by Allyn Jackson
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 .
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.