#### by Deanna Haunsperger and Stephen Kennedy

*The first thing that strikes you is that she’s charming. Only later
do you realize that she is probably the smartest person you’ve ever
met. Somehow, you don’t expect mathematical genius and a charming,
genuinely warm personality to coexist. Yet, here sitting down to
lunch with us is an existence proof: Professor Ingrid Daubechies of
Princeton University, MacArthur Fellow, member of the National
Academy of Sciences, the mother of wavelets. Fuzzy-haired,
bespectacled, quick to smile and a world-class storyteller, she’s
telling us about growing up in a coal-mining town in Belgium in the
sixties.*

“I grew up in a very small town. My father worked in a coal mine. He was an engineer in a coal mine, and of course while I was growing up I didn’t notice, but in retrospect coal-mining towns are very special kinds of small towns. I mean, there is just one big employer, and he controls the whole life, even the social life, of the town. My mother, well, for her generation in Belgium it was not common to have university education, but she did. She had expected to have a career, but after marrying my father she didn’t. Partly because there was no opportunity, but also because in this very small paternalistic town, two generations behind the wide world, it just was not done for wives of engineers to work. There was one wife who worked, she was a nurse, and everybody knew that that was why her husband never got a promotion. It wasn’t said that way, of course, and even then they couldn’t write things down that way. So my mother didn’t work.

“I remember when we were little she did a lot with us kids. As we grew up we became more independent because she wanted to give us our independence, but she was also very bitter at not having any bigger framework for her own life. She went back to college when I went to university. We had moved by then. She went for a different degree because her first degree, in economics, had become obsolete — she hadn’t worked for twenty years. She got a second degree in criminology and worked for about twenty years as a social worker. She worked with troubled youths, trying to monitor them, and to help them, and to give them decent lives.

“She had met my father while they were students at different universities at a meeting that brought together students from universities in Belgium. She met him, and later they decided to get married. They married in 1952, and he started work at the coal mine. At this coal mine they would only hire engineers who were either married or engaged to be married very soon. They did not want trouble with single men around.

“The region where my father’s parents were born and lived was a very poor region. The reason my grandfather didn’t work in the coal mines was that his mother really wanted one of her children not to go down the coal mine because that was the time when you would die young if you went into the coal mines. They didn’t know how to prevent black lung disease; everybody died young. The coal mines were just disputing the fact that it was anything to do with work in the mines, so you didn’t get any compensation either. My grandfather had been sickly when he was little and he was the first-born, so his mother wanted him to work elsewhere. Generations in my family are very long, so I’m talking the end of the 19th century when my grandfather was born. He left school when he was nine. I said this was a poor region, and this was before child-labor laws. His mother, my great-grandmother, had arranged a job for him in a big glass factory; he was in packing. But then, these things are incredible, by the time he was 14, he had in various accidents lost a finger and an eye in the glass factory. I think of my children, and I think — how is this possible? My father always says of his father that he was really very smart. He went to evening school at some point. He was in packing all of his working life, and he became a foreman. At some point, for a very complicated delivery, they had to make a case the inside of which was the intersection of two cylinders, but he had to make it in wood, of course, and fold it out of plywood. He had tried with ellipses and somehow it never fit, so he actually went back to evening school to study mathematics in order to learn how to do that. Another foreman had explained how to do it, but he wanted to know what it looked like. If you took a cylinder and then unfolded it, what would it look like? How do you actually compute that? So, he went back to school.”

#### An education in physics

Ingrid attended the Dutch Free University in Brussels where she studied physics. She held a research position in physics at that same university until 1987 when she came to the United States. Today, though, she asserts, “I’m a mathematician.”

“My father was always interested in mathematics, and he was always interested in explaining things to me, and I liked it. I would ask questions. I would usually get answers which were much longer than I hoped for, so I am trying with my children to yes, give answers, but maybe not go beyond so far. I remember liking mathematics when I was little, but I actually did major in physics. I think I majored in physics because it was my father’s dream to become a physicist; he explained to me things about physics. He went to extra Open University courses whenever he could. Sometimes they would organize a series of physics lectures, and he would go.

“Physics just seemed to be a very noble choice because of my father’s influence. It was something intermediate between what I really wanted — mathematics — and what my mother really wanted — which was that I would have become an engineer. She was a bit worried about all this science. She thought scientists were like artists, they really cannot make a good living. An engineer can always find a good job.

“With a free choice, I think I probably would have chosen mathematics. I don’t know. I liked physics very much. I especially liked some physics classes that we had. At some point I was considering switching between math and physics, and I decided to stay in physics because of one particular course which I thought was wonderful, in which we were going beyond geometric optics. If you go to the Kirchoff–Fresnel theory of optics, you actually see, and can compute, that a lens, in fact, computes a Fourier transform, which I think is wonderful. I think this is mind-boggling, that a lens would, in fact, compute a Fourier transform and this is used in some optical computing. This was marvelous. In fact, this course was really a course in applied mathematics. It was labeled as a physics course, and it was wonderful, and so I stayed with physics. I don’t regret it. As a result I have learned a whole lot of things that I wouldn’t have learned in a standard math curriculum. And the math that I wanted and needed I have learned by myself anyway.

“I think I think like a mathematician; I switched from theoretical particle physics to more mathematical physics because I felt that people who were really good at particle physics had an intuition about which I had no clue. I felt like I could learn how to read those papers, but it was like learning a language without understanding the meaning of the words, which I didn’t like at all. It’s hard to describe how I think. Even in analysis, I don’t think in formulas. Although when I work something out, I do compute a lot. I have some kind of mechanical or geometrical way of thinking, I don’t really know where that comes from.

“Anyway, in Belgium, undergraduate education is really different from here in that you track very, very early on. When you register for the university, you have to say what you’re going to major in. So you get very few courses outside your major or outside things related to your major. For physics, you get a lot of math, you get some chemistry, but you don’t get any liberal arts courses. I think you could go sit as an auditor in some of these courses, but really there’s no time; you don’t choose your own courses. You say ‘I will major in physics,’ and then the courses are specified except that in later years, you have some choices, you get to choose one of four. In the first two years everything is completely chosen for you, and it’s quite a heavy schedule. It’s a heavier schedule than I see here, but the result is that you can do much less independent work. I think a schedule where you put together a combination yourself and where you’re encouraged to do a lot of independent work is actually better.

“I was tracked with physics, so I had a lot of math courses, especially the first two years. And when I had majored, I had seen a lot of physics courses that would be at graduate level in the States because you cannot cram four years full of physics courses and not get to that level. Things are not organized so much by semester as they are for a whole year, so many courses were a full year. In the third year, that was really the heaviest year, we had 13 different physics courses, and we had 5 weeks of labs. Lots of that physics I have forgotten.

“At the graduate school level, however, in most universities in Belgium, you don’t get courses any more. There is a movement there now to get some what-they-call third-cycle courses which are graduate courses, but mostly you’re left to learn extra things on your own or with your advisor. You’re assigned or find a research topic; right away when you arrive you start working on papers. You also have a teaching schedule, and you’re expected to have your PhD in 5 or 6 years. I never had a teaching schedule because I had a special fellowship, but on a teaching assistantship, you would teach (be in the classroom) 30 hours a week. I would typically have 8–10 hours a week. The things you would be teaching would be problem sessions. Of course, you were there for the students, and typically the only person there for the students to ask questions from because they would not dare to approach a professor. You don’t have control over what you’re doing; you’d be given a list of problems, and you’d answer questions about them. It was very frustrating because it meant you had zero input into what you did. That persists even geophysicist Jean Morlet and mathematical after the PhD, for a while.”

#### A new paradigm

Wavelets are everywhere these days. Wavelets are a new method for encoding and compressing information (see box). They are being used in image compression (the FBI’s files of approximately 200 million fingerprints are being converted to wavelet-compressed electronic images), also sound and video compression, medical imaging, and geological exploration. Ronald Coifman of Yale University used wavelet techniques to remove the noise from a century-old recording of Brahms playing one of his own compositions.

The most exciting thing about wavelets might just be the way that they are drawing together people and ideas from so many different fields of science: mathematicians, physicists, geologists, statisticians, computer scientists, engineers of all kinds. In fact the history of the idea has roots in all of these fields and more. Yves Meyer has identified precursors to the idea in mathematics, computer science, image processing, numerical analysis, signal processing, studies of human and computer vision, and quantum field theory. The short version of the history has geophysicist Jean Morlet and mathematical physicist Alexander Grossmann introducing the idea in the early 1980s. Meyer and Stéphane Mallat pieced together a mathematical framework for wavelets in the mid-eighties. In 1987 Daubechies made her famous contribution of a family of wavelets that are smooth, orthogonal, and equal to zero outside a finite Thus, in a stroke, accomplishing what everyone supposed impossible and making wavelets very much more applicable.

“How did I get started on wavelets? For my PhD work, I had worked on something in quantum mechanics which are called coherent states. This is a tool to understand the correspondence between quantum mechanics and classical mechanics. So you try to build functions that are well-localized, that live in Hilbert space, but that correspond as closely as you can with being in one position, in one momentum in classical mechanics. I had worked in Marseilles with Alex Grossmann, and Alex was one of the people who really started the whole wavelet synthesis. There are roots in pure mathematics, in many different fields, but the synthesis really, I feel, was created by Alex Grossmann and Jean Morlet. Morlet was not happy with what was called the wavelet transform, and he wanted something else, so he invented an algorithm. But he had no mathematical theory behind it. Alex Grossmann realized there was an analogy with the coherent state formalism in quantum mechanics. He recast it in terms of a group representation and then people started saying ‘But we’ve been doing that all along in a different context.’ And they were right; they had been doing it all along. So that’s what made the synthesis happen that made the jump from mathematicians to engineers.

“I knew Alex Grossmann very well from my thesis work, and I was looking for something else to start working on. This was a time when I had many changes in my life. Before my husband, I had a long relationship, and I had just left him, and I was looking for something else, for changes. So I changed research topics. I started working on wavelets in ’85. I started working on wavelets and I met Robert [Calderbank, the mathematician to whom she is married] within a period of six weeks in ’85.

“At this time geologists were doing windowed Fourier transforms, so they would window [look at a short time segment of a signal], then Fourier transform [approximate that bit of signal with combinations of sines and cosines]. This means that for very high frequency things that live in very short time intervals — these are called transients — if you’ve determined your window to be this wide, then you need a whole lot of high frequency functions to capture that behavior. So, you could of course make your window very narrow, but then you don’t capture a lot of them. Morlet didn’t like that aspect of the windowed Fourier transforms, so he said ‘After all, I’m using location and modulation, let’s do it differently. Let’s take one of these functions that have some oscillations and let’s put that in different places and squish it so that I have a different thing,’ really wavelets. Now he didn’t really formulate this precisely. Actually the name comes from there because in geology when you have different windows, which determine the shape of these multi-layered functions, they call them wavelets. So he called his transform ‘a transform using wavelets of constant shape’ because the other ones didn’t have constant shape. If you adjust the window, or you modulate this way, then of course they look different. In his case, the wavelets look the same, they were just dilated versions. He called them wavelets of constant shape, but then once they left that field, there was no other thing around called wavelets, so he just dropped the ‘of constant shape’ to the great annoyance of geophysicists because in their field it has another meaning.

“Wavelets were not really something that were a trend in geophysics, Morlet just came up with it. In harmonic analysis, people had been looking at, not exactly the same way, but something similar for ages. It goes back to Littlewood–Paley theory [circa 1930], and even the integral transform formula that Grossmann and Morlet wrote, because Morlet had no real formulas, is a transform that you find in Calderón’s work in the sixties. So in some sense they had reinvented the wheel. In another very real sense they had looked at it completely differently. For Calderón it was a tool to carve up space into different pieces on which he would then use different techniques for estimates. Grossmann and Morlet gave these wavelets some kind of physical meaning, in a certain sense, viewing them as elementary building blocks which was a different way of looking at it. And Yves Meyer later told me that when he read those first papers by Grossmann, it was very hard for him because it was a different style. It took a while before he realized in what sense it was really different, because at first you see the formulas there, and you say, ‘well, yeah, we’ve been doing that for 20 years,’ but then you realize that here was a different way of looking at it: a new paradigm shift.”

#### A link in a chain

To most mathematicians it appears that wavelets sprang full-grown from the foreheads of Grossmann, Morlet, and Daubechies and then were immediately grabbed by engineers and scientists. It is unusual for a piece of mathematics to find so many applications so quickly. Daubechies with her ability to talk the languages of physics and engineering and mathematics is, to a large degree, responsible for the building of so many bridges between the groups.

“In pure mathematics the idea was developed starting in the thirties, then in greater detail in the sixties. It was a very powerful tool which lived in a relatively small community in mathematics, and outside the small community, I felt it spread rather slowly. For example, in quantum mechanics I think some of these techniques would have been useful to mathematical physicists earlier than they penetrated. I think it’s because through Grossmann and Morlet there were intermediate people. I treasure every single electrical engineer I meet and with whom I can talk. I’m interested in talking with them, I think many mathematicians aren’t, but I am. Even so, I find it hard to talk with many of them because we’ve been trained in completely different ways and the words mean different things. But I have found some I can talk to and I think it’s very valuable when they are also interested in talking with me. I think it’s easier for me because of this physics background I have and because I have learned at least some of their language. I think Alex Grossmann played a very important role that way. I have met Jean Morlet several times, I think he’s a very interesting man, but I find it very hard to talk with him. I mean, of course, not talking socially, but to really understand his ideas. Because it’s not even that I can see that here’s an idea and I know that I don’t understand the formal mathematics, it’s that I don’t even understand the idea: that I don’t even notice or can’t tell if there is something there or not.

“The problem is you don’t know what are ideas and what aren’t: he probably knows the different layers of what he’s saying, but for me it’s impossible. Alex Grossmann can talk to him. It’s good to have a chain of people. I think so. And I think that’s the role that I played. In some sense you could say that I didn’t discover anything that anybody didn’t know because there’s this one aspect of the mathematical roots, but then there’s the other one which has to do with an algorithm for implementing the whole thing. If there wasn’t an algorithm, then none of this would be happening anyway. But that algorithm existed in electrical engineering: it’s called sub-band filtering. There was no connection with any of the pure mathematics, and I don’t know that that connection ever would have been made if it weren’t for this chain of people. I mean, once this connection was made, then you had mathematicians interested in hearing about the algorithms and electrical engineers interested in hearing about the mathematics.

“I’m a mathematician. I feel like one. I feel like a mathematician, but I am very much motivated by applications. I like to go off on a mathematical tangent, but I like to get back to applications. So in that sense, I’m an applied mathematician. At one time, and still today for some people, applied mathematics meant only certain types of results obtained by solving certain types of partial differential equations. I’m not that type of mathematician at all. So I very much feel I’m an applied mathematician, but what I apply is functional analysis, rather than PDE theory. Actually, there’s no such field as applied mathematics: I think there are subfields within mathematics and that, as a mathematician, you always really like it when different subfields get into contact with each other. I think that virtually all of these mathematical subfields can have contacts with applications, so in some sense I’m an applied harmonic analyst, not an applied PDE person, but one can just as easily be, say, an applied number theorist.

“Ideally, I think there’s an important place for pure mathematics and an important place for pure mathematicians; I see my role as identifying and bringing to more pure mathematicians than myself very interesting problems coming from applications. I think that’s an important role to play and that it is good for pure mathematics. There was a while when pure mathematics wasn’t open to this, but I think that mathematicians are starting to open up more. It’s very important to remember that whole fields in pure mathematics have come from applications. That doesn’t mean that all the pure math that was done in that area can therefore be described as applied. No, it’s just mathematics. But it also suggests that it’s very well possible for other fields of mathematics to start to be fostered by applications. I mean applied mathematics is not just learning some nice mathematics and not being upset by getting your hands dirty on some problems where things are not as neat but they will have an application — it’s also identifying opportunities for mathematical thinking, which can lead to other fields of mathematics. I don’t know which other fields. I can’t predict. I mean it’s the ones that you cannot predict that are the most interesting.”

#### A life in mathematics

Problems are the lifeblood of mathematics and Daubechies, like most mathematicians, has several going at once. All the ones she tells us about come from real world applications. It’s quite clear listening to her explain her problems that she is a phenomenal teacher — the explanations are so clear and the problems sound so exciting that we’re itching to get out of the interview and get to work on them.

“Before this meeting I was at a molecular biology meeting. I got really interested in people who can add proteins; for some proteins they know in which order all the atoms go, but then they don’t know what the thing will look like, and they have to ‘solve it’ to know what it looks like, and these are really important to understand their functions. There are groups that try to predict the form of proteins from energetic computations from just the formula, and they have it organized so that they have a way to objectively compare how good their predictions are with what the grand truth is by finding out about proteins that will be solved. I mean you can reasonably predict when things will get solved, but are not solved yet. So they use all those formulas, and they all work on it: they have a deadline. At some point it’s clear when the thing will be solved, and they say ‘now, you have to submit.’ And you submit at that time, and it gets compared with the grand truth, and in that way you can score different prediction programs.

“Okay, so one guy was explaining about his prediction program: because it’s just too big a space to exhaustively search, he searched in a multi-resolution way. He first tried to build a coarse model, then find the best coarse model given, then build it up from there. Now his first-level coarse model, was indeed very coarse, but he was putting it on a regular lattice, and his method was doing very well. But it struck me that if we have a good idea of how to compress, how to find subdivision schemes for curves, we could look at all the proteins that they know, and try to find that subdivision scheme that will be adapted to the protein world. I’ve been always looking at smoothness; they don’t care about smoothness. These proteins actually do all kinds of strange things. But one could try to find a subdivision scheme that would adapt to their goal, and that could give you a good idea of what kind of coarse things to start from. So that’s something that hasn’t started, but something I’m very excited about and I hope to work on this spring.

“Another thing that I’m very involved in is understanding the mathematical properties of coarsely quantized but very oversampled audio signals, modeled by so-called band-limited functions. There are really neat links to dynamical systems. I’d like to do that for other wavelet transforms. I think we can do it. If we can do it, I think it’s going to have very useful applications, plus I think mathematically it’s going to be very interesting. Already for the band-limited functions it’s much more interesting that I had expected a year ago.

“I have a graduate student with whom I work on applications of wavelets to the generation and compression of surfaces. People represent surfaces with triangulations with tens and hundreds of thousands of triangles, so you’d like to compress that information. Well, you can do that via multi-resolution, and then you can wonder what kind of wavelets are associated with that. Then you can think about smoothness. In some applications, smoothness again is very important. So that’s another project.

“What else? I look at my students and collaborators because everything I work on I work on with a collaborator. I’m still working with one of my former students on a way of using frames for transmitting information over multiple channels, but that’s very theoretical work, and I’m not sure how close it’s going to get to applications. I have the impression I’m forgetting something.

“I love to talk about mathematics and so I enjoy all the courses I teach. I teach regular undergraduate math courses, I developed a course of mathematics for non-math majors. For many students in our calculus classes this will be their last contact with mathematics. I don’t think this is a very good idea. Many of them are really not turned on by calculus, and it’s hard to get a really meaningful application in a course if you also want to teach calculus tools. You can try to fit in some applications, but they really feel very contrived. The real applications of calculus are all the physics and other math courses, but they won’t ever see that, and they’re not interested in seeing that. I wanted to get mathematical ideas across without teaching technique.

“I call the course Math Alive; in two-week units we visit different concepts. I do one on voting and fair share, and one on error correction and compression, and I have one on probability and statistics, and one on cryptography, and there’s one (actually not taught by me, it’s a course we co-teach) on Why Newton Had to Invent Calculus, that’s a unit students have more trouble with. Then there’s one on dynamical systems and population explosion, it gets into population models. I’ve enjoyed this course a lot. I’m teaching it this spring for the fifth time, and I’d like to document it so that it can be taught by somebody else the next time, because that’s the best way to prove a concept, if it can be done by somebody else. “I want these students to go away knowing that mathematics is really important, that it turns up in lots of things where they may be impressed by the technology, but they don’t realize there’s very deep mathematics in there. I want them to see that mathematics is neat in that you really solve a problem. You think your way out of it. And basically, if they remember that, that’s fine. I’m sure that when people meet historians, they don’t say, “you must know all the dates.” They know it’s something else. Well, they don’t know that it’s something other than balancing a checkbook in mathematics. I’d like them to know that.

“I also teach graduate courses — often a starting graduate course in wavelets, sometimes a more advanced course. I enjoy teaching undergraduate courses more than graduate courses. Not because I don’t like teaching graduate courses, but because at graduate study level, the starting graduate courses I can see work well, but I think an advanced graduate course works better as a reading course than as a lecture course.”

#### An American life

Daubechies is married to Robert Calderbank, a distinguished British mathematician. They have two children, Caroline and Michael. If you’re wondering what it must be like to have a genius for a mom, well, it sounds a lot like having a mom.

“I go to my children’s school and the tables are in groups and the classroom is full of wonderful things. In my school days classrooms might have had some stuff, but we had little desks which were all lined up in rows, maybe that’s the way things were here as well in the sixties. I think the new way is much more fun. It may well be different in Belgium now; I haven’t visited the elementary schools. All my elementary and secondary education was in single-sex schools. That was the way it was in Belgium at that time. I went to public schools, but all public schools at that time were gender-separated.

“I have mixed feelings about that. At the time I thought it was a bad thing; it’s better if you don’t see another gender as a different species because at some point you will start looking for a companion, and if you haven’t really met any people of the other gender until you’re 18, it’s very artificial. So I didn’t like it at the time, but then after I got to university I realized that people in classrooms were less likely to ask me to give n answer than some boys that were there. At least in the beginning. After a while, when they knew I was interested, then it was different. But you always felt that there was a bigger hurdle to get over as a girl to get noticed than as a boy. I hadn’t thought about that in great detail, but then coming to this country and hearing all the debate here about it — I can see how it might be good for some girls to have separate gender schools. In an ideal world there would not be this effect, but given it exists, I can see how it might help in building self-assurance in girls. My daughter goes to public school and it’s mixed gender, and she’s happy, but I’m wondering if at some point there might be a problem. I think she’s a smart little girl, and if at some point I feel that because she is in a mixed-gender school she is not getting as much of an opportunity, I might consider a single-gender school. I haven’t, she’s only seven, and it’s not an issue at this time. It’s something that ten years ago I would not have thought I would ever consider, but now I would.

“I don’t know if going to all-girl schools had an effect on me. My parents always made it clear I could do anything. It didn’t occur to me until I went to university that people could think I was less good at something because I was a girl. I think I was very fortunate, because at that age, you’re too old to take that prejudice seriously, and when you encounter someone with that attitude, you think, ‘You’re a jerk.’

“As I said, my parents, especially my father, really influenced my education. So did popular psychology actually. When I was little the prevailing theory was that it wasn’t good to mix languages too early. You might really confuse children and then they wouldn’t really be able to use any of the languages in great depth and it would leave marks on them the rest of their lives. My parents were in an ideal situation to bring my brother and I up bilingually, because they spoke French to each other and we lived in the Flemish part of Belgium, where people speak Dutch. But since there was this myth that mixing languages early was not good, they decided to bring us up in Dutch, which is my mother’s tongue; my father’s fluent in it since he went to school in Dutch.

“Theories having changed now; I bring my children up bilingually in Dutch and English. I thought it would be too hard doing it in a language that is not my mother tongue. My French is fluent, but in the beginning especially it took quite an effort to have a bilingual household because my husband is British and he didn’t speak Dutch. He has learned together with the children. So I speak English with him, but Dutch with the children.

“My husband since he has learned Dutch would like to have practice speaking, but my children won’t allow it. They roll on the floor when he tries it. They think it’s quite incredibly funny. My son went through a stage where when my husband would say something in English that he knew was of interest to me too, he would turn to me and translate for me.

“English is their first language because they go to school in English. So in Dutch they sometimes have more difficulty finding words, but I really try to encourage them to find the words rather than switch to English, and so sometimes before my daughter tries to tell me something, she’ll say ‘how do you say that word in Dutch?’ They had been talking in school about different languages, and she really wanted to tell me, so she said, in Dutch ‘How do you say ‘Dutch’ in Dutch?’ But at the end of the sentence the name was already there.

“Our son is talented in math, and I think our daughter might be too. We, of course, like to stimulate that when we ask questions, but we are not pushing them hard. What we are pushing is that they have to cooperate and work at school and do the best they can, not just in math, but in everything. In fact, I’m more concerned about writing and things like that. Trying to get ideas in an organized way on paper I think is important in mathematics as well as elsewhere.”

“I like to go to my children’s school and help out, especially on science day. When I’m there I’m not that woman professor mathematician, I’m Michael’s mom and Carolyn’s mom, and I like that. I like that.”