## Maths is (also) for women

As a little girl, Ingrid Daubechies felt she was always a bit different from her friends. Before the age of 6, she was already familiar with complex mathematical concepts, and when she couldn’t sleep she did not count numbers as others do, but instead mentally computed powers of 2.

As a young and promising scientist, she entered the Vrije Universiteit Brussel [1] at 17. After completing her undergraduate studies in physics she worked at both the CNRS Center for Theoretical Physics in Marseille and her own alma mater, to develop the material for her thesis in quantum mechanics (1980). During the first years of her career, and until 1987, she remained at Vrije Universiteit, but then headed to the US to become a Member of Technical Staff at AT&T Bell Laboratories.

During a visit in 1986 to the Courant Institute of Mathematical Sciences [2], an independent division of New York University [3], Daubechies made her most important discovery, in the field of wavelets (wavelets are mathematical functions useful in digital signal processing and image compression as well as in many other branches of applied and pure mathematics). After her move to the US, she continued in this direction, which eventually led her to engage in more interdisciplinary work. From 1994 to 2010, she was active in the Program in Applied and Computational Mathematics at Princeton [4]. In 2011, she moved to Duke University [5], where today she serves as James B. Duke Professor of Mathematics, nurturing interests that cover mathematics as well as arts and other fields.

She is a global authority on techniques to decompose and analyse images on a mathematical basis. She is the recipient of numerous awards, and became the first female full professor of Mathematics at Princeton. Today, she serves as the first woman president of the International Mathematical Union [6] (IMU). She visited Trieste during the 2 to 21 June joint ICTP-TWAS School on Coherent State Transforms, Time-Frequency and Time-Scale Analysis, Applications [7], for which she also served as co-director.

Talking with TWAS staff writer Cristina Serra, Ingrid Daubechies explained why more women should engage in mathematics.

**There's a common assumption that women are less good than men at mathematics. What could be the reason for this, assuming it is true? **

I disagree with this view – *completely*. There is a highly variable percentage of women in academia and in departments of mathematics across Europe. Differences are so enormous that it becomes obvious that it has something to do with cultural habits, which differ from one nation to another, and not with intelligence. I have a very cynical colleague who says that the number of women mathematicians who are in the academia is inversely proportional to some average of the amount of money and prestige that this job can grant: If there is little money and no prestige, there you’ll find more women. I agree: These aspects seem to play a much larger role than being smart.

**You are the first female president of the International Mathematical Union (IMU). This proves that the glass ceiling can be broken: Have you been elected for your curriculum or for your semi****nal discovery?**

My election as IMU president had little to do with my specific scientific achievements. Of course, there is a general level of distinction that the IMU Nominating Committee looks for. Beyond that, it has to do with the need to find a person who really cares for the mathematical community, who wants to create networks. Many mathematicians believe mathematical talent is distributed more or less uniformly around the globe, and the IMU cares about education in developing countries. This is not just about spotting extremely rare top geniuses, but also about fostering the growth of strong, healthy maths communities that interact productively with the whole mathematics world. Raising awareness about and trying to remediate the scarce number of women in mathematics is, to me, part of that whole package.

**As the IMU president you must be in a powerful position. How does it feel?**

I do not to see this position in terms of power, but in terms of service. I like to work with people rather then steer them. I think it’s important to find out what people like to do: Sometimes I feel I must push harder in a situation, but that is still just to change it for the better, if others indeed agree. Most of the time my position is just about building consensus, talking with people, coming up with new ideas and see whether they can work.

**Did you develop this passion for mathematics early in your life?**

I always wanted to understand things and I never had problems with mathematics, because it has always been a tool that I used to do physics. Later in my life, as I became interested in not only physics, but also in engineering and other subjects, I used the mathematics that I knew to do the rest. If I have always wanted to learn more it was because different types of problems require different types of mathematical tools. I use and develop maths to approach problems. Sometimes that involves developing new mathematical tools.

**You have joined a number of European and American universities, like Vrije Universiteit in Belgium, Rutgers, Princeton, and now Duke. Did you experience major differences between the two academic systems?**

When I left Europe, almost 30 years ago, life in US academies was much more fluid than here. Young scientists in Europe were not given responsibility for their research, there was always a boss to control everything. Even for their teaching activities young scientists couldn’t apply for funding. Now the situation is much more fluid, but it wasn't then, in Europe. I’m very happy at Duke: It’s a place where scientists can easily engage in interdisciplinary work. I find this attitude very helpful work now I work on applications of mathematics to fields like art history and art conservation. At Duke they like this a lot and I find great enthusiasm around.

**As a mathematician your name is linked to the field of “wavelets in image compression”. What is it about? **

Wavelets are used in signal processing. A high-resolution digital image is composed of a huge number of tiny pixels (a pixel is the smallest controllable element of a picture represented on a screen), each of them a small square of one constant colour or grey level (if the image is not in colour). Numbers can quantify the exact colour or grey level of the pixels, so that the image corresponds to an enormous array of numbers. Yet most pixels in an image are very similar to their neighbours; although of course there are also many places in the image (wherever something interesting happens) where a pixel differs appreciably from at least some if the neighbouring pixels, these locations are still in the minority. A wavelet decomposition of an image exploits that aspect. Wavelets decompose the image into building blocks of different scale that, together, describe what's going on in the image. Simplifying a lot, this approach tells you where you need to put lots of detail (because pixels differ a lot from their neighbours) and where not, in image analysis.

**What exactly do you do in the application of image analysis to art conservation?**

We analyse old paintings with imaging techniques, to provide artists and people who restore old paintings with tools to understand the nature of the masterpiece from yet another point of view than what they are already using. The first step is talking to people to find out what kind of questions they have. The interesting thing is that we often do not have ready-made solutions for them, so their questions also pose interesting challenges for us.

**Can you make an example?**

We recently applied digital image-processing to the famous Ghent Altarpiece, a polyptych located in the Saint Bavo Cathedral [8] in Ghent, Belgium, which is composed of 12 panels, eight of which are hinged shutters. Dating back to the 14th century, it is considered not only a masterpiece, but also a key element in understanding the art history of that period in Western Europe.

Like most paintings of its age and materials, it has many small cracks or breaks in the paint layer, caused by differences in the way the paint layers and the underlying wood support react to temperature and humidity changes. We carry out virtual restorations of high-resolution digital versions of paintings by automatically detecting and removing cracks, which turns out to be remarkable challenging.

In the case of the Ghent Altarpiece we combined three detection methods to make a map of all the cracks. Then we virtually inpainted these cracks, thus reconstructing a sharper view of, in this particular case, the letters in a medieval book depicted in the background on one of the panels. This made it possible for paleographers to decipher many more words than the 2 they could read before; as a result they can now unambiguously identify not only that the painter referred to a particular text by Thomas of Aquinas about the Annunciation, which turns out to be especially appropriate for the panel in question.

**Where does maths come in, in this case?**

Basically everywhere. To analyse the Altarpiece, digitized at an incredibly high resolution, we designed mathematical algorithms to detect the cracks, to fill then in, and to detect letters in the “restored” writing on the book. In other projects connecting image analysis and art restoration, we develop techniques that will relieve art conservators from some of their more tedious tasks; after we develop the algorithms, we passed them on to other software experts, to turn them into user-friendly packages for professional art renovators. This has led to new interdisciplinary collaborations that are giving interesting results.

**Is maths an important subject for scientists from developing countries? What is your experience on this?**

Mathematics is very popular in developing countries – it has a great appeal because it is so neat – you literally solve problems and build approaches by just the power of thought. What it is sometimes surprising to me is that people prefer theoretical to applied mathematics – I think it is best to always develop both: applied mathematics helps in building strong STEM education, and will benefit other sciences and engineering; theoretical mathematics is the essential foundation, on which everything is built and that makes it possible to extend ideas useful in one direction to apparently completely different frameworks.

The good news is that mathematics is spreading thanks to the Internet and to the attitude that mathematicians have to make their work available (about a quarter of all papers are now available on Internet even before they are accepted for publication). I think this is great because it makes literature accessible to all. Also, maths is cheaper to do than other sciences, as it needs less investment in terms of labs and equipment.

As far as my experience with developing countries, I lived in Madagascar for three months, which is not very long, but nevertheless was an eye-opening experience. I still have links with people there, and I'm trying to help the local department of mathematics to further develop.

**What is your impression of TWAS and its role in the developing world?**

I think TWAS is wonderful in what it does for developing countries, especially the fellowships programme.