Celebratio Mathematica

Ingrid Daubechies

Interview with Ingrid Daubechies

by Allyn Jackson

Photo courtesy of Duke Today

In­grid Daubech­ies was born in Houthalen-Helchter­en, Bel­gi­um, in 1954. She showed math­em­at­ic­al tal­ent at an early age and as­pired to fol­low in her fath­er’s foot­steps to be­come an en­gin­eer. But his love of phys­ics also in­flu­enced her, and she stud­ied that sub­ject when she at­ten­ded the Free Uni­versity in Brus­sels. Later as a PhD stu­dent there, she grav­it­ated to math­em­at­ics and fin­ished her thes­is in math­em­at­ic­al phys­ics in 1980 un­der the dir­ec­tion of Alex Gross­mann.

Daubech­ies is one of the pi­on­eers of the concept of wave­lets, which brought about a re­volu­tion in data ana­lys­is and stor­age. This re­volu­tion de­pended cru­cially on her break­through in con­struct­ing wave­lets hav­ing an or­thonor­mal basis with com­pact sup­port. Her work made wave­lets in­to a prac­tic­al tool that could be put to im­me­di­ate use. Today wave­lets are used in a wide range of ap­plic­a­tions across sci­ence and en­gin­eer­ing.

Daubech­ies stands in the van­guard of a dec­ades-long trend in which math­em­at­ic­al meth­ods have be­come per­vas­ive across a broad swath of hu­man en­deavor. She has worked on ap­plic­a­tions of math­em­at­ics not only to areas that are tra­di­tion­ally close to the field, such as phys­ics and sig­nal pro­cessing, but also to more-dis­tant areas like evol­u­tion­ary bio­logy and art res­tor­a­tion. Her many col­lab­or­a­tions are en­abled by her lively curi­os­ity and power­ful tech­nic­al ex­pert­ise, as well as her strong in­tu­ition about which math­em­at­ic­al meth­ods fit with which prob­lems.

After po­s­i­tions at the Free Uni­versity of Brus­sels, Bell Labor­at­or­ies, and Prin­ceton Uni­versity, Daubech­ies took her cur­rent po­s­i­tion as the James B. Duke Pro­fess­or of Math­em­at­ics and Elec­tric­al and Com­puter En­gin­eer­ing at Duke Uni­versity. She has re­ceived a host of hon­ors, in­clud­ing two Steele Prizes from the Amer­ic­an Math­em­at­ic­al So­ci­ety (for ex­pos­i­tion in 1994 and a sem­in­al con­tri­bu­tion to re­search in 2011), the Na­tion­al Academy of Sci­ences Award in Math­em­at­ics (2000), a John D. and Cath­er­ine T. Ma­cAr­thur Found­a­tion Fel­low­ship (1992–1997), and the In­sti­tute of Elec­tric­al and Elec­tron­ics En­gin­eers In­form­a­tion The­ory So­ci­ety Golden Ju­bilee Award for Tech­no­lo­gic­al In­nov­a­tion (1998). She was elec­ted to both the US Na­tion­al Academy of Sci­ences (1998) and the US Na­tion­al Academy of En­gin­eer­ing (2015).

The fol­low­ing in­ter­view was con­duc­ted by Allyn Jack­son in late 2018.

Life in a coal mining town

Jack­son: You were born and grew up in Bel­gi­um. What lan­guage was spoken in your fam­ily home?

Daubech­ies: My par­ents are in a minor­ity of Bel­gians in that one of them grew up French-speak­ing, the oth­er Dutch-speak­ing. My moth­er is Flem­ish but after a few years of her early school­ing in Dutch, her moth­er tongue, she switched to the French-lan­guage school sys­tem and did the re­mainder, in­clud­ing all of her ad­vanced school­ing, in French. Many Flem­ish par­ents wanted their chil­dren to get ahead and so sent them to French-speak­ing schools, think­ing that, with im­pec­cable French, they would have a bet­ter op­por­tun­ity at good jobs.

My fath­er — and this was very un­usu­al at that time — al­though he was French-speak­ing, did all his high school stud­ies in Dutch. That was be­cause his par­ents were liv­ing then in the Flem­ish part of the coun­try. His fath­er had been black­balled for uni­on activ­ity and was offered a job only when people with his qual­i­fic­a­tions could not be found for a new plant in Flanders.

So my [pa­ternal] grand­fath­er and his fam­ily were liv­ing in a small French-speak­ing com­munity that was built by the com­pany he worked for in the middle of a Flem­ish part of the coun­try. When World War II broke out, my fath­er could no longer com­mute to board­ing school be­cause there were no more trains. So he went to Flem­ish schools, and as a res­ult he speaks flu­ent Dutch. He is one of the very few French-speak­ing people out­side Brus­sels who speak flu­ent Dutch.

Later when my fath­er found a job in a Flem­ish part of Bel­gi­um, my par­ents de­cided they would bring up their chil­dren in Dutch. In the early 1950s there was a the­ory that it was not good for chil­dren to have early bi­lin­gual­ism. My par­ents con­sul­ted psy­cho­lo­gists about this and de­cided to bring their chil­dren up uni­lin­gually and to in­tro­duce us to the second lan­guage only later. My fath­er had hated board­ing school with a ven­geance, so he did not want to send his chil­dren to a French-speak­ing board­ing school. Since we were go­ing to go to school in Dutch, they de­cided that Dutch would be the lan­guage at home.

Jack­son: But your par­ents spoke French to each oth­er?

Daubech­ies: Yes — es­pe­cially when they didn’t want the chil­dren to un­der­stand! My fath­er later said that he had found by ex­per­i­ence that this was one of the most power­ful mo­tiv­at­ors for chil­dren to learn the oth­er lan­guage.

My fath­er is a min­ing en­gin­eer. When school­teach­ers told my grand­par­ents — who were blue-col­lar work­ers, skilled blue-col­lar, but still — that they should let their son go to col­lege be­cause he was really smart, they thought of only two pos­sib­il­it­ies: min­ing en­gin­eer or doc­tor. My fath­er couldn’t stand the sight of blood, so be­com­ing a doc­tor was not in­dic­ated. My grand­fath­er worked in a glass fact­ory, not a coal mine, but their whole back­ground was in the coal min­ing re­gion in the Bor­in­age. The fam­ily was liv­ing in Flanders, but all their re­l­at­ives were in the Bor­in­age, very close to a min­ing school. My fath­er could study there and stay with re­l­at­ives, and it would be much cheap­er. My grand­par­ents had to save to­wards their re­tire­ment, so they made a deal with my fath­er. He would get a high­er edu­ca­tion, and as a res­ult, they would count on him to help them with their re­tire­ment, which he did. Even so, they needed to do it cheaply, so my fath­er stayed with re­l­at­ives and stud­ied on the kit­chen table.

There was nev­er any thought of ma­jor­ing in any­thing oth­er than min­ing en­gin­eer­ing. At that time coal was still strong in Bel­gi­um. He did in­tern­ships in tin min­ing in Congo, which was then a Bel­gian colony, and he had a one-year job in Al­ger­ia, in iron min­ing. But he al­ways thought he would settle in Bel­gi­um, where there was noth­ing but coal to mine. There were two coal re­gions in Bel­gi­um, one in the Bor­in­age with very old mines, and one with young­er mines in Limburg, which is part of Flanders and is Dutch-speak­ing. So he got a job in Limburg, and that’s where they settled.

Coal min­ing towns have one huge em­ploy­er. Be­fore min­ing starts, of­ten there is noth­ing, no vil­lage or any­thing. So a new com­munity starts. Al­though there is an eco­nomy that builds up, the coal com­pany plays an enorm­ous role in the loc­al so­cial life. The com­pany took on at­ti­tudes that even at the time seemed out­dated. For in­stance, when they in­ter­viewed a single young man, they would ask wheth­er he was en­gaged and would get mar­ried soon. They wanted nicely settled couples; they didn’t want to have the dif­fi­cult situ­ation of young single men not find­ing op­por­tun­it­ies to meet wo­men. Not only that, but the men were sup­posed to marry “suit­able” young wo­men. The fact that my moth­er was study­ing at col­lege was viewed as a very good thing. They liked the wife to be edu­cated, al­though they did not want the wife to work. My fath­er had one col­league whose wife who did work. She was a pe­di­at­ric nurse, which is usu­ally ac­cept­able for wo­men. That col­league nev­er got a pro­mo­tion, and every­body knew that it was be­cause his wife worked.

Jack­son: They thought he didn’t need more money?

Daubech­ies: No — it was just not done. He was clearly not con­ven­tion­al. You shouldn’t rock the boat.

Jack­son: That is strange, when you think of it now.

Daubech­ies: Or even by the stand­ards then. If she had been a start-up en­tre­pren­eur, that would have been sur­pris­ing, but she was a nurse look­ing after chil­dren! But wives of en­gin­eers were not sup­posed to work.

Jack­son: Your moth­er did not work, though she in­ten­ded to?

Daubech­ies: Yes, she fully in­ten­ded to work but didn’t. When my broth­er and I were little, she was very in­volved with us. As we be­came more in­de­pend­ent, she star­ted to be very frus­trated by her lim­ited ho­ri­zon. She ac­tu­ally had a word for this in Dutch, “uit­ged­roogd”, which means parched. She said “I feel my­self parched” — in­tel­lec­tu­ally parched. I would come home from school and ask my broth­er, “How are things?” And he would say, “She’s parched again.”

Jack­son: She came out and talked about it, she didn’t keep it in­side.

Daubech­ies: Oh no, my moth­er doesn’t keep things in­side. My moth­er is very ar­tic­u­late.

“This is going to be really ugly”

Daubech­ies: When I was 9 years old, my fath­er moved from one mine to a dif­fer­ent one, where he be­came the head of per­son­nel. The mines were very deep, about 1 kilo­met­er deep. If you said you worked at a mine, the first ques­tion would be, “Un­der­ground or above ground?” Pre­vi­ously he had worked un­der­ground, but his new job was above ground. That mine closed 3 or 4 years later, which caused im­mense ri­ots. When you at­tract lots of people to come work and settle and start fam­il­ies, and then you say that you go­ing to cut off these 5000 or 6000 jobs — and when even a year earli­er, there was no talk about clos­ing so that people could slowly pre­pare — there will be a lot of in­dig­na­tion. So there were demon­stra­tions.

I re­mem­ber one even­ing my fath­er came home from a meet­ing pri­or to a big demon­stra­tion where the po­lice had been plan­ning how to con­tain it. He looked com­pletely ashen. He said, “This is go­ing to be really ugly, be­cause the po­lice will be armed with real bul­lets.” For some demon­stra­tions, the po­lice would fire blanks, but this time they de­cided they could not do that. Giv­en the emo­tions, and giv­en the num­ber of people, he said he ex­pec­ted there would be people killed. And in­deed there were. Part of the demon­stra­tion was in front of our house. My fath­er told us, “Please stay in the back of the house, please don’t even peek around the cur­tains, don’t show there is any­body even in the house.” I was 13 years old at the time.

There is a fant­ast­ic feel­ing when you have a big mass of people — you feel this broth­er­hood with every­body. And I feel that too, but I also feel that it’s a very fright­en­ing emo­tion. It’s not an emo­tion I have felt com­fort­able with, ever — not since then. The emo­tion it­self does not ne­ces­sar­ily have to do with the cause you are in but with be­ing in this group of people. And it eli­cits a sim­il­ar emo­tion in the op­pos­ing group. I think it’s a dan­ger­ous thing.

Jack­son: Did your fath­er take part in that demon­stra­tion?

Daubech­ies: He did not. But his heart was on the side of the demon­strat­ors. Look, he came from a blue-col­lar back­ground.

Jack­son: And he was the head of the per­son­nel and knew the work­ers.

Daubech­ies: The head of per­son­nel of the com­pany that is clos­ing the fact­ory! With­in a few months, his hair turned white.

We then moved, and my fath­er looked for a job out­side the coal sec­tor. He thought that all the mines would close very quickly, but the oth­er mines were shored up by the gov­ern­ment. Al­most all his col­leagues served out their whole work­ing lives in coal mines. But he de­cided he would leave coal mines, and he be­came the CEO of a wall­pa­per print­ing com­pany. We moved to a dif­fer­ent town, and that’s where I went to high school.

That’s when my moth­er was “parched”. When we were in the min­ing re­gion, she had built up a so­cial circle. Part of the pa­ter­nal­ist­ic ap­proach of the coal com­pan­ies dic­tated that the wives are not sup­posed to work, but it also sub­sid­ized activ­it­ies that would en­gage these wo­men. For ex­ample, my moth­er de­signed sets for the loc­al theat­er com­pany, which was ac­tu­ally of very high qual­ity and once won an award. But now she was out of all that. Maybe also my fath­er look­ing for jobs brought home that she was eco­nom­ic­ally de­pend­ent.

Later, my fath­er was asked to dir­ect an­oth­er wall­pa­per print­ing com­pany as well, near Brus­sels. So he spent half his time near Brus­sels, and half his time in the town where we were liv­ing. When I had to go to col­lege, it was nat­ur­al for us to move to Brus­sels, be­cause my fath­er would still be on the road just as much. My moth­er ap­plied to col­lege again, be­cause her de­gree was com­pletely ob­sol­ete by then.

Jack­son: So she went back to col­lege at the time you star­ted col­lege?

Daubech­ies: Ex­actly. She had wanted to do art his­tory, but for some com­plic­ated, stu­pid, leg­al­ist­ic reas­on, her earli­er de­gree — which was in book­keep­ing — did not al­low her to en­roll in art his­tory. She looked at what de­grees she could start in, and there was one in crim­in­al justice. So she did that. She came in­to the work force at age 50 and had a ca­reer for 15 years with the justice de­part­ment, as a so­cial work­er for youth. Some­times kids came to the at­ten­tion of the justice de­part­ment be­cause either they had been in­volved in petty crimes them­selves, or be­cause their par­ents had been in­car­cer­ated and they needed some­body to fol­low their cases. So that’s what she did. She would in­ter­view the kids and their par­ents or foster par­ents and write re­ports about their situ­ations.

Jack­son: That’s really im­press­ive to start do­ing that at age 50. Did she en­joy the work?

Daubech­ies: Yes, she did, al­though it was really tough. My moth­er is a tough wo­man. Some of those kids were trouble for a good reas­on. One of them be­came a seri­al mur­der­er. Some­times she had to go to bad neigh­bor­hoods where she didn’t feel safe, so she would ask for po­lice sup­port.

Multivariable calculus at age 11 — “just following rules”

Jack­son: Who gave you early in­flu­ences in math­em­at­ics? Was your fath­er in­ter­ested in math­em­at­ics?

Daubech­ies: My fath­er real­ized while at col­lege that there were many oth­er things that people did when go­ing to col­lege. He really would have liked to be a phys­i­cist. In the coal min­ing towns there were some­times guest lec­tures on phys­ics, and he would al­ways go to those. For one of them he bought the Feyn­man Lec­tures on Phys­ics. He dis­covered the Open Uni­versity on tele­vi­sion, and he took some classes in that. He talked to me about phys­ics — some­times at great­er length than I had bar­gained for! For him, math was a tool to do things, so when he taught me math, he taught me about the mech­an­ics of those tools, which are fairly simple.

For many years, my par­ents held it up as a fant­ast­ic thing that I was do­ing mul­tivari­able cal­cu­lus when I was 11 or so. But I was just fol­low­ing the rules, which are very simple. I didn’t have the un­der­stand­ing of it that I have now. So later I real­ized this pride that they had was not so war­ran­ted. At the time I half en­joyed it and half didn’t en­joy it. I didn’t en­joy be­ing singled out as Ex­hib­it A. I no­ticed at some point that it made it so­cially more dif­fi­cult for me, and then I star­ted down­play­ing it very much.

Jack­son: Be­cause you were a girl?

Daubech­ies: I don’t think it was be­cause I was a girl. I don’t think I ever had that feel­ing be­fore I got to col­lege that be­cause I was a girl I should be less com­pet­ent. I went to an all-girls school be­cause that’s what pub­lic schools were like in Bel­gi­um at that time. I just didn’t like be­ing singled out as the smart one, the nerd.

Jack­son: Be­ing the brain, and only the brain.

Daubech­ies: Yes — that coupled with the fact that my so­cial an­ten­nae are not al­ways ac­cur­ate. I am very em­path­ic, I think, but of­ten I don’t un­der­stand the so­cial un­der­cur­rents in com­plic­ated con­ver­sa­tions. I take things very much at face value. That’s something I still have trouble with. That’s why I don’t have good polit­ic­al in­stincts. For ex­ample, in meet­ings where people have dif­fer­ent opin­ions, I just blurt out what I think and think it’s ob­vi­ous that cer­tain courses of ac­tion are much bet­ter than oth­ers, and I get very frus­trated when not every­body sees that. It’s only later that I real­ize what was go­ing on.

Jack­son: You men­tioned go­ing to an all-girls school. Look­ing back on that now, do you think that had a par­tic­u­lar ef­fect on you?

Daubech­ies: At the time I thought it was ri­dicu­lous that an edu­ca­tion­al sys­tem would keep genders sep­ar­ated. They were ex­pec­ted at some point to start meet­ing — shouldn’t they get to know each oth­er? But it had the side-ef­fect, of which I was not aware at the time, that there were no sub­jects that you would feel, as a girl, were not ap­pro­pri­ate to be good in, or that you were not ex­pec­ted to be good in. You were all girls — some were bet­ter at some things, and some were bet­ter at oth­ers.

At­tend­ing an all-girls school has an in­flu­ence, but some­times a big­ger in­flu­ence is not so much what is said as all the things that are not said. Meet­ing one per­son who thinks you should not be good in something be­cause you are a girl — well, cer­tainly in my case that could nev­er have had a bad in­flu­ence. But be­ing in sur­round­ings where every­body sub­lim­in­ally makes that as­sump­tion, that could have a bad in­flu­ence. A gen­er­al ex­pect­a­tion, a cloud of ex­pect­a­tion — 

Jack­son: — an un­spoken ex­pect­a­tion?

Daubech­ies: Yes, I think that is more dev­ast­at­ing — and more per­ni­cious and harder to deal with — than the oc­ca­sion­al very ex­pli­cit re­mark. The very ex­pli­cit ones are rare. I don’t know that they can have such a big im­pact. I’ve al­ways been kind of ornery, so when I got to uni­versity and met the at­ti­tude of, “How can you be good in this, you are a girl?” I thought, “Well, you’re a jerk.”

I was at a small uni­versity, the Vrije Uni­versiteit in Brus­sels. I en­rolled in phys­ics, and in the first 2 years we had many classes in com­mon with the math ma­jors. I was very good at math, and the math pro­fess­ors gen­er­ally as­sumed that people who stud­ied phys­ics had gone for that be­cause they didn’t want to do the harder courses of math­em­at­ics. I was among the top per­formers in math, but I was not a math ma­jor. When we had prob­lem ses­sions, they would give some starred prob­lems, and only the math ma­jors had to do those. Be­ing ornery, I would start by do­ing the starred prob­lems. The teach­ing as­sist­ants would say to me, “You don’t have to do those, you are a phys­ics ma­jor.” And I would say, “So what? I want to do it!”

“You’ll live in the gutter!”

Jack­son: How did you end up choos­ing phys­ics for your ma­jor?

Daubech­ies: I had al­ways thought I wanted to be­come an en­gin­eer, be­cause my fath­er was an en­gin­eer, and I was clearly in­ter­ested in what he was in­ter­ested in. My fath­er felt that phys­ics is much more in­ter­est­ing than en­gin­eer­ing. Still, I wanted to be­come an en­gin­eer, and my moth­er en­cour­aged me in that.

In my fi­nal high school year, I went to vis­it vari­ous uni­versit­ies. At that time, if you had a high school de­gree from a Bel­gian school and you were Bel­gian, you could just en­roll in any uni­versity; they didn’t have a lim­ited num­ber of slots. The tu­ition was low, so my par­ents clearly could af­ford it. So I could choose where to go.

At the be­gin­ning I vis­ited en­gin­eer­ing de­part­ments. The uni­versity in Ghent has a very prom­in­ent en­gin­eer­ing de­part­ment and is es­pe­cially known for civil en­gin­eer­ing, at least it was at that time, the late 1960s, early 1970s. That’s the place where an im­port­ant dis­cov­ery about con­crete was made. They dis­covered that, if you don’t just add re­bar to the con­crete, but when you pour the con­crete you put pres­sure on it and use the re­bar to tense it, then the con­crete is sig­ni­fic­antly stronger.

The uni­versity was very proud of this dis­cov­ery, and they had big ma­chines to test the strength of con­crete — pow! and pow! again. I was not im­pressed at all. I didn’t see my­self do­ing that kind of thing. So at the oth­er uni­versit­ies, I vis­ited phys­ics. We learned about black­body ra­di­ation and ab­sorp­tion spec­tra. I thought this was won­der­ful and de­cided to ma­jor in phys­ics. My moth­er was very up­set. “Phys­ics!” she said. “En­gin­eer­ing — now that’s a pro­fes­sion. With phys­ics you might just as well choose to be an artist. You won’t make a liv­ing, you’ll have to live in the gut­ter!”

It was strik­ing that there was nev­er any ex­pect­a­tion from either my fath­er or my moth­er that I would live by mar­ry­ing and be­ing provided for.

Jack­son: Why was that ex­pect­a­tion ab­sent?

Daubech­ies: I don’t know. Maybe it’s be­cause my moth­er had so much re­gret­ted not hav­ing had a ca­reer her­self. I think my par­ents must have giv­en it some thought them­selves, but they didn’t dis­cuss it with me.

My moth­er said, “You’ll end up in the gut­ter — but at least you might still change your mind.” In Bel­gi­um you had to do an en­trance ex­am­in­a­tion if you wanted to study en­gin­eer­ing. I passed the ex­am but didn’t change my mind. A few days be­fore the uni­versity opened for the fall, the first-year stu­dents would go there, de­clare a ma­jor, and get a stu­dent card. Since I had taken the en­gin­eer­ing ex­am, the man star­ted to write down en­gin­eer­ing as my ma­jor. I said, “I’m not do­ing en­gin­eer­ing, I’m do­ing phys­ics.” He said, “But you passed the ex­am — and you had the highest score!” I said, “I am still do­ing phys­ics” — to the great re­gret of my moth­er! Later she rue­fully ad­mit­ted that I did man­age to make a liv­ing!

It happened that I was in the same co­hort as Jean Bour­gain, who en­rolled in math­em­at­ics, at the same uni­versity. Be­cause the phys­ics stu­dents had so many classes in com­mon with the math ma­jors, we saw a great deal of each oth­er. There were three stu­dents who were very good in math: me, Jean, and an­oth­er stu­dent who I com­pletely lost track of and who was the daugh­ter of one of the pro­fess­ors. The three of us would talk math to­geth­er some­times. I promptly de­veloped a crush on Jean. I had not met many boys at all, but he was cer­tainly the very first I met who was really smart. But I don’t know that he ever no­ticed at that time wheth­er I was fe­male or male! So my crush was not re­cip­roc­ated. And then I got over it.

Jack­son: How did you make the de­cision to go on to a PhD?

Daubech­ies: Many people who do a PhD in Bel­gi­um stay in the same uni­versity where they did their un­der­gradu­ate de­gree. The stu­dents say, “There is still so much that our pro­fess­ors know, so we can still learn from them.” And the pro­fess­ors say, “If we let our best stu­dents go else­where, then how will we get good stu­dents to work with?” I didn’t know that in oth­er parts of the world stu­dents were ex­pec­ted to go to a dif­fer­ent place for the PhD.

People told me there would be a pos­sib­il­ity to get a fel­low­ship to do a PhD at the Vrije Uni­versiteit in Brus­sels. In Bel­gi­um at that time you were already con­sidered an em­ploy­ee once you got a re­search or teach­ing as­sist­ant­ship. It’s a tem­por­ary job, but there was the ex­pect­a­tion that after you got a PhD your job might be made per­man­ent. It was a good fel­low­ship, and I made at least as good a liv­ing as I could have if I had looked for an entry-level job else­where. My moth­er said that, as an en­gin­eer I would have earned much more. That prob­ably wasn’t wrong — but still.

It wasn’t so much that I made a de­cision of go­ing for a PhD as that the pos­sib­il­ity was offered to me, and it meant I didn’t have to go hunt­ing for a job else­where. So I took it.

Jack­son: Jean Bour­gain did the same thing, right?

Daubech­ies: Yes, he also had the pos­sib­il­ity to do a PhD at the Vrije Uni­versiteit, and that’s what he did. Both of us got fel­low­ships from the Bel­gian equi­val­ent of the NSF [Na­tion­al Sci­ence Found­a­tion in the U.S.].

Jack­son: You have one young­er broth­er. What did he end up do­ing?

Daubech­ies: He stud­ied ro­mance lan­guages. He would have liked to study his­tory, and he was dis­suaded by people say­ing that it would be very hard to find a job. I think he should have just done it. He stud­ied ro­mance lan­guages in­stead, with the idea that, be­cause Bel­gi­um is a bi­lin­gual coun­try, there are al­ways jobs for people who know oth­er lan­guages, es­pe­cially teach­ing jobs. But teach­ing was nev­er something that really suited him. Also, the kind of job that he ended up do­ing, he could have done hav­ing stud­ied his­tory just as well as ro­mance lan­guages. He works in re­tail in Lon­don. For a long time he worked for Carti­er, and now he heads the Wedg­wood sec­tion for sev­er­al House of Fraser stores, which are high-end lux­ury stores in Eng­land.

Part of what made his life com­plic­ated when he was young­er was that he is gay, and my par­ents were not very tol­er­ant of gay people. It’s hard to come out as a gay per­son, but what is much harder is be­ing in­flu­enced as a kid by at­ti­tudes that your par­ents had, and then dis­cov­er­ing that you are something that you’ve been taught to des­pise. That’s really hard.

Jack­son: When did you find out he is gay?

Daubech­ies: When he ran away the first time — he ran away sev­er­al times. The first time, he went to Par­is but came home after about a week. He was go­ing to en­roll in the For­eign Le­gion, of all things! He had the idea of re­mak­ing his life, and in the For­eign Le­gion, you could even give a dif­fer­ent name — you could start your life over again. While sit­ting in a church, gath­er­ing his cour­age, he was no­ticed by a priest, who talked with him and fi­nally called us. My par­ents had found a note some­where men­tion­ing Par­is, so they had gone there to look for him. But of course Par­is is a big place, how could they find him? I was at home, so I got the phone call. I couldn’t reach my par­ents — this was be­fore cell phones. So I went to Par­is and brought him back.

The second time he dis­ap­peared, he was gone for two years. After about a year, just as I was go­ing to go off to a postdoc in Prin­ceton, I got a post­card from him, from Am­s­ter­dam, that gave a post box num­ber. So for a year from the States I would write to this post box num­ber, send him things for his birth­day, and so on. In the middle of my postdoc, I had to go to a con­fer­ence in Europe, so I took an ex­tra week. I wrote to the post of­fice box num­ber and said I’d be in Am­s­ter­dam. I didn’t know Am­s­ter­dam well, but I knew the Rem­brandt paint­ing The Night Watch was in a mu­seum there. So I said that on a cer­tain day I would sit in front of The Night Watch all day. And he came. So that’s how we found each oth­er again.

A polyglot advisor

Jack­son: You form­ally had two ad­visors for your PhD, but mainly you were work­ing with Alex Gross­man. Can you tell me about him? He sounds like quite an un­usu­al char­ac­ter.

Daubech­ies: He is.1 He was born in Zagreb. His fath­er was Jew­ish, his moth­er was not. Dur­ing the war, he and his fath­er hid in Italy. Even though Italy was one of the Ax­is coun­tries, it was — at least un­til Ger­many really took over the coun­try — much more friendly to Jews and much safer than Croa­tia was. After the war Alex went to Har­vard and stud­ied phys­ics. Between the war and col­lege, he was dia­gnosed with tuber­cu­los­is. At that time, if you had tuber­cu­los­is, you were sent to a san­it­ari­um in the moun­tains in Switzer­land, if your fam­ily could af­ford it, and his fam­ily could. Every­body was bored stiff there, be­cause all they were try­ing to do all day was to get bet­ter. Alex learned Lat­in there, and to this day, he reads Lat­in for en­ter­tain­ment. He is an in­cred­ible poly­glot. I said to him, “I must be one of the few people you work with, with whom you do not con­verse in their nat­ive tongue.” This was after I’d heard him speak in Itali­an and Ger­man and French and Eng­lish. He thought a bit and said, “No, no, I’ve worked with T. T. Wu, and he’s Chinese, and I don’t speak Chinese.” I felt it made my point!

My work­ing with Alex came about quite by ac­ci­dent in a sense. The pro­fess­or with whom I was work­ing in Bel­gi­um had sev­er­al fel­low­ships in the same year and re­cruited sev­er­al very bright young people. He didn’t have time to work with all of us, so he tried to find, de­pend­ing on our in­terests, col­leagues he knew who could work with us. My good friend Christine De Mol went to Gen­oa, and she worked with Mario Bertero. I went to Mar­seille to work in the­or­et­ic­al particle phys­ics with Ray­mond Stora, a very prom­in­ent phys­i­cist who was the dir­ect­or of the in­sti­tute there. But there were im­mense in­tern­al polit­ic­al prob­lems at that time, and the in­sti­tute split in two. So he had no time to work with me.

I was shar­ing an of­fice with Percy Deift, who was there fin­ish­ing his thes­is. I was at a loss. I read as much math­em­at­ics as I could, be­cause I was really in­ter­ested in it. Percy said, “You came here to work on the­or­et­ic­al phys­ics, but I no­tice that the books that you want to read are all math books. You should do math­em­at­ic­al phys­ics.” He said that a great place to do it is Prin­ceton. But I had a boy­friend then, and it wasn’t the right time in our re­la­tion­ship for a long sep­ar­a­tion. When I told Percy I couldn’t go to Prin­ceton, he in­tro­duced me to Alex Gross­mann.

Jack­son: What did your thes­is end up be­ing about?

Daubech­ies: It was on Weyl quant­iz­a­tion. When you do quantum mech­an­ics, there is a cor­res­pond­ence with clas­sic­al mech­an­ics: Ob­serv­ables that were func­tions of po­s­i­tion and mo­mentum earli­er be­come op­er­at­ors in a Hil­bert space. Things that de­pend on po­s­i­tion be­come mul­ti­plic­a­tion op­er­at­ors; mo­mentum be­comes the gradi­ent op­er­at­or. It’s easy to see what you get when you have something like a Hamilto­ni­an, but not if you have more com­plic­ated func­tions that de­pend on po­s­i­tion and mo­mentum in a more in­tric­ate way. Her­mann Weyl gave a pre­scrip­tion for how you could turn those more com­plic­ated func­tions in­to op­er­at­ors. The trans­fer from func­tions to op­er­at­ors is a very tricky one, so you have to re­strict your­self to cer­tain func­tions. If you tried to do all pos­sible func­tions, you would get all kinds of in­con­sist­en­cies — func­tions could tend to a lim­it func­tion in func­tion space and their equi­val­ents in op­er­at­or space could tend to a lim­it in op­er­at­or space, but the lim­its might not cor­res­pond. It’s the worst thing — they are worse than not con­tinu­ous. Be­ing not con­tinu­ous might mean that the im­ages don’t con­verge, but in fact you can find ex­amples where they do still con­verge, but they con­verge to the wrong thing! So I stud­ied the math­em­at­ic­al prop­er­ties of that cor­res­pond­ence, us­ing a lot of co­her­ent-state tech­niques.

Alex nev­er gave me a thes­is prob­lem. I worked on a num­ber of things, and we wrote pa­pers. At some point I had enough volume for a thes­is, and the vari­ous things I had done were linked to­geth­er in many dif­fer­ent ways. I de­fen­ded my thes­is in 1980. That was be­fore word pro­cessing for math, so you typed math on an IBM Se­lec­tric type­writer. The way it worked in our group was that, if some­body’s thes­is needed to be typed, every­body would vo­lun­teer to type. You would write it in longhand, and while you were still writ­ing, every­one else would type. The ma­chines were avail­able only at night when the sec­ret­ar­ies wer­en’t there! I don’t do it any­more, but for a long time after that I would still would write math pa­pers longhand. I felt the writ­ing was bet­ter be­cause I was edit­ing as I wrote.

“Talk through your tears”

Daubech­ies: Right after my thes­is de­fense I went to a sum­mer school in Erice in Si­cily on math­em­at­ic­al phys­ics, and I met El­li­ott Lieb. I had ap­plied for a NATO fel­low­ship, and asked him if I could use it to vis­it Prin­ceton. He said “Sure,” be­cause he was very in­ter­ested in the co­her­ent state es­tim­ates. I in­ten­ded to go for 6 months, but I ended up spend­ing 2 years in Prin­ceton.

El­li­ott gave me a prob­lem to work on. He didn’t tell me that it was a prob­lem on which he had been think­ing hard and not been mak­ing much head­way, so I didn’t real­ize how hard it was. I made no pro­gress on it. I was ab­so­lutely, com­pletely dis­cour­aged, think­ing, “Here I’ve been giv­en this in­cred­ible chance, and I’m just blow­ing it.” I went to see Percy Deift, who was in New York, and told him this. He said, “Well, have you told El­li­ott?” I said, “I don’t know that I can tell him. If I do, I’ll just burst out in tears.” And he said, “Well, then you should tell him, and if you burst out in tears, then you talk through your tears.”

Jack­son: He sug­ges­ted that?

Daubech­ies: Yes, and ac­tu­ally it was very good ad­vice. I did talk to El­li­ott, and I did burst in­to tears, and I did talk through my tears. I told him I was not mak­ing any pro­gress. I said, “Look, I want to do something. I don’t want this to be a big fail­ure.” It was great how El­li­ott re­acted. He sat me down and fetched me a glass of wa­ter. There was some­body else who wanted to see him, and he told them to come back an­oth­er time. And then he gave me a dif­fer­ent pro­ject, and that worked much bet­ter.

I learned a great deal from El­li­ott, be­cause I learned taste in prob­lems. Be­fore, I was work­ing on prob­lems be­cause they were ques­tions that oth­er people had and I wanted to solve them. I hadn’t really thought about a big­ger pic­ture. Now I tell my stu­dents: You should work on something that is of in­terest not just be­cause it was there and you think you can solve it, but be­cause you can see the story you can tell about how it fits in­to a big­ger pic­ture. Per­haps it is something that is known to be tech­nic­ally hard and many people looked to do it, and you can tell the story of what they looked at and did dif­fer­ently. Think about a big­ger edi­fice — don’t just try to pro­duce a little pa­per. That is something I learned in Prin­ceton.

Jack­son: What is “math­em­at­ic­al taste”? What qual­ity do the good, in­ter­est­ing prob­lems have?

Daubech­ies: They are prob­lems that you can in­terest oth­ers in, that will ap­peal to more than just your­self, that mean something to the com­munity. In com­mu­nic­at­ing math­em­at­ics, there are in­tric­a­cies and de­tailed as­pects that are hard to con­vey on the prin­ted page. When you talk with people, there is a whole lot that goes on in body lan­guage, in draw­ings, and in meta­phors that helps com­mu­nic­a­tion. In talk­ing with oth­ers, you learn a view of the land­scape in which you are work­ing. As you learn that land­scape, you see the paths that people have fol­lowed and you start see­ing areas that have not had as much ex­plor­a­tion. Some­times that’s for a good reas­on, namely, there is noth­ing in­ter­est­ing there! Oth­er times you can see that there might be some in­ter­est­ing things to ex­plore. But in or­der to make that men­tal map, you have to im­merse your­self in it. You have to see it as something of which you should make a men­tal map, not just as a heap of prob­lems.

Jack­son: Go­ing back to your postdoc in Prin­ceton — did you work with any­one else while there?

Daubech­ies: I also worked with John Klaud­er, who Alex Gross­mann had sug­ges­ted should be on my PhD thes­is com­mit­tee. John was a the­or­et­ic­al phys­i­cist at Bell Labs who worked on co­her­ent states. He gave me a the­or­et­ic­al phys­ics prob­lem, but I worked on a very math­em­at­ic­al as­pect of it. That was ac­tu­ally very in­ter­est­ing and gave me some pa­pers too. So when I went back to Bel­gi­um, I could look back at a very suc­cess­ful postdoc, where I had worked on two dif­fer­ent top­ics and got­ten good res­ults on both.

This was a time when it was very hard to get ten­ure in Bel­gi­um. We didn’t have po­s­i­tions like as­sist­ant pro­fess­or. You were a re­search­er de­pend­ent on a big pro­fess­or un­til you got ten­ure — and even then, you might still be de­pend­ent.

Jack­son: Could you move up in the same in­sti­tu­tion?

Daubech­ies: Yes. But I was on a fel­low­ships from the Bel­gian equi­val­ent of the Na­tion­al Sci­ence Found­a­tion. I needed that be­cause there were no va­can­cies for per­man­ent po­s­i­tions. There wer­en’t many spots, and there was a lot of back­room deal­ing between uni­versit­ies about them. Our uni­versity was small and would get one of those spots per year. Vari­ous dis­cip­lines with­in a uni­versity would vie with each oth­er, each con­vinced that its can­did­ate was much more im­port­ant than the oth­ers. So I was already think­ing I might look abroad for a job, when I did get a per­man­ent po­s­i­tion.

The birth of wavelets

Jack­son: You were still work­ing in math­em­at­ic­al phys­ics at this time [1984].

Daubech­ies: That’s right, but I had star­ted work­ing with Alex Gross­mann on wave­lets. The trans­ition was kind of seam­less. There were ap­plic­a­tions to sig­nal ana­lys­is rather than phys­ics, but the tools were very sim­il­ar. So this was not for either him or me a big change.

Jack­son: Can you tell me about how you first got in­volved in wave­lets?

Daubech­ies: I went to vis­it Alex when I had got­ten back from the States, and he told me about his work with Jean Mor­let, who was a geo­phys­i­cist. I thought it was very in­ter­est­ing, and there were im­me­di­ately prob­lems to look at. We had the sub­lim­in­al be­lief that there would not be nice or­tho­gon­al bases for wave­lets, be­cause for the win­dowed Four­i­er trans­form, you can show that there are no nice or­tho­gon­al bases. So we built a frame­work in which you could work com­pu­ta­tion­ally very eas­ily with wave­lets. The fam­il­ies of vec­tors we used were not or­tho­gon­al bases; they were not lin­early in­de­pend­ent. We called that “pain­less nonortho­gon­al ex­pan­sions”.

Jack­son: There is a pa­per with that title.2

Daubech­ies: Yes. That’s a joint pa­per between Alex, Yves Mey­er, and my­self, though I hadn’t met Yves Mey­er at that time. Alex said that there were some ideas that he had got­ten from con­ver­sa­tions with Yves Mey­er, and so it should be a triple-au­thor pa­per. Then later I met Yves Mey­er, on a vis­it to Mar­seilles.

Jack­son: Later you did find or­thonor­mal bases for wave­lets. When did you find them?

Daubech­ies: Yves Mey­er had picked up on our sub­lim­in­al be­lief that there were no or­thonor­mal bases. So he tried to see wheth­er he could prove or­thonor­mal bases wer­en’t pos­sible in the wave­let case. He de­rived a whole lot of con­di­tions that would have to be sat­is­fied if there were a basis. Then he found something that sat­is­fied all those con­di­tions, which was kind of mi­ra­cu­lous. You had to have mi­ra­cu­lous can­cel­la­tions for it to work out. The in­ter­pret­a­tion of that changed when Stéphane Mal­lat met Yves and de­veloped with him what be­came mul­tiresol­u­tion ana­lys­is, which is a hier­archy of ap­prox­im­a­tion spaces — you look every time at the dif­fer­ence in­form­a­tion as you go from one ap­prox­im­a­tion to the next.

That in­tro­duced a struc­ture that ex­plained all the mi­ra­cu­lous can­cel­la­tions. But they again had an im­pli­cit as­sump­tion, namely, that you could have a nice fi­nitely sup­por­ted basis only if you gave up smooth­ness — if you wanted smooth­ness, you had to go to an in­fin­itely sup­por­ted basis. In­deed, if you start with spline ap­prox­im­a­tion spaces, the bases have to be in­fin­itely sup­por­ted. That means you have to trun­cate, and it be­comes cum­ber­some.

I said, “Look, I want those nice al­gorithms, but I don’t want it to be something trun­cated from a much nicer, in­fin­ite basis of sup­port. Can I make it fi­nite?” And the an­swer was yes. Over a peri­od of a few months I worked very hard to make it work.

Jack­son: What was the key idea? How did you fi­nally solve the prob­lem?

Daubech­ies: There were people in com­puter vis­ion who had already been look­ing at fil­ters where you in­ter­pol­ate and re­fine scale after scale, but with a fi­nite num­ber of fil­ter coef­fi­cients. They wanted smooth­ness, and they had found em­pir­ic­ally that with some fil­ters it worked much bet­ter than oth­ers. They were won­der­ing, what makes them so spe­cial? How can you get smooth­ness? How can you prove that you get that smooth­ness?

So I looked at that prob­lem. At the same time, mul­tiresol­u­tion ana­lys­is came, and I could see that a sim­il­ar mech­an­ism was at work there. So I built in a tool that was go­ing to give me smooth­ness with fi­nite fil­ters. But I didn’t know yet that it was go­ing to cor­res­pond to the func­tion­al, ana­lyt­ic frame­work. Maybe I could build the fil­ters, but did they mean something in terms of func­tion spaces? I put the whole thing on its head. In­stead of start­ing from a basis and then de­riv­ing fil­ters from them, I said, “I want those fil­ters, and I want them to be fi­nite. What con­di­tions should they sat­is­fy?” I found those con­di­tions, and then I back-tracked. It took a while to prove it, but it all worked.3

Jack­son: How did you come in­to con­tact with the people in com­puter vis­ion?

Daubech­ies: I was vis­it­ing Cour­ant be­cause I had met my hus­band Robert [Calderb­ank] in the mean­time. We had met at a sci­entif­ic meet­ing in Bel­gi­um. Robert was at Bell Labs, so I wrote to Percy Deift at Cour­ant and asked about a vis­it­ing po­s­i­tion there. I ap­plied and got the po­s­i­tion. At Cour­ant I talked with people in com­puter vis­ion.

Echoes from everywhere

Jack­son: When people star­ted de­vel­op­ing the the­ory of wave­lets, they real­ized there were a lot of echoes of wave­lets in pure math­em­at­ics.

Daubech­ies: Yes — ac­tu­ally we knew that as soon as Yves Mey­er got in­to the busi­ness, be­cause he knew that whole back­ground in pure math­em­at­ics. In fact there were echoes from every­where, from co­her­ent states in quantum phys­ics, co­her­ent states in grav­it­a­tion, in com­puter vis­ion, in elec­tric­al en­gin­eer­ing, with what are called quad­rat­ure mir­ror fil­ters. Many dif­fer­ent things linked to­geth­er, and of course wave­lets also tied in very nicely with work in har­mon­ic ana­lys­is. But I still be­lieve that the power­ful stuff would not have spilled out from that field without the in­put from all these oth­er dis­cip­lines. It’s be­cause Alex Gross­mann had found a way of look­ing at what Jean Mor­let was do­ing in geo­phys­ics and built the­ory for it. That the­ory then turned out to be re­lated to oth­er math­em­at­ic­al de­vel­op­ments, and then we could use that whole ma­chinery to prove the­or­ems about wave­lets. Har­mon­ic ana­lysts had the the­or­ems in their little corner of math­em­at­ics. I don’t know that any­body would ever have pen­et­rated to all the rest, if it hadn’t been for Alex Gross­mann.

Jack­son: He was an im­port­ant bridge-build­er. How was he able to do that?

Daubech­ies: He knew many people, like Jean Mor­let. Mor­let had tools he had cre­ated for sig­nal pro­cessing for oil ex­plor­a­tion. He was not very ar­tic­u­late, and for me it was im­possible to un­der­stand him. But Alex did un­der­stand him, and he put Mor­let’s ideas in­to a frame­work that made them un­der­stand­able to many oth­er people.

The idea of us­ing dif­fer­ent scales is used har­mon­ic ana­lys­is. Alex dis­cussed this with Yves Mey­er, who real­ized there was something new there. There were oth­er people who said, “They just re­dis­covered something that we knew all along” — which is the stand­ard way of re­act­ing in situ­ations like that. But Yves Mey­er said, “Yes, in a cer­tain sense they are writ­ing things we knew all along, but they write them in a dif­fer­ent way and read the for­mu­las dif­fer­ently from how we read them.”

From a co­her­ent states point of view, what you were do­ing was tak­ing in­ner products with little build­ing blocks, which had two la­bels, and then re­con­struct­ing a mix of po­s­i­tion with those coef­fi­cients that had two la­bels. You had build­ing blocks that you su­per­posed in or­der to build oth­er stuff. By con­trast in har­mon­ic ana­lys­is, they were con­volving with things of dif­fer­ent scales. So the two la­bels, which were very ex­pli­cit in the way Alex Gross­man and Jean Mor­let wrote things, were hid­den in the way har­mon­ic ana­lysts worked — they had one para­met­er, which was scal­ing, and the oth­er para­met­er was hid­den in the con­vo­lu­tion. There was not that same view of de­com­pos­ing in­to build­ing blocks and writ­ing the sum of build­ing blocks. So try­ing to do it dis­cretely and find­ing an or­thonor­mal basis would have been much less ob­vi­ous. But it later turned out that, pri­or to Mey­er’s work, a har­mon­ic ana­lyst, Jan-Olav Strömberg, had built an or­thonor­mal basis, al­though nobody ac­tu­ally paid at­ten­tion to it.4

Jack­son: In an in­ter­view,5 Mey­er said he went to a talk by Strömberg on this sub­ject but was think­ing about something else at the time and didn’t pick up on what Strömberg was say­ing.

Daubech­ies: Yes, with hind­sight, he should have. He only real­ized that later.

Jack­son: Your 1988 pa­per giv­ing an or­thonor­mal basis for wave­lets con­tains a lot of tables of coef­fi­cients. Those tables would not or­din­ar­ily ap­pear in a math pa­per. How did that hap­pen?

Daubech­ies: As you said there were echoes of wave­lets in many dif­fer­ent dis­cip­lines, in­clud­ing elec­tric­al en­gin­eer­ing, which I be­came aware of only as I was writ­ing up the pa­per. I wanted these tools to be used, and I knew that the en­gin­eers wer­en’t go­ing to read all the math­em­at­ic­al the­or­ems I was de­riv­ing. But I thought that if I provided tables that were fa­mil­i­ar to en­gin­eers, they would try it. I in­sisted on those tables — not just provid­ing a way of com­put­ing such tables, but the ac­tu­al tables. I also wanted the fig­ures. The pa­per was very long, and journ­als don’t like long pa­pers. But I in­sisted, and I am glad that the journ­al agreed be­cause the tables and fig­ures made the pa­per much more im­pact­ful. I knew this be­cause I had been look­ing at the en­gin­eer­ing lit­er­at­ure.

Jack­son: Was that part of the ap­peal of work­ing on wave­lets, to reach re­search­ers out­side of math­em­at­ics?

Daubech­ies: For me that’s part of the ap­peal. I didn’t know it then, but I know it now, that I really love talk­ing with people in dif­fer­ent fields and find­ing the right math­em­at­ic­al frame­work to think about their prob­lems. The ap­proach is not: these are the tools I have, so let me trans­form the prob­lem in­to something in which I can use those tools. In­stead, the ap­proach is: this is what they care about, so what kind of math­em­at­ics do I know of that would fit this? And then you can start learn­ing the math­em­at­ics.

In ad­di­tion to sig­nal ana­lys­is, I now also work on prob­lems in shape re­cog­ni­tion, where we need dif­fer­en­tial geo­metry. I hadn’t done dif­fer­en­tial geo­metry since gradu­ate school, and that was a while ago! So I had to learn more dif­fer­en­tial geo­metry. But if the prob­lem re­quires it, then that’s what you do.

Once Eu­genio Calabi was on a vis­it­ing com­mit­tee at Prin­ceton, and he said, “Do you know what the dif­fer­ence is between a pure math­em­atician and an ap­plied math­em­atician?” And I said, “Pro­fess­or Calabi, I am very in­ter­ested to hear how you would an­swer that!” He said, “A pure math­em­atician, when work­ing on a prob­lem and get­ting really stuck, de­cides to nar­row the scope a little bit, in or­der to get out of the prob­lem. An ap­plied math­em­atician, when get­ting stuck, says: time to learn more math­em­at­ics!” I really feel that’s true — not the char­ac­ter­iz­a­tion of the pure math­em­atician, but the char­ac­ter­iz­a­tion of the ap­plied math­em­atician.

Babies and Bell Labs

Jack­son: You gave a series of CBMS [Con­fer­ence Board of the Math­em­at­ic­al Sci­ences] lec­tures in 1990, which be­came your book Ten Lec­tures on Wave­lets.6 Can you tell me about the CBMS con­fer­ence?

Daubech­ies: The per­son who had pro­posed me as the speak­er was Mary Beth Ruskai. I met her for the first time just after my PhD de­fense. She was in math phys­ics. I met up with her again when I was a postdoc at Prin­ceton. In the CBMS con­fer­ence format, the main speak­er gives sev­er­al morn­ing lec­tures, and then there are lec­tures by oth­ers in the af­ter­noon. That worked out very nicely for the sub­ject of wave­lets then, be­cause there were so many dif­fer­ent dir­ec­tions in which wave­lets were spread­ing and from which they were get­ting in­put.

I wrote most of that book while I was preg­nant with my daugh­ter and im­me­di­ately after she was born. Ac­tu­ally, I no­ticed something when I had my first child, my son, in 1988. After I gave birth, I found that there was a peri­od of sev­er­al months, something like 6 months, where I couldn’t do new, cre­at­ive work math­em­at­ic­ally. I was in an in­cred­ible pan­ic, be­cause I thought, “Oh my god, this is how I make my liv­ing, and it’s gone.” I didn’t tell any­body, be­cause I was too afraid. But it did come back, and I was re­lieved and could then talk about it. I ex­pec­ted this might hap­pen again when I had my daugh­ter, and it did. No new cre­at­ive ideas about math­em­at­ics would come, but I found I could still have cre­at­ive ideas about writ­ing. So I wrote a good part of the book after in the 6 months after she was born.

Jack­son: Did you have no math­em­at­ic­al ideas just be­cause you were tired, or was it more than that?

Daubech­ies: I have no idea. I would still get fant­ast­ic math­em­at­ic­al ideas when I was nurs­ing, so it was not the pre­oc­cu­pa­tion with the child. It was just that for about 6 months after de­liv­ery, no new math­em­at­ic­al ideas would come. I have since told oth­er wo­men, “Look, this cer­tainly hasn’t happened to every­body I know, but it happened to me, so if it hap­pens to you, don’t worry about it — I pan­icked the first time it happened, but it did come back.”

Jack­son: In 1994 you went to Prin­ceton and were the first wo­man to be in the math de­part­ment there.

Daubech­ies: The first ten­ured wo­man — there had been wo­men as­sist­ant pro­fess­ors be­fore. When I ac­cep­ted the po­s­i­tion, I wanted to start with a leave, be­cause I wanted to fin­ish some pro­jects while I was still at Bell Labs. And I had a small baby. So I ac­cep­ted in 1993, took leave, and star­ted in 1994.

Jack­son: Was that when Bell Labs broke up?

Daubech­ies: It was a little bit after. The break­up in­to AT&T and Lu­cent is something that my hus­band lived through but I had already left by then.

Jack­son: How was the at­mo­sphere at Bell Labs?

Daubech­ies: I liked it a lot. It’s not be­cause the at­mo­sphere wasn’t good that I left. It’s be­cause I was con­cerned that it was too good to be true and wouldn’t last! Also, I missed teach­ing and missed be­ing in a place where there were people oth­er than sci­ent­ists or en­gin­eers. The irony is, once I was at Prin­ceton, it took me 10 years to have enough time in my sched­ule to ac­tu­ally meet with people oth­er than sci­ent­ists and en­gin­eers!

But Bell Labs had a won­der­ful at­mo­sphere. It was nice and friendly, and it was con­du­cive to in­ter­est­ing col­lab­or­a­tions. In fact I was very dis­ap­poin­ted when I went to aca­demia that people seemed to work to­geth­er less than at Bell Labs. People don’t of­ten col­lab­or­ate with­in de­part­ments but in­stead have col­lab­or­at­ors else­where. At Bell Labs, man­age­ment put some thought in­to en­cour­aging col­lab­or­a­tions. Math­em­at­ics was split in­to smal­ler de­part­ments, which were more ho­mo­gen­eous re­search-wise, but you were nev­er in an of­fice sit­ting next to some­body from the same de­part­ment. The staff was mixed to­geth­er a bit.

Bell Labs had a tra­di­tion star­ted by Henry Land­au. Every Thursday just after lunch there was Henry’s Sem­in­ar, which was a prob­lem sem­in­ar. It was very in­form­al. Some­times it would be a vis­it­or, some­times it would just be the Bell Labs math­em­aticians them­selves. The idea was that you would give some back­ground on a prob­lem that you hadn’t solved, and then there would be dis­cus­sion. It was through a talk I gave in Henry’s Sem­in­ar that I star­ted a col­lab­or­a­tion with Jeff Lagari­as, which led to sev­er­al pa­pers. Every­body went to Henry’s sem­in­ar, and there you met people from all kinds of dis­cip­lines.

It was rumored — and I later found out this was true — that when Bell Labs did salary re­views, re­search­ers would get ad­di­tion­al cred­it for col­lab­or­a­tions. So if two re­search­ers wrote one pa­per to­geth­er, each would get more than half cred­it for that pa­per. There was nev­er any bick­er­ing about “Did this per­son really con­trib­ute to the pa­per?”, which there some­times is in uni­versit­ies. There was much more col­lab­or­a­tion between people with­in the or­gan­iz­a­tion at Bell Labs than I have seen in any aca­dem­ic de­part­ment.

The mystery of deep neural networks

Jack­son: Do you have a fa­vor­ite ap­plic­a­tion of wave­lets, something that you found es­pe­cially sur­pris­ing, un­usu­al, beau­ti­ful…?

Daubech­ies: I don’t know — by now, wave­lets have be­come such a stand­ard tool. Wave­lets are a very nat­ur­al way to do things, though in a sense, they are nat­ur­al only when you have com­pu­ta­tion­al tools. You can graph wave­lets, you can de­rive many prop­er­ties of them, you can prove all kinds of things about them, but you don’t have a closed-form for­mula for them. You graph them com­pu­ta­tion­ally, you com­pute things on a dense lat­tice, and you plot the in­di­vidu­al points, and the plot is close to the graph of a con­tinu­ous func­tion, so it’s mean­ing­ful. But they are not ei­gen­func­tions of some dif­fer­en­tial op­er­at­or. They are a dif­fer­ent way of build­ing spe­cial func­tions.

There are oth­er ways too, like com­pressed sens­ing and dic­tion­ary learn­ing, where you try to find from data the ways in which you want to com­pute with them. Now we have deep neur­al net­works and the mir­acle of how they work, and we are still try­ing to un­der­stand that.

Jack­son: Are you work­ing on deep neur­al net­works?

Daubech­ies: Yes, I have been work­ing with Ron DeVore on this.

Jack­son: It’s not really known why deep neur­al net­works work. Do you think there is a math­em­at­ic­al the­ory that could be de­veloped to ex­plain that?

Daubech­ies: Oh, ab­so­lutely. If things work, they work for a reas­on. We just don’t know the reas­on yet. I think math­em­aticians should be work­ing on this. No ana­logy is per­fect, but when Four­i­er trans­forms were first de­veloped, people were very sus­pi­cious of them — why did they work? There was a reas­on they worked, and we know the reas­on now.

We math­em­aticians are the people who train ourselves to get to the mar­row of why things work. I get very im­pa­tient when I hear math­em­aticians say, “Oh, neur­al net­works are not something we should think about, it’s not pretty, it’s all dirty en­gin­eer­ing stuff.” But there is something go­ing on there. Earli­er mod­els did not ob­tain these res­ults, and new mod­els do. There must be good reas­ons for that. Once we un­der­stand those reas­ons, we will be able to do much more.

Jack­son: Do you have any ink­lings about what kind of math­em­at­ics might work?

Daubech­ies: If I did, I would have done it already! I think it will take a while. But that’s not a reas­on not to work on it.

Jack­son: There are eth­ic­al ques­tions that come up when deep neur­al net­works are used to make de­cisions about, for ex­ample, deny­ing people in­sur­ance. What do you think about such eth­ic­al ques­tions?

Daubech­ies: There are many situ­ations where you trust cer­tain people more than oth­er people be­cause ex­per­i­ence has shown that they are more of­ten right, and more of­ten make the right de­cision. Sim­il­arly, you should use net­works only if they have been thor­oughly vet­ted for the kind of prob­lem you are go­ing to ap­ply them to and make mo­ment­ous de­cisions with. You shouldn’t do that lightly. I wish that we would un­der­stand bet­ter what deep neur­al net­works do be­fore us­ing them, but I am not say­ing we should not use them un­til we un­der­stand. They have really made an im­mense, quant­it­at­ive dif­fer­ence in areas where we had struggled for quite a while. But a deep neur­al net is not a sil­ver bul­let. For every ques­tion where you are go­ing to really ap­ply them, you must test them thor­oughly. We have a com­pu­ta­tion­al black box, of which we know too little, and we have to study and un­der­stand it bet­ter. But you can do things with it already. And as for prob­lems they might cause — we have seen the stock mar­ket col­lapse be­cause of com­pu­ta­tion­al is­sues long be­fore there were deep net­works!

Art restoration: How cool is that?

Jack­son: Can you tell me about your re­cent work with art res­tor­a­tion?

Daubech­ies: That uses im­age pro­cessing to help with art res­tor­a­tion or con­ser­va­tion. One of my re­cent pro­jects was a vir­tu­al re­ju­ven­a­tion of an al­tarpiece painted by Frances­cuc­cio Ghis­si. This is not something mu­seums would nor­mally have done, but it made a dif­fer­ent ex­per­i­ence for the mu­seum vis­it­ors. In that case we be­came part of an ex­hib­it — the ex­hib­it ac­tu­ally changed nature through what we did and be­came a much more in­ter­est­ing and much more vis­ited ex­hib­i­tion.7

Again, part of this work is listen­ing to the con­cerns the con­ser­vat­ors have and dis­tilling in­ter­est­ing pro­jects out of that. Some­times these pro­jects are not as deeply math­em­at­ic­al as in, say, shape re­cog­ni­tion, but there are still in­ter­est­ing ques­tions. For in­stance, right now we are work­ing with the Na­tion­al Gal­lery on a fam­ous por­trait by Goya of a wo­man in a man­tilla.8 She is sit­ting there very proud, but if you look at the can­vas with x-rays, with in­frared, or with x-ray fluor­es­cence, you see a ghost of a dif­fer­ent por­trait un­der­neath. You would like to vir­tu­ally “peel off” the top por­trait and get a bet­ter view of the one un­der­neath.

In some cases of paint­ings on wooden planks, 19th and early 20th cen­tury con­ser­vat­ors planed the plank down un­til it be­came quite thin (for in­stance, to re­move wood de­cay or worm dam­age), after which a lat­tice­work of hard­wood would be put onto the back to make it more ri­gid again. If you x-ray something like that, all you see is the lat­tice­work, so you can’t for ex­ample ana­lyze brush-strokes. But you can try im­age pro­cessing to first re­move the lat­tice­work ef­fect from the x-ray. Some­times that means ad­apt­ing ex­ist­ing tools, and some­times the prob­lem is so chal­len­ging that it re­quires a com­pletely dif­fer­ent ap­proach. There are in­ter­est­ing ques­tions, and if you have good an­swers, they will have an im­pact — like be­com­ing part of an ex­hib­i­tion. How cool is that?

These prob­lems are chal­len­ging for the un­der­grads, gradu­ate stu­dents, and postdocs who work on them. I also like this area be­cause the re­search is not mo­tiv­ated by com­mer­cial in­terests. A lot of com­puter graph­ics is mo­tiv­ated by video game de­vel­op­ment and things like that. Art con­ser­va­tion has less money for soft­ware de­vel­op­ment. The work has a big im­pact, but it is not a com­mer­cial im­pact. We typ­ic­ally ask for a grant to de­vel­op a pro­to­type and then pay pro­fes­sion­al de­velopers to port it to something that can be used by art con­ser­vat­ors. If you wanted to get your in­vest­ment back by selling it, that’s not go­ing to hap­pen. So we very much like to do this with­in the open-source frame­work.

Jack­son: How did you make the con­nec­tion with the mu­seums?

Daubech­ies: Rick John­son, who is an en­gin­eer at Cor­nell, has al­ways been in­ter­ested in art and no­ticed that they didn’t use im­age pro­cessing. So he brokered a deal with the Van Gogh Mu­seum, say­ing that if they made data avail­able, he would find im­age pro­cessing teams who could show what they can do with that data. So we got data about Van Gogh’s paint­ings and tried to ana­lyze things like brush­work, com­pos­i­tion, and col­or choices.9 That pro­duced enough res­ults that people then came with more ques­tions.

I really feel you can find al­most every­where in­ter­est­ing ques­tions on which a math­em­at­ic­al ap­proach has in­ter­est­ing things to say. You want to find the es­sen­tial bones of the prob­lem, to find the skel­et­on of it.

Jack­son: Since you got your PhD, ap­plied math­em­at­ics has changed greatly. Can you tell me about the changes that you have seen?

Daubech­ies: The very first time I sent a pro­pos­al to NSF, I sent it to Of­fice of Ap­plied Math­em­at­ics. I didn’t even get any re­views of it. I just got an as­sess­ment say­ing, “Well, it looks very in­ter­est­ing, but it’s not ap­plied math­em­at­ics.” We are talk­ing here late 1980s, early 1990s, and even then it was an old-fash­ioned point of view. That as­sess­ment saw ap­plied math­em­at­ics only as cer­tain types of res­ults con­cern­ing cer­tain types of PDEs. And I def­in­itely wasn’t do­ing that. I’ve nev­er been a PDE per­son.

Today many few­er young math­em­aticians are hung up on the di­vi­sion between pure and ap­plied math­em­at­ics. I don’t really see the di­vi­sion either. The ex­cite­ment, the pleas­ure, and the en­gage­ment I feel are ex­actly the same for an ap­plied pro­ject or a much purer pro­ject. There is no dif­fer­ence. People who have a Pla­ton­ic point of view that pure math­em­at­ics is something that ex­ists to be dis­covered seem to feel that ap­plied math­em­at­ics is something you build, and pure math­em­at­ics is dis­covered. I don’t be­lieve that. I be­lieve all math­em­at­ics is something we build in or­der to make sense of things we in­tu­it or ob­serve, or of ana­lo­gies or frame­works that we see in dif­fer­ent con­texts.

Broadening the appeal of mathematics

Jack­son: Can you tell me about the Math Alive course you de­veloped at Prin­ceton?

Daubech­ies: The idea is that, in the Amer­ic­an uni­versity sys­tem, most stu­dents are re­quired to take one math course. If they are not in­ter­ested in be­com­ing sci­ent­ists or math­em­aticians, they of­ten take a low-level cal­cu­lus course. But at a place like Prin­ceton, stu­dents have had cal­cu­lus be­fore. Those who take it just to ful­fill the re­quire­ment usu­ally hated cal­cu­lus in high school and hate it all over again. And if you think about it, cal­cu­lus is not the right vehicle for this re­quire­ment. Cal­cu­lus is de­signed to teach you lots of tech­niques for oth­er stuff that you are go­ing to do in math­em­at­ics or in sci­ence. Low-level courses do a few ap­plic­a­tions of cal­cu­lus, but they are fake ap­plic­a­tions. They feel fake too, so stu­dents are asked to sus­pend their dis­be­lief. And stu­dents who are not go­ing to go in­to math­em­at­ics or sci­ence see no reas­on to sus­pend dis­be­lief.

I felt we should do the com­plete op­pos­ite. We should make a course in which you get stu­dents to see a vari­ety of dif­fer­ent kinds of math­em­at­ics and make them reas­on, without try­ing to teach skills, so that they would get an im­pres­sion of what math­em­at­ics is about. They don’t need the tech­nic­al skills, but the abil­ity to reas­on and fig­ure something out just by think­ing about it — that is a skill that is in­cred­ibly power­ful and can be used in many dif­fer­ent con­texts. That’s what math­em­at­ics is about — see­ing pat­terns in one con­text that you can use in a dif­fer­ent con­text.

I talked about this with Henry Pol­lak, who was then re­tired from Bell Labs but who still oc­ca­sion­ally came there. He told me about vari­ous books, like For all prac­tic­al pur­poses, which had a lot of the top­ics that I wanted to do. But I was not happy with the math­em­at­ic­al depth. The idea was not to just do tour­ism. I wanted to have a mod­u­lar course that would vis­it dif­fer­ent top­ics and really re­quire stu­dents to do some math­em­at­ic­al reas­on­ing, not just to see it.

So for in­stance, we make them ex­per­i­ence RSA and how factor­ing large num­bers be­comes more dif­fi­cult as the num­ber gets big­ger and big­ger. We provide a pro­gram that does the factor­ing, and they are asked to put in products of big­ger and big­ger primes. They can then see that it takes more and more time. We also design small codes that the stu­dents have to break.

That course was very suc­cess­ful, al­though nobody in the math de­part­ment liked to teach it! So I taught it, and when I went on leave it wasn’t offered, even though there was a big de­mand. Now the Prin­ceton math de­part­ment brings some­body in for 3 months a year, who did it as a postdoc and liked teach­ing it. He’s now at Ox­ford, but he comes to Prin­ceton for 3 months a year to teach this course.

Jack­son: Nobody in the math de­part­ment would take it on?

Daubech­ies: Some­body like Man­jul Bhar­gava might have done it, but Man­jul de­veloped his own course, which is more pure math, for that same pub­lic. I found a big dif­fer­ence at Duke. When I came to Duke, I told people I had de­veloped this course, which we now call Math Every­where, and offered to teach it. There were im­me­di­ately oth­er fac­ulty mem­bers who were in­ter­ested in teach­ing it, and ac­tu­ally I haven’t taught it my­self for sev­er­al years now. At Prin­ceton, the math fac­ulty was per­fectly okay with the course be­ing taught by some­body as long as it was not they who had to do it.

At some point, when Charles Fef­fer­man was de­part­ment chair at Prin­ceton, he asked me to put to­geth­er a com­mit­tee on broad­en­ing the ap­peal of the math ma­jor. When we star­ted, I told the com­mit­tee that it was really im­port­ant to do this, be­cause we had to coun­ter­act the repu­ta­tion the math de­part­ment had. People asked, “What repu­ta­tion is that?” And I said, “It’s the repu­ta­tion that we are in­ter­ested only in un­der­gradu­ates who already know that they are go­ing to be­come pro­fes­sion­al re­search math­em­aticians.” And they said, “Isn’t that the case?” Well, that was ex­actly the prob­lem! If you are in­ter­ested only in teach­ing people who will be­come pro­fes­sion­al math­em­aticians, then you should have not many more people in a math de­part­ment than you have in a French lit­er­at­ure de­part­ment.

Jack­son: The at­ti­tudes at Duke are dif­fer­ent?

Daubech­ies: Yes, they are very dif­fer­ent. They take ser­i­ously the im­port­ance of teach­ing math­em­at­ics to people who need to un­der­stand it but who might not need to go in­to all of the de­tails that a math ma­jor would need.

Fiber bundles, connections, and lemur teeth

Jack­son: Earli­er you men­tioned your work in shape re­cog­ni­tion. Can you tell me about that?

Daubech­ies: Re­search­ers in bio­lo­gic­al mor­pho­logy want to be able to com­pare the shape of teeth and bones and to quanti­fy sim­il­ar­it­ies and dif­fer­ences. On these shapes they find points that are ho­mo­log­ous, that cor­res­pond on dif­fer­ent sur­faces. Once they have say 20 of those points, they take that 20-tuple from one ob­ject and move it by ri­gid trans­form­a­tions to lie as close as pos­sible to the cor­res­pond­ing points on a dif­fer­ent sur­face. Then they look at the sum of the squares of the dis­tances between cor­res­pond­ing points. That’s called the Pro­crustes dis­tance, and they use it to do stat­ist­ics. They want to be able to do this in a more auto­mat­ic way, so that it would be less sub­ject­ive and would take less time. So for the last ten years or so, we have been work­ing on this.

We wanted to in­cor­por­ate in­to the ana­lys­is the fact that there is a whole col­lec­tion, so that we don’t com­pare only pairs of teeth. Bio­lo­gists sel­dom com­pare just pairs; typ­ic­ally they con­sider a whole col­lec­tion when they place their land­marks, and they draw on their whole back­ground know­ledge, which they get from hav­ing stud­ied and un­der­stood an even lar­ger col­lec­tion. We wanted to in­cor­por­ate that know­ledge.

The col­lec­tion is thought of as a man­i­fold, in which each point is one of the teeth in the col­lec­tion. You might have little neigh­bor­hoods that con­tain very closely re­lated in­di­vidu­als, and then lar­ger neigh­bor­hoods that con­tain the spe­cies, and even lar­ger neigh­bor­hoods con­tain­ing re­lated spe­cies, and so on. But we do be­lieve there is enough small vari­ation that you really have a man­i­fold-like ob­ject, or maybe a uni­on of man­i­folds. That’s the man­i­fold we were try­ing to de­scribe and on which we want to un­der­stand the Pro­crustes dis­tances.

But if you think a bit fur­ther — and that’s a step that one of my gradu­ate stu­dents at Duke did — you have a base man­i­fold, which is the man­i­fold on which each one of these teeth is one point, but each one of them is really a little sur­face. So you have a fiber bundle. There is a map­ping from fiber to fiber, so you have a con­nec­tion. This whole lan­guage of fiber bundles, which comes from dif­fer­en­tial geo­metry, be­comes very nat­ur­al to de­scribe what’s go­ing on.

The gradu­ate stu­dent I men­tioned, Tin­gran Gao — who is now a Kruskal In­struct­or at Uni­versity of Chica­go — real­ized that the fiber bundle frame­work can help de-noise data. The reas­on is that, if you have a con­text for the data, you can de-noise it in a much more ef­fect­ive and ac­cur­ate way.

Jack­son: Are an­thro­po­lo­gists us­ing these tools?

Daubech­ies: Yes, evol­u­tion­ary an­thro­po­lo­gists. For the past sev­er­al years, we have met with them more or less weekly. We have used the tools in par­tic­u­lar to ana­lyze data about lemur teeth.10

Jack­son: Have they learned the math­em­at­ics be­hind the tools?

Daubech­ies: They have learned enough math­em­at­ics that they can par­ti­cip­ate in dis­cus­sions and we can ex­plain some things to them. They don’t want to learn any of the com­plic­ated tech­niques, just like we don’t want to learn all the names of all the bones! But we listen enough to each oth­er to be able to have a dia­logue.

“We are not going to worry about this”

Jack­son: How would you char­ac­ter­ize your math­em­at­ic­al think­ing? Do you think geo­met­ric­ally? Or do you think in terms of ana­lys­is?

Daubech­ies: I have a visu­al way of think­ing. I make visu­al meta­phors. I kick things around, I move things — I don’t know wheth­er that is geo­met­ric think­ing, it may not be.

Jack­son: Do you find you need to write things down?

Daubech­ies: Yes, though not al­ways in words. But I do find I need to write. To un­der­stand things, I write and scribble and doodle and re­write. To help me con­cen­trate I start by writ­ing again in ab­bre­vi­ated form what I already know.

As a child my daugh­ter was dia­gnosed with AD­HD [at­ten­tion-de­fi­cit and hy­per­activ­ity dis­order]. When I saw the tests that the eval­u­at­or made her do, and he ex­plained to me how the res­ults led to the dia­gnos­is, I said, “But I think I do some of these things.” The eval­u­at­or ques­tioned me a bit, and then said, “This is not a dia­gnos­is, be­cause that would need much more time, but I think it is in­deed prob­able that you have AD­HD too.” Then I turned to my daugh­ter and said, “We are not go­ing to worry about this.”

I have lots of things that flit through my head — and ac­tu­ally I find that it gives me much more hu­mor in life! But it also means that when I want to con­cen­trate, I need to turn it off and to de­cide, “Now I am go­ing to con­cen­trate.” I can con­cen­trate in­cred­ibly deeply then, but I need to de­cide to turn it on. It’s not that I have no at­ten­tion. Ran­dom things come through my head, and I find that is ac­tu­ally use­ful, though I un­der­stand that if I had it to a much more pro­nounced ex­tent, it would not be use­ful.

Jack­son: How did your daugh­ter end up? What does she do now?

Daubech­ies: She’s a data ana­lyst for a chain of dis­count stores. She ma­jored in math­em­at­ics, but she likes stat­ist­ics and pro­gram­ming a lot, and she is us­ing all those skills and prob­lem-solv­ing.

Jack­son: So she has no prob­lem with con­cen­tra­tion.

Daubech­ies: No, ab­so­lutely not.

Jack­son: Why was she brought in for eval­u­ation in the first place?

Daubech­ies: Be­cause in class she was of­ten dis­trac­ted, and the teach­er said she might need to be tested. It was not that AD­HD was mak­ing it im­possible for her to con­cen­trate. She can de­cide to con­cen­trate, and then she is do­ing fine. She just needed to get enough stim­u­la­tion to de­cide that it was in­ter­est­ing to con­cen­trate.

Jack­son: And your son? Did he have any­thing like that?

Daubech­ies: No, not at all. My son is much more like my hus­band. But he ma­jored in math­em­at­ics too — also to our sur­prise. I thought our daugh­ter would ma­jor in bio­logy, and our son in fin­ance or eco­nom­ics. But they both picked math­em­at­ics. I’m very proud of my son — well, I am very proud of both my chil­dren — but our son be­came a high school teach­er in math­em­at­ics in an in­ner city school in Chica­go on the south side. I’m very proud of that.

Jack­son: That’s not an easy job.

Daubech­ies: No, it isn’t. He really likes the job and has been do­ing it for 8 years now. The thing he finds a prob­lem is the at­ti­tude of the city of Chica­go and the state of Illinois to­wards teach­ers.

Jack­son: Teach­ers who have a good back­ground in math­em­at­ics are very much needed.

Daubech­ies: Yes. He is a very smart kid. We were still at Prin­ceton when he de­cided to be­come a teach­er. People would ask me, “So what is your son do­ing?” They knew he had gone to the Uni­versity of Chica­go. When I said he was go­ing to be a high school teach­er, you would see this fleet­ing ex­pres­sion on their faces, as if they were think­ing, “Oh my, but that was a smart kid, how did this go astray?” And I would get very an­noyed!

Jack­son: I re­mem­ber when I first met you, your chil­dren were small. You said, “I don’t do any­thing be­sides math­em­at­ics and tak­ing care of my fam­ily.”

Daubech­ies: That was the case then. And I had a hus­band who did the same. Some­times wo­men ask me how I have man­aged hav­ing a ca­reer and a fam­ily. I tell them that one thing that really helps is to choose your hus­band well. You have to talk about it be­fore­hand and see what his im­pli­cit as­sump­tions are. Be­fore Robert, I had an­oth­er long re­la­tion­ship with some­body who liked hav­ing an in­ter­est­ing fiancée, but who ac­tu­ally wanted his wife to be much less in­ter­est­ing. He hadn’t ar­tic­u­lated that — if he had, then of course we would nev­er have had a re­la­tion­ship! But that’s what it amoun­ted to.

Jack­son: But you didn’t marry that per­son.

Daubech­ies: No. In Bel­gi­um at the time, wo­men lost some civil rights when they got mar­ried. For in­stance, a single wo­man could open a bank ac­count by her­self if she were over 18, but if she was mar­ried, then she needed the au­thor­iz­a­tion of her hus­band — or he would at least be no­ti­fied. I didn’t feel that was something I was go­ing to sub­ject my­self to.

Jack­son: You have had a long ca­reer in math­em­at­ics. What are your cur­rent thoughts about the status of wo­men in the field?

Daubech­ies: First of all, to me it’s ob­vi­ous that it’s a ques­tion more of cul­ture and at­mo­sphere than of in­nate tal­ent or in­terest. When you look at a map of Europe and con­sider the per­cent­age of wo­men in aca­dem­ic jobs in math­em­at­ics, coun­try by coun­try, that per­cent­age var­ies so much from one coun­try to the next, even though the ge­net­ics don’t really change so much. But the at­mo­sphere does. An­oth­er thing that changes is how well re­mu­ner­ated an aca­dem­ic job is, and also how much prestige it has. So for in­stance, in Por­tugal, where there is little money and not much prestige, you have lots of wo­men. In Switzer­land, where there is lots of money and lots of prestige, you have al­most none. It might not be so stark every­where, but still I think that over­all this pic­ture is cor­rect.

At­mo­sphere does play a role, and I think it could be stud­ied. Dif­fer­ent sub­fields with­in math­em­at­ics have dif­fer­ent num­bers of wo­men. I won­der if a so­cial sci­ent­ist who really knows how to study friend­li­ness, level of col­lab­or­a­tion, and so on, would find a cor­rel­a­tion there. But you still would not know if it is cause or ef­fect. I think it is clearly something cul­tur­al, and it is more subtle than just say­ing, “We want more wo­men.” If wo­men choose less of­ten to go in­to aca­dem­ic jobs in math­em­at­ics, it’s not be­cause they don’t like math­em­at­ics, but it’s be­cause with math­em­at­ics you can do so many more things than just go­ing in­to aca­demia in math­em­at­ics. And maybe some of these oth­er things are more at­tract­ive to them.

Jack­son: One math­em­atician I talked to said that wo­men look at math­em­aticians as a group and see many nerdy people and say, “I don’t want to be in that group.”

Daubech­ies: But if you look at all the math­em­aticians you know, don’t you think that the very nerdy ones are a minor­ity?

Jack­son: Yes — and the guy who said this was def­in­itely not nerdy!

Daubech­ies: I think math­em­at­ics is a field where you have a high­er in­cid­ence of suc­cess­ful nerds than in many oth­er fields, be­cause we are tol­er­ant of it. But that doesn’t mean be­ing a nerd is a pre­requis­ite, not by a long shot. It’s still something you have to over­come. You can over­come it bet­ter in math­em­at­ics than in oth­er fields.