# Ingrid Daubechies

Fellow: Awarded 2010

Field of Study: Mathematics

Competition: US & Canada

Fellow: Awarded 2010

Field of Study: Mathematics

Competition: US & Canada

Born in Belgium, Ingrid Daubechies studied physics at Vrije Universiteit Brussel (B.S., 1975, Ph.D., 1980) and worked as a Research Assistant (1975-84) and then Research Professor (1984-87) in its Department of Theoretical Physics before moving to the United States to take up a position as Technical Staff Member in the Mathematical Research Center at AT&T Bell Laboratories. During part of her seven-year tenure at Bell Labs she also held an appointment as a Professor of Mathematics at Rutgers University. She left both positions in 1994 to join the Mathematics Department and Program in Applied and Computational Mathematics at Princeton University; she directed that program from 1997 to 2001. Currently she is the William R. Kenan Jr. Professor at Princeton, a post she was appointed to in 2004.

Ms. Daubechies’ many innovative solutions to problems in physics and mathematics and derivation of practical applications for scientific discoveries earned her many important awards: the Louis Empain Prize for Physics (1984); the Steele Prize for Exposition (994) and Ruth Lyttle Satter Prize (1997) from the American Mathematical Society; IEEE Information Theory Society Golden Jubilee Award for Technological Innovation (1998); National Academy of Sciences Medal in Mathematics (2000); the Eduard Rhein Foundation 2000 Basic Research Award for the invention, mathematical advancement, and application of wavelets (2000); the Gold Medal of the Flemish Royal Academy of Arts and Sciences (2005); the ICIAM Pioneer Prize (2008); and, most recently, the Onsager Medal from the University of Trondheim. In addition she has been awarded honorary doctorates from the Université Libre de Bruxelles (2000) and Universiteit Hasselt (2008), both in Belgium; Université Polytechnique Fédérale, Lausanne, Switzerland (2001); Université Pierre et Marie Curie, Paris (2005); the Universita degli Studi di Genova, Italy (2006); and the University of Trondheim, Norway. She is an elected Fellow of the American Academy of Arts and Sciences and the National Academy of Sciences, and a Foreign Member of the Royal Netherlands Academy of Arts and Sciences, the Académie des Sciences in Paris, and the Royal Belgian Academy of Sciences. In 1992, she was given a MacArthur Foundation “genius” grant. Beginning in January 2011, she will begin a four-year term as President of the International Mathematical Union.

Scientific Trajectory: A Short First-Person Narrative

Although I obtained my Bachelor’s degree in Physics, and my Ph. D. degree in Theoretical Physics, I am now generally considered to be a mathematician. I am often asked what made me switch from one field to another. In fact, if switch there was, it didn’t result from a conscious decision—it happened entirely naturally.

I have been fascinated since childhood by how things work, why natural phenomena we observe are the way they are. I must have been four or five, very fond of a little tiger doll dangling from the rearview mirror in my parents’ car, when I discovered, one evening as we were driving home after a visit to my grandparents, that its beautiful orange color had just bleached away, and it now looked a sickly yellow. My father explained that it really was just as orange as always, but that in the orange light of the neon streetlamps, even white objects looked orange, and so the difference between orange objects and white objects was much reduced, which gave me the impression the doll had lost its color. The same would happen, he explained, if you looked at the world through a colored sheet of glass; everything white would then look colored as well. I only half believed him, but had to admit the tiger was back to normal the next morning. Not too long later, however, I observed that if I looked at the neon lights through the green sunscreen at the top of the car’s windshield, the light looked dimmer, but its color was still the same orange. This time my father had to try to explain that most light has many colors mixed in it, but that neon light was monochromatic—I don’t think I understood that explanation until much later.

I was also fascinated by how things can be made, as I am to this day—weaving, ceramics, knitting, sewing, sculpting, cooking are activities I enjoy a lot, because they make something new, or rather they transform their materials or ingredients, and create something that may have been only intuited or seen with the mind’s eye before. Sewing clothes for my dolls was so interesting: you could make non flat surfaces from flat fabric, by cutting out pieces and sewing seams.

The way I experience mathematics is very much the same; mathematics helps me coax out of given materials something else, a new way of looking at things, of understanding, that maybe wasn’t there before.

For my Ph.D. I worked on Mathematical Physics, a branch of physics that tries to justify all the statements it makes about physics in mathematically rigorous terms. Some arguments made by theoretical physicists are not “completely” true, that is, the researchers sometimes “overlook” some mathematical difficulty in the midst of their argument, confident that in the end, it will all come out in the wash. Their end results still make physical sense because they (almost always) have an unerring instinct to choose, when there are choices to be made in mathematically troublesome situations, the one solution that will not lead to trouble. Mathematical physicists are the people who worry about this—after all, theoretical physicists are the first to admit that “the book of nature is written in mathematics.” If, when you really tried to do things entirely properly, the correct approach would lead in a different direction altogether, then that would be seriously upsetting. (Despite this, most theoretical physicists feel that mathematical physicists are too obsessive about mathematical rigor.) So Mathematical Physics tries to find the ways in which you have to construct the basic mathematical framework so that you can tackle these difficult physics situations, in quantum mechanics, or in statistical mechanics, in a way that is mathematically entirely satisfactory and transparent (to experts, at least), without having to make leaps of faith. I published a series of papers in Mathematical Physics on several topics [1-25, 27, 29]. (The bracketed numbers throughout refer to my list of publications.)

From Mathematical Physics, I learned that it is possible to construct different mathematical frameworks to study the same phenomenon, and that each framework has its value; often the different approaches highlight different features. I had done my Ph.D. in Europe (nominally in Belgium, where I held my fellowship, but working a lot with Dr. Alex Grossmann, whose home base was Marseille, France); this was followed by a two-year postdoctoral visit to the U.S., working with John Klauder and Elliott Lieb, still in Mathematical Physics. Returning to Europe, to take up a position as Research Fellow (Aangesteld Navorser) and getting in touch again with Alex Grossmann, I was swept up in the enthusiasm with which he had tackled a new topic, using techniques from quantum mechanics to study audio signals and seismic traces.

This new technique, which later turned out to be related to similar ideas from many directions, but which nevertheless led to a new synthesis, a new paradigm, became known as the wavelet transform. It was exhilarating to find that the tools I had learned to master so well could be used in signal analysis, and provided a different point of view from that of the engineers working in the area. In Quantum Mechanics, one is constantly thinking in two domains (position and momentum) that are simultaneously valid, although not quite compatible, so it is important to know how to juggle them both; in Electrical Engineering, the incompatibility of the two points of view (time and frequency, in the signal analysis case) is dealt with by carrying out all the reasoning and all the constructions almost exclusively in one domain (the frequency domain). I found that when I asked “why” and “how” questions in signal analysis, the answers with which I myself came up were often not the same as the standard ones, and in some cases my answers were better. This was exciting, of course, and led to my first work on wavelets [26, 28, 30-34].

The years 1981-87 were very exciting; people from many different fields found themselves contributing to the new development, and the annual wavelet conferences were multidisciplinary delights. By 1986 we had obtained, thanks to the work of Stephane Mallat and of Yves Meyer and his students, a good mathematical understanding of the framework that made orthonormal bases of wavelets possible. The framework was very elegant, very beautiful—many pieces that seemed puzzling before had all fallen into place—exactly like in a high-quality jigsaw puzzle. But in order to apply this framework to concrete signal analysis, you had to skew things a bit, as if you had some bits glued onto or taken away from individual puzzle pieces; after this, you could still see clearly the underlying neatness, but the pieces didn’t quite fit so nicely anymore. It is something that mathematicians often take for granted, that a mathematical framework can be really elegant and beautiful, but that, in order to use it in a true application, you have to mutilate it: well, they shrug, that’s life—applied mathematics is always a bit dirty. I didn’t agree with this point of view, and wanted to start from the requirement that the orthonormal basis of wavelets that would be the end product of the mathematical analysis could be implemented without mutilation. To my surprise and delight, this approach did work, and I came up with wavelet bases that could be implemented without any violation, and that are now used by many people in many applications [35].

The paper [35] was written in 1987, which was also the year that I moved permanently from Belgium to the U.S.A. because in May of that year I married my (still) husband, who was based in the U.S., and we wanted to start a family. In July 1987 I started in a new job, as a researcher at AT&T Bell Laboratories.

In the next few years, I developed, with collaborators, mathematical techniques to study these new species of wavelets in more detail; they were not defined in the usual way, by explicit functional expressions, but instead were defined by the algorithm generating them, which meant that more work was needed to characterize all their properties [38, 39, 41]. We also developed other bases of wavelets with other properties [44, 45, 47-50, 53, 54]. One of these papers [47], combined with a technique published in a slightly later paper [77], became the basis for the wavelet transforms used by the JPEG2000 algorithm, which is widely used for compression of images on the internet, and is the basis of digital cinema in Europe and North America.

In 1993, I accepted a position in the Mathematics Department at Princeton University, where I was the first tenured woman professor. (In my view, this was more something they should have

been ashamed of than a reason for celebration.) I had enjoyed my time at Bell Labs very much, but I felt I missed a university atmosphere; during my time at Bell Labs, I spent six months at the University of Michigan, and also took a leave of absence to teach at Rutgers. After moving to Princeton, my research continued on wavelets and other mathematical approaches to signal analysis, generally called time-frequency analysis [55-90].

In 2002, I became interested in analog-to-digital conversion of signals. Before they can be transformed, via wavelets or otherwise, so as to give rise to efficient compression or transmission, the data, typically in a continuous form, have to be digitized. We have all heard about the immense advantages of digital signal processing—but how do we get the data digital in the first place? It turns out that this is much less straightforward than one would think, and that constructing the widgets that carry out this conversion is still as much an art as a science, because very little is understood thoroughly in this area, mathematically speaking. Together with Ron DeVore, then on sabbatical in Princeton, I obtained the first non-trivial mathematical results in this area [91], which I continue to mine [99, 100, 122], and which provided Ph.D. topics for two of my first five graduate students at Princeton.

This turned out to be but the first of many “branchings out” that resulted from being in the more diverse scientific environment at a university, as compared to an industrial research lab. Here is a list of different areas in which I have worked, and in several cases, continue to work in with postdocs and graduate students:

**Computer Graphics**: Computer graphics is one of the areas in which academic advances translate the fastest to “real life,” if computer animation and video games can be called so. It poses very interesting mathematical problems, which amount to characterizing smoothness properties for function defined by algorithms rather than by analytic expressions or functions

[80, 86, 90, 96].

**Astrophysics**: Strictly speaking, this was not really a foray into a different field, since it still amounted to signal analysis, but now of a different type of data, namely astrophysical observations [111]. It did provide, however, a nice application of an algorithm for which my collaborators and I were the first to prove convergence in general [101].

**Learning Theory**: This is an extremely interesting field, in which I expect to continue to be active for a while, and where better mathematical framing is needed. I got involved through

the determination of my graduate student Cynthia Rudin, co-advised by Rob Schapire, an expert in the area, and then got captivated by the problems. I now have other students

working in this field as well [103, 104, 106, 107].

**Neuroscience** (more precisely, functional **Magnetic Resonance Imaging**): This is really signal analysis again, but of a highly specialized area, and it led to a very interesting (and ongoing) collaboration [110, 118].

**Geophysics**: Together with graduate students and a postdoc, I am involved here in a global tomography problem, where the goal is to reconstruct, from seismic data observed around the globe, for every major earthquake of the last 30 years, the deviation from (approximate) spherical symmetry of the distribution of the Earth mantle’s constituents (mantle = from the surface to halfway the center of the Earth). We hope to see both thin hot plumes rising from the core under volcanic hot spots, for example in Hawaii, but also the more gradual upwelling (so-called “superplumes”) under areas such as Southern Africa [115 and papers in preparation].

*Image Analysis for Art History*: this is a very exciting project, in collaboration with the Van Gogh Museum in Amsterdam, the Kröller Müller Museum in Arnhem (also in the Netherlands), the Museum of Fine Arts in Brussels, Belgium, and the Museum of Modern Art in New York. We aim to develop techniques that will assist art historians in tasks like authentication, dating, quantifying transition stages in an artist’s style and technique, among others [121 and papers in preparation].

**Centre ValBio**: The island of Madagascar is a rich resource for researchers in many fields. Especially after the 1986 discovery in the Madagascar forest of a golden bamboo lemur—a supposedly extinct species—by anthropologist Patricia Walker of SUNY Stony Brook, the students and scientists flocking to Madagascar overwhelmed the facilities there. To address this problem, a consortium of universities established Centre ValBio. This research center, situated in Ramonafana, a national park in the southern half of Madagascar, was inaugurated in 2003; since then more research facilities have been built, so that it is now possible for foreign researchers to visit for an extended period with a modicum of comfort. (This is by no means luxurious: it means that visitors can sleep under a roof instead of in tents, and can take showers.) Centre ValBio produced the additional benefit of stimulating the local economy, both through the increase in the tourism trade made possible by the enhanced facilities and through the employment of local residents as support staff and in other capacities.

I want to achieve two different goals with my visit to ValBio: conduct mathematical research connected with several of the biological and ecological projects at ValBio; and help the nearby University of Fianarantsoa (created in 1988) set up an applied mathematics program, in association with its fledgling Centre for Ecology-Oriented Science and Technology.

**Mathematical Research at ValBio**

Together with my postdoc Dr. Yaron Lipman, I already have an ongoing collaboration with Dr. Jukka Jernvall, of the Center for Biotechnology at the University of Helsinki, and his collaborator Douglas Boyer (now at Brooklyn College, CUNY). The mathematical problem on which Yaron and I are working concerns the characterization and comparison of 2D surfaces in 3 dimensions. Concretely, we want to define a notion of distance between surfaces that measures how similar (or dissimilar) the surfaces are, and not the physical distance one could find with a measuring stick. From this point of view the distance between two copies of a surface, where one of them is enlarged, rotated and shifted, whereas the other isn’t, would still be zero. Mathematically, it is not too hard to define such a distance, and there do indeed exist several candidates, of which the best-known is probably the Gromov-Haussdorff distance; in practice, they are very hard to compute efficiently. We have succeeded in defining such a distance that can be computed very efficiently, and we have applied this distance algorithm to surfaces obtained by Dr. Jernvall’s group by scanning molars of several species of lemurs, as well as to other data sets obtained by scanning bones or fragments of bones. We have found that classification algorithms, based on our automatically determined distances between surfaces, outperform those that use human-defined landmarks. Madagascar, and more particularly ValBio, is one of the sites where Dr. Jernvall does his field work, and I expect that he will be there for a significant part of the period during which I will visit. In an environment where we have daily contact and discussions, Dr. Jernvall and I expect that not only our present project will see significant progress, but also that many more joint projects will arise—our present collaboration grew out of one such conversation, after I visited Madagascar for ten days in the winter of 2007-08, during which I met Dr. Jernvall for the first time.

My second scientific project is in collaboration with Dr. Patricia Wright, the lemur expert whose discovery of the golden bamboo lemur was the direct incentive for the creation of Ramonafana as a National Park, and who has been carrying out most of her fieldwork in and around Ramonafana for over twenty-five years. We have talked about possible ways in which to model a “block” of rainforest, and the possible trajectories through the vegetation for animals of different sizes and agilities. Such a model would be useful for many other settings besides the rain forest; indeed Dr. Mimi Koehl (a member of the Department of Ecological and Evolutionary Biomechanics at UC Berkeley and a 1988 Guggenheim Fellow in Biology and Ecology) has expressed interest in it for the study of corals. It is, however, a project that is hard to tackle in the abstract, because many different approaches could be adopted, and the parameters are crucial for choosing the most suitable approach. For instance, one could view the spatial block as empty space, in which are placed obstacles of many different types that prevent or slow down the passage for different animals depending on their size. This could be tackled computationally, with many different realizations for given densities of the different obstacles, and analyzed statistically. Another completely different approach would be to model the different corridors themselves, and distribute those through the block of space, and study where and how corridors for different animals intersect. When we tried to get our heads around the different possibilities, on two visits I made to Stony Brook in the past year, it became clear that many of the questions to which I needed answers before starting to model in earnest required examining the field conditions through the eyes of the modeler, assisted by the biologists. It will thus be significantly helped by my going to Madagascar for a longer period, during Dr. Wright’s field season there, which is in the Fall. (Spring is the rainy season.)

**Applied Mathematics at the Centre for Ecology-Oriented Science and Technology at the University of Fianarantsoa**

ValBio has had enormous economic impact already on the local villages. Ecotourism and local employment at the Centre are helping build a local economy that depends on the health of the present remnant of rain forest, and values keeping it as pristine as possible, instead of burning it for the sake of growing a rice crop for a few years, until the soil is depleted and a further piece of land needs to be acquired. In the long run, it is desirable that not only the workers at the research center but also the scientists steering and directing the research be local people, who have, after all, the greatest stake in the future of their environment. Moreover, it is likely that only if the local people are involved on an equal footing with foreign scientists that the National Park and its research mission can be truly sustainable. Dr. Wright is working with the nearby University of Fianarantsoa, which has started a Centre to train a next generation of local scientists and engineers who will take part in and help set the research agenda for ValBio. I will participate in this effort at the Centre by setting up its mathematics curriculum and helping to select mathematically talented students who will be trained in other countries and then return to strengthen the Centre by sharing what they’ve learned.

National Academy of Sciences , 1998