Interview with a mathematician: Ingrid Daubechies

Ingrid Daubechies is the James B. Duke Professor of Mathematics and Electrical and Computer Engineering at Duke University. Her doctoral work was in physics, focusing on the mathematics of quantum mechanics. She is best known for her discovery of compactly supported continuous wavelets, now called Daubechies wavelets. Her work has found broad application to signal and image processing, and more recently to art restoration and detecting art forgeries.

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Ingrid has a luminous academic career, and she has many accomplishments and awards. She won the Leroy P. Steele Prize, the Ruth Lyttle Satter Prize, and the Steele Prize. She was the first woman to receive the National Academy of Sciences Award in Mathematics, and she’s former MacArthur Fellow, Guggenheim Fellow and member of both the National Academy of Sciences and the National Academy of Engineering. She is a Fellow of the American Mathematical Society, and past president of the International Mathematical Union from 2011 to 2014. Most recently, Ingrid was awarded a major grant from the Simons Foundation for 1.5 million US dollars. Recently on December 7, 2016, Ingrid received the Golden Medal of Flanders for her merits for Flanders, the Dutch speaking part of Belgium.

I was thrilled when Ingrid readily responded to my request for an interview. Ingrid was gracious with her time and gave truly thoughtful answers to my questions. In my view, she is one of the most impactful working mathematicians, and among the most prominent women mathematicians alive today.

Enjoy!

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AB: When did you first realize you had a talent for mathematics?

ID: It didn’t come suddenly. I was always interested in mathematics, and how and why things work and fit together.

AB: You also had early interest in physics, and that was the subject of your doctorate.

ID: Yes. I don’t have a degree in mathematics.

AB: Was there a person or teacher who influenced your mathematical development?

ID: My father took great pains to answer my scientific and mathematical questions if he knew the answer.  I had a physics teacher in high school who I very much liked, who went into explaining. Most people would find that the teachers who influenced them were those who went beyond the textbook or recipes. The best teachers are those who convey understanding and insight, and stimulate questions. I had a mathematics teacher in high school who was also like that.

At university, I considered switching my major from physics to mathematics. But then I took a course on Fourier optics that was absolutely beautiful. If you look at what a lens system does mathematically, then it turns out that it computes a Fourier transform, but in an analogue way. I thought that was really beautiful. That made me decide to stay in physics. The professor for that course was Roger Van Geen; he passed away years ago, but he was a very distinguished Engineering professor (he was a physicist) at my alma mater. All my training, both undergraduate and graduate was done in Belgium. I came to the US for the first time as a post-doc.

AB: Was it your own interest in physics and mathematics that lead you to a doctorate?

ID: In Belgium, it was the case that if you go for a PhD, then you often do that at the university where you did your undergraduate degree. Professors are on the lookout for smart students, and if they have funding with which they can support a graduate student, then they contact students to ask if they are interested in a PhD. That is what happened to me.

Coming to the US, I’ve been exposed to the idea that it’s a good thing to change schools between your undergraduate to your graduate degrees. I do think now that is good advice. When I tell students in Belgium that, however, they say that they can still learn so much from their professors, and they ask, “why should we change?” I respond that apart from learning material, there is also the learning of an outlook, or way of approaching things that you subliminally pick up from your professors. It is good to learn new approaches from other people. They don’t believe me, by and large.

AB: You’ve worked in both industry at AT&T Bell Labs, and in academic settings at universities such as Princeton and now Duke. How would you compare the culture of both settings?

ID: I would preface my answer by saying that AT&T Bell Labs was the closest environment to an academic group outside academia, and the furthest from a consumer product oriented industry lab that I know. AT&T Bell Labs was big then, with about 90% of work focused on development, and 10% on research. I was on the research side.

Paradoxically, I had more time for my research at Bell Labs then I would at a university: for example, I had no teaching or committee work. As time went on in my career, I have assumed more responsibilities in my profession nationally and internationally. That has left me less time.

On the other hand, I like teaching, so I missed the contact with students at the labs. I did miss contact with academics and intellectuals other than scientists or engineers. There were fantastic engineers and scientists at Bell Labs, but I thought it would be nice to meet art historians, political scientists, and philosophers. I realized that progress in my career meant that when I returned to academia I had less time than before, and it took as much as ten years before I had the breathing space where I could find the time talk to non-STEM academics.

Every university has its own atmosphere. Princeton is a fantastic place and one of the oldest universities in the US, while Duke is a younger institution. At Duke, I experience the administration as much more attuned to the faculty. Princeton is a place directed by an inner circle.  For example, at Duke, there are more creative approaches to offers to incoming faculty from other institutions.

AB: One of your most famous discoveries concern wavelets. What is a wavelet? How do wavelets apply to image and signal processing?

ID: Many things of interest to mathematicians or engineers are very complex and have several complicated structures layered on top of one another. As we do in other things in life, we like to take things apart into more elementary building blocks so we can unravel this rich structure.

To do such unravelling, we view these complex things as a superposition of much simpler ones. You could imagine trying to work on a screen or tablet with graphical tools putting down broad strokes of color, and then gradually adding finer details. What you are doing is constructing a complex drawing with elementary tools or action.

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Ingrid’s definitive text on wavelets, published by SIAM.

Decomposing into simple building blocks can be done in many different ways in mathematics. Wavelets are one way that emerged gradually as a powerful tool in this direction both theoretically and computationally. It is powerful because it can capture many different scales. For example, in an image or photograph, there are large color planes, and small details such as edges or textures. For large color planes, you might want to use a building block that gives that feature. As you go to smaller and smaller detail, you try to take more localized ways of making changes to your image. Wavelets are mathematical building blocks that allow you to decompose complicated things in such a multiscale fashion.

Figure 5. Daubechies wavelet on ECG signal 
A Daubechies wavelet on an ECG signal.

Wavelets are computationally very useful, as you can start with your image and think of the fine details as all the information you lose when you blur the image. All that information is fine and local. You can do that again and blur even more, and so on. It is like peeling off layers, and each layer has its own building blocks. It gives you a fast way algorithmically of making a decomposition.

AB: Wavelets also have applications in JPEG files?

ID: Older JPEG files use a discrete cosine transform, but JPEG 2000 (used for many current internet applications) uses wavelets.

AB: Your work spans both mathematics and its applications to science and engineering and beyond. What do you think the role of mathematics is in the sciences and engineering?

ID: Mathematics is what you need to quantitatively and accurately describe phenomena in science. Mathematics is the discipline we invented to be able to do that. When people describe the “unreasonable effectiveness of mathematics”, they seem to say mathematics is more effective than other things.  But we really have nothing else to do that. I am not saying there aren’t other things in life. There are tons of things in life in which mathematics plays no role. But to accurately and quantitatively describe things in science and engineering, mathematics is what you need.

We have built many mathematical frameworks as we realized that certain ways of understanding things are useful in more than one context. That is when you start abstracting, giving names to mental concepts in one field or another.  That is how you build mathematics.

At some point you start enjoying the mathematics—just the building of these mental concepts. That is what pure mathematicians do: they work only with those abstract concepts. It is similar to when artists play with materials and build things that others enjoy.

AB: Congratulations on receiving the Math+X award from the Simons Foundation, which comes with a 1.5 million dollar grant. Can you tell us how you came to receive this award and how you plan to use it?

ID: The Simons Foundation has, for a number of years, given these awards. They ask universities to nominate someone, and the nominee has to write a five-page description of how they see themselves, their research, and what they want to do. I am fascinated by applications to engineering and other fields; I am also fascinated by current developments in machine learning. Although I am in awe of many of the achievements of machine learning (such as recognizing faces), I think there is a challenge in understanding why these machine learning algorithms work so well.

Right now in machine learning if you give the algorithm lots of data that is labeled (for example, representing one species or another), it will churn through these and learn how to distinguish type A from type B. In the end, you have something that is very complicated, but we haven’t gained much insight into why A is different from B. In other situations, we don’t have lots of type A or B data. I would like to understand machine learning at a higher level and use that in our scientific investigations. I described some of those in my proposal. I work with biologists, art historians, and so on, and I described those collaborations and they liked it.

AB: In my recent interview with Nassif Ghoussoub, he mentioned that more mathematicians need to be working on machine learning.

ID: Some mathematicians feel that if something is too complicated with no nice description, then you cannot do good mathematics with it. However, I feel that if something works and achieves results and we don’t understand it, then we need to work on it. I agree that we need more mathematical work on machine learning.

AB: Would you explain your work and interest in using mathematics to restore art and detect forgeries?

ID: I had a colleague who was interested in art, and who thought that image analysis could be used for art history and conservation. He convinced people interested in image analysis and art historians to create joint workshops, which really were quite interesting. The first application was to detect forgeries; although we had some good results, that was not the most interesting of the projects I’ve worked on in this direction. Once they saw what we could do, others would approach me with other projects.

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Portrait of Ingrid by Charlotte Caspers.

For example, some projects were trying to getting a better impression of a painting hidden under another painting, or removing cracks virtually from a painting to get a better reading of the work. The projects were mathematically and engineering-wise quite challenging, and different from other fields. These provided excellent projects for my students, and provided interesting results for art historians and conservators. We also use machine learning for these projects.

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Ingrid lecturing on applications of mathematics to art restoration.

AB: What role does beauty play in mathematics? Do you have views as mathematics as an art or science (or neither/both)?

ID: Think about what it means to find something beautiful. For me, I recognize in myself a feeling of joy, awe, and reverence. It gladdens the heart and lights up the eye.

Mathematics is beautiful as it elicits the same emotions in us. We may recognize it as elegant or different. It is awe-inspiring and joy-inspiring at the same time. For example, consider the experience you have after understanding something arduous, glimpsing truth, or a way of doing things in a new way… with all these things you get that feeling.

An element of surprise is also important. Beauty always has surprise. I don’t find it strange to talk about mathematics as beautiful. It’s not just mathematics; biologists come up with extraordinarily elegant ways of testing a theory or devising an experiment. Economists when checking a theory have to come up with ways to test a hypothesis, and some of these are quite elegant. Beauty comes up in many abstract disciplines.

To me, the fine arts (music, painting and so on) are a short cut to part of the human emotional experience. It gives us access to part of ourselves on our emotional plane. Mathematics is experienced in the intellectual plane, and is a tool to do things intellectually. In that sense, mathematics and art are different, but this notion of finding beauty in mathematics is on the emotional plane.

Saying whether mathematics is an art or science is a way of trying to capture something with a sound bite. Using only one word to describe it simplifies things too much.

AB: The state of women in mathematics and STEM is of active interest these days, with many mathematics departments not even reaching 20% of their tenured faculty as women.  What advice would you give to young women studying mathematics? Also, what advice would you give to Departments to help achieve gender parity?

ID: Most people agree that women are just as good as doing mathematics as men. If you look at the papers that are written, it would be hard to decide from the math whether a man or woman writes it. The number of women in mathematics depends enormously on what country you consider. Even in Europe, neighbouring countries can have completely different ratios of men and women mathematicians. I think this more of a cultural thing than anything else.

If you are a young woman, you should not believe anyone who tells you that mathematics is not a job for women. To become a professional mathematician, you have to really love mathematics. As an academic mathematician, the thing you will get the most recognition for is your research, which can be a challenging thing. You spend ages trying to figure things out. When you figure it out you are overjoyed, but then you realize you want to understand it better. As you understand it better, you feel like a moron for not having seen it sooner. Research has fantastic highs that are not very frequent, and they don’t last a long time. If you love mathematics, then that is good enough for you. You can be a good mathematician as a man or woman.

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Ingrid (in her role as President of the International Mathematical Union) at the ceremony for the first female Fields medalist Maryam Mirzakhani.

Today we had a funny experience, where the faucet was broken down on the floor of our Department, and we had a plumber come and repair it. I’ve never thought about it, but it was the first time in my life I met a female plumber. Have you seen many female plumbers in your life?

AB: Not too many.

ID: I remarked about it to her, and she said she hears it all the time.

AB: Maybe you were the first female mathematician she’s met.

ID: She didn’t say that. There are many other pursuits in life that woman can excel in. We are accustomed to certain cultural things.

I remember a joke from a while ago when Margaret Thatcher was prime minister in England. A kid on television was asked what he wanted to do. He said maybe fireman or astronaut. They asked him “Not prime minister?”, and the boy responded “Nah, that’s a woman’s job.” I hope these cultural shifts are persistent, because we react to what we see.

AB: I just wrote about Emmy Noether getting a statue. And you were involved this year in unveiling a plaquette of her for the ICM Emmy Noether Lecturer.

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Emmy Noether plaquette by artist Stephanie Magdziak.

ID: Yes, and the likeness of her on the plaquette is very good. And there is a bust of her in Munich. It’s not very representational of her, however.

AB: I’d like to close with looking forward. Given your broad interests, what would you say are some of the major directions for mathematics and its applications in the future?

ID: I don’t like to make broad predictions. For me, machine learning is my last big project. On the other hand, when I was at the IMU, I was trying to help developing countries foster a more mathematical culture. I think it will help them with their economic development, as it will train better engineers and scientists. I think having good mathematicians helps the economy. There are economists that have studied the impact of having good mathematics training, and who feel there is a definite connection. I hope the mathematics community will help that effort in the coming decades.

Mathematics is also at a very exciting time, with so many cross connections happening between areas of mathematics and other fields. It is a very fun and fertile time to be a mathematician.

Anthony Bonato

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