son: When you were already at Penn State?

Ban­yaga: No, it star­ted earli­er. I came to Penn State in 1984. The pro­gram star­ted when I was at Har­vard and Bo­ston Uni­versity and con­tin­ued at the be­gin­ning of my time at Penn State. But after a while, the pro­gram just stopped. I don’t know why.

Jack­son: By that time were there PhD math­em­aticians there who could take stu­dents in Rwanda?

Ban­yaga: No, there were no Rwandans and not even any for­eign­ers.

Jack­son: So you were the only pos­sible ad­visor for a math PhD there.

Ban­yaga: Yes. I took a few people from Rwanda — one or two to Bo­ston Uni­versity, and an­oth­er to Penn State. I was suc­cess­ful with one of them. He got his PhD in 1991 from Penn State with me.9 Since then, he has re­mained in the US and is at Flor­ida In­ter­na­tion­al Uni­versity.

Jack­son: Did the PhD pro­gram in Rwanda die out be­cause you could not spend so much time there?

Ban­yaga: That was one reas­on, and second, there were no stu­dents. There was no in­terest. Maybe it was pre­ma­ture.

Jack­son: Why was there a lack of in­terest? Did people not un­der­stand what kind of ca­reers math­em­at­ics might lead to?

Ban­yaga: Prob­ably. Any­way, it didn’t work.

Jack­son: You are not able to do any­thing to sup­port math­em­at­ics in Rwanda, but you have done this in oth­er coun­tries in Africa.

Ban­yaga: Yes. I found an­oth­er Afric­an coun­try that I con­sider my coun­try, Ben­in. I have gone there al­most every year for al­most 20 years. In my adult life, I know more of Ben­in than Rwanda. My adult life in Rwanda is zero. I was not there.

Jack­son: What dif­fer­ences do you see between Ben­in and Rwanda?

Ban­yaga: The sim­il­ar­ity is that they are both very small coun­tries. The dif­fer­ence is that Ben­in is more open. The typ­ic­al Ben­inois is like a French guy — a guy who shouts, who laughs. A typ­ic­al Rwandese is just the op­pos­ite — quiet, doesn’t shout.

Jack­son: What do you do when you go to Ben­in?

Ban­yaga: I give lec­tures, I talk to stu­dents, I do re­search, I col­lab­or­ate to write pa­pers. In 2017, I pub­lished a book con­tain­ing a sum­mary of lec­tures I have been giv­ing in Ben­in for sev­er­al years.10 The Ben­in guy who wrote it with me was my as­sist­ant. This is one of the jobs I have to do when I go to Africa — to take some­body and train him, to help him and work to­geth­er with him.

Jack­son: Have you got­ten PhD stu­dents from Ben­in also?

Ban­yaga: Yes. I have sev­er­al. One who wrote his thes­is11 in 2012 is now at the Uni­versity of Buea in Cameroon. I am proud of him. He is very good. He just wrote to me this morn­ing; he is try­ing to help my cur­rent stu­dent in Ben­in.

Jack­son: What do you see as the pro­spects for de­vel­op­ment of math­em­at­ics in Africa? What’s needed? What po­ten­tial do you see?

Ban­yaga: Today there are more math­em­aticians out­side Africa who are will­ing to go there, and there is more fund­ing. For in­stance, the Ger­man Re­search Found­a­tion [Deutsche Forschungs­ge­meinsch­aft] is fund­ing sev­er­al chairs in sev­er­al uni­versit­ies in Africa. This is won­der­ful. Also, the in­sti­tute in Ben­in12 now has a Cen­ter of Ex­cel­lence fun­ded by the World Bank. I am an ad­visor for this cen­ter. There are op­por­tun­it­ies, there is more fund­ing, and there are young people who are ready to work. The situ­ation is much bet­ter than 20 years ago.

Jack­son: In the US, there is a lot of con­cern about the need to en­cour­age more un­der­rep­res­en­ted stu­dents in the so-called STEM [sci­ence, tech­no­logy, en­gin­eer­ing, and math­em­at­ics] fields. How does this is­sue look to you? Do you see im­prove­ment?

Ban­yaga: In gen­er­al, I don’t see much im­prove­ment. For in­stance, in the par­tic­u­lar case of Penn State, we have very good wo­men math­em­aticians. But black math­em­aticians — for many years there was only me. In 2001, we hired a black math­em­atician from Seneg­al. Every­body said, “The black pop­u­la­tion in math at Penn State went up 100 per­cent!” Black Amer­ic­an stu­dents, I don’t see any; Latino stu­dents, I don’t see much. We have a lot of Asi­an stu­dents, es­pe­cially from China. But that is the trend every­where.

Jack­son: Math de­part­ments every­where in the US struggle with this. The num­bers of un­der­rep­res­en­ted stu­dents are so small, that even if the de­part­ment wants to en­cour­age them, they just don’t get the stu­dents.

Ban­yaga: Ex­actly. In con­trast, Afric­an math­em­aticians go much more of­ten to Europe, es­pe­cially France and Ger­many.

But I think Africa it­self has po­ten­tial to de­vel­op. They have opened more good schools and more cen­ters. They have help from Europe and the US. The Si­mons Found­a­tion also has a pro­gram, called Math in Africa. I was on the com­mit­tee for that. I am really op­tim­ist­ic. I think the situ­ation should im­prove.

Jack­son: There are also the AIMS cen­ters.13

Ban­yaga: Yes. I have vis­ited the AIMS in Seneg­al sev­er­al times and dir­ec­ted sev­er­al mas­ter’s de­grees there. The stu­dents were very good. AIMS Seneg­al also has a Hum­boldt Fel­low­ship that funds a chaired pro­fess­or­ship. The chair now is an ex­cel­lent Sene­g­alese math­em­atician [Mouhamed Moustapha Fall]. And I just heard a col­league say­ing that the Hum­boldt Found­a­tion is also fund­ing a chair at the AIMS cen­ter in Rwanda.

Jack­son: Yes, there is an AIMS in Rwanda now.

Ban­yaga: I think it’s the new­est one.

Jack­son: What is it like to be a black man in the United States today?

Ban­yaga: Per­son­ally, I don’t feel any­thing bad or any pres­sure. Ac­tu­ally, I feel bet­ter in the US than in France or Ger­many. Here, I am not a stranger. There are many black Amer­ic­ans.

In the small town in which I live, there are very few Afric­an-Amer­ic­ans. But there are no prob­lems, no ra­cism. Every­body looks after them­selves, and nobody both­ers any­body else. I feel much safer here than I would in Rwanda! There is no prob­lem in day-to-day life.

Jack­son: And when you look na­tion­ally, more broadly, what do you see?

Ban­yaga: Na­tion­ally I think the situ­ation is get­ting bet­ter. Even with Trump! I think the First Step Act, which he signed in­to law, has helped make the in­car­cer­a­tion laws that put a lot of blacks in­to jail more gentle.

Jack­son: You have a very pos­it­ive out­look.

Ban­yaga: Yes, ab­so­lutely. Next year, I have a sab­bat­ic­al leave, I will go to Africa again. Everything looks pos­it­ive.

Jack­son: I won­der if your pos­it­ive out­look comes from your child­hood, run­ning around the ba­nana plant­a­tion.

Ban­yaga: Yes, it is con­nec­ted.

Jack­son: That sounds like it was a very happy time, very free; you had a lot around to in­terest you, grow­ing your own ba­nana trees and look­ing at nature.

Ban­yaga: Ex­actly. And I must say that I have been really lucky, to go to Switzer­land by mis­take, and find Hae­fli­ger, and find geo­metry, and meet so many math­em­aticians and do my work. I feel my­self a lucky guy. Here at Penn State, I found someone I have been work­ing with for more than 15 years, Dav­id Hur­tu­bise. We wrote a book to­geth­er.14

Jack­son: It’s nice to have a col­lab­or­at­or at your own in­sti­tu­tion.

Ban­yaga: Yes, it’s great.

Jack­son: I wanted to ask you what your chil­dren do.

Ban­yaga: My eld­est daugh­ter is mar­ried and takes care of her fam­ily. She has a bach­el­or’s de­gree in Eng­lish from Penn State. My second daugh­ter is a pat­ent law­yer. She did her doc­tor­ate in law at Emory and a chem­ic­al en­gin­eer­ing de­gree at the Uni­versity of Mary­land. Now she works for the gov­ern­ment, in Den­ver, Col­or­ado. My last child, my son RK, has an MBA from Penn State. My wife Ju­dith also has an MBA and a mas­ter’s de­gree in French lit­er­at­ure, both from Penn State. She has been a re­altor since 1996.

RK is in­ter­ested in something very un­usu­al, crypto­cur­rency. You have heard about that?

Jack­son: I have, but I can’t say I un­der­stand it! I can’t un­der­stand why people buy crypto­cur­ren­cies.

Ban­yaga: I don’t un­der­stand it either!

Jack­son: But it can be luc­rat­ive.

Ban­yaga: Or it might make you very poor!