1- What is your research domain?
2- How did you come to do mathematics?
3- How do you see your job as a teacher-researcher?
4- What does it mean to you to be a mathematician?
5- Are there mathematicians that influenced you?
6- What does it mean to you to be an invited speaker at the ECM?
7- What role does collaborative work play in your work?
8- Mathematics, even when it is called applied, is a fundamental science. What does this mean to you?
1- What is your research domain?
My research essentially revolves around computational sciences, in a variety of theoretical and applicative directions. I have both developed concrete methods and addressed more fundamental questions about the compromises between precision and complexity. For example: how many bits are needed to encode an image with a certain precision? How many operations are required to perform a numerical simulation with a certain margin of error? How many measurements are required to reconstruct a signal with a satisfactory quality? All these questions are at the heart of many technological issues. Addressing them in terms of mathematical modeling allows us to try and grasp the fundamental theoretical limits in these compromises and also to understand if the concrete methods we propose allow us to reach these limits.
The challenge of information loss at the heart of computational sciences
A central problem in the computational sciences is the transition from continuous to discrete. It is a question of moving from a phenomenon that involves an infinite number of variables or parameters to a phenomenon that can be described with a finite number of values, for example on a computer. The simplest example of discretization is the sampling of a function. We keep a finite number of values and we ask the question: to what extent are these values informative? Do they allow us to have sufficient information on the object we had at the beginning for the targeted application? What is at stake at the moment of transition from continuous to discrete is the loss of information.
The concept of optimal method
Since by nature, by discretizing, we lose information, we lose the precision of the phenomenon, one of the fundamental questions around the discretization processes is how to quantify the compromise between complexity and precision. What makes one method better than another? To determine this, we can consider a mathematical model that will describe the class of objects we are trying to approximate, be it a signal, an image or a physical phenomenon, and consider the way we choose to measure the accuracy. The question that arises is that of the optimality of the methods: how can a numerical method that we propose be qualified as optimal for this class of objects?
Fundamental barriers
When I try to understand, for a class of mathematical objects, what is an optimal method, what interests me is to try to identify what are the limits, the fundamental obstructions that cannot be overcome. In a sense, this approach is tinged with a certain pessimism: it means that for a problem we have a theoretical limit. But if the fundamental barriers that we manage to identify mathematically block us, they also allow us to say: when a concrete method reaches these performances, it is an optimal method, since we know that we cannot do better. Let's say that a mathematical calculation proves that the human body cannot run a 100 meters in less than 9.5 seconds. If an athlete develops a technique that allows him to run in 9.5 seconds, we could say that he has an optimal running method.
Linear and non-linear methods
In the numerical methods we consider, we often distinguish between the so-called linear methods, where the result of the calculation depends linearly on the data - if we multiply the data by two, the result of the calculation will itself be multiplied by two - and the non-linear calculations, where such proportionality does not occur. Non-linear methods are often more complex but can be shown to have better performance on certain classes of objects that are of practical interest. For example, when processing images, or numerical simulation of physical phenomena such as shock waves, we discretize the phenomenon adaptively by allocating more information or computational resources on the edges of the images or on the transition zones. Such a treatment naturally leads to so-called non-linear approximation methods. For these more sophisticated methods, there are also fundamental obstructions.
The curse of dimensionality
What brought me to these problems were wavelet bases, which are a certain way of representing complex signal functions and which have proved to be particularly useful for image compression. But these questions quickly went beyond the particular framework of wavelet bases. What I have been working on for the last ten years is a way to numerically approach phenomena described by functions of a large number of variables, and where an obstruction called the curse of dimensionality will appear. It refers to the fact that in order to reach a given precision in the capture of a phenomenon depending on a large number of variables, the complexity of the discretization will explode exponentially as a function of the dimension. To make an analogy, think of someone trying to open a safe where there are small knobs with numbers ranging from 0 to 9. If there is one wheel, there are only ten trials to be made, if there are 2 wheels, one would have to explore 10 x 10 combinations, if there are ten wheels, one would have to explore 10 x10 x 10 x... etc. combinations. A similar problem arises for exploring a high-dimensional space finely enough.
Getting around the curse: Ockham's razor
These questions are frequently encountered in the physical and technological world, where we are constantly confronted with the study of phenomena that depend on a large number of variables. The question is then to know if there are additional properties that will allow us to bypass the curse of dimensionality. They can be properties of regularity: the objects depend on many variables but vary little. It can be the notion of sparsity: the phenomenon depends only on a small number of hidden variables that are not known from the start. This is sometimes called Ockham's razor principle, a scientific postulate that leads to favoring the explanation of a phenomenon involving the smallest possible number of explanatory variables. The question is then: how to identify more precisely the mathematical properties that allow us to bypass the limits of the curse of dimensionality? To what extent are these mathematical properties verified for phenomena occurring in practical applications? How can these properties of regularity and parsimony be found in certain physical phenomena? Does the real world naturally produce properties of regularity and sparsity?
Small data vs big data
What I do, the methods I develop, have connections with data science, but the framework in which I place myself is not that of big data. I am in a small data scenario with data that are difficult to acquire. As an example, take the study of the performance of an engine as a function of variables such as the mixture of the different fuel components injected into it, but also properties such as the geometry of its configuration. For each possible combination of variables, we can do the physical experiment of running the engine, or do it by a numerical simulation, and we will be confronted with the problem evoked previously with the example of the safe: we will have to do tests with potentially a very large number of combinations and, for each combination, either a physical or numerical experiment. However, we cannot afford a very large number of physical experiments and digital experiments. We are therefore not in the field of big data but in that of small data, in the sense that each piece of data is costly in terms of experimental or computing time.
2- How did you come to do mathematics?
As far back as I can remember, I have been attracted to this discipline. In France, scientific studies are centered around physics and mathematics.I realized that what interested me in physics was the passage to the equations and in fact the mathematics part in it. This led me to make a choice that turned out to be quite natural.
3- How do you see your job as an academic?
I cannot embrace my profession as a researcher without leaning it against that of teacher. I have a double role, as the custodian of a scientific corpus that I disseminate through teaching, and as a researcher who makes a very modest contribution to the scientific edifice. For me, these two roles go together. Moreover, teaching often leads to good research ideas.
4- What does it mean to you to be a mathematician?
What is interesting about doing mathematics is that one works with a common language that allows one to interact very quickly with colleagues of all cultures and nationalities, as well as at the disciplinary interfaces - although you need to keep in mind that not all disciplines have the same familiarity with mathematical language. This is the open dimension of mathematical language. On the other hand, that language is also isolating: one cannot talk about what one does with one's family and friends as easily as if one were a researcher in medicine, literature or biology. You often come up against a wall. But that isolation can be a kind of haven or refuge, which is nice. I suppose researchers in other disciplines might also feel the same way. And then doing mathematics is also being exposed to beauty, a very particular beauty. Research in mathematics is one of the professions where people most often use the word "beauty". A few years ago, Yves Meyer gave a conference on "the true, the beautiful and the useful in mathematics". Do we do mathematics for truth, for beauty, or for utility? I believe that of the three, beauty is essential. I often tell students, because we are in a world where we have to justify ourselves all the time, to appreciate the beauty of what we are doing. The other components, truth and usefulness, are only strengthened by it.
5- Are there mathematicians that influenced you?
More than by individuals, I am above all marked, in the history of mathematics, by great developments, which are collective adventures. I find it fascinating to watch how things are built. Sometimes one has the impression that they are gradually revealed, like a monument that was hidden and that reveals itself. Take the history of Fourier's series. It is a permanent back and forth movement between very applied questions and fundamental developments. We start from the very concrete problem of representing and calculating the solutions of physics equations modeling the propagation of waves or heat. The study of Fourier series then infuses all the development of a rigorous framework for analysis in the 19th century, with the theory of measurement - what does the notion of "almost everywhere" mean - up to absolutely fascinating links with the theory of numbers, of which we see an example with the mathematical theory of quasicrystals. And finally we have very applied returns with signal theory and the development of alternative representation methods such as wavelet bases. But if I have to mention names of people who have marked me, it will inevitably be colleagues whom I have had the chance to work with and who, in addition to being mathematicians of great talent, have also marked me as human beings: Yves Meyer, Ingrid Daubechies and Ronald DeVore.
6- What does it mean to you to be an invited speaker at the ECM?
A big quality generalist conference such as the ECM is for me the opportunity to confront my work and ideas with the international community in fields other than mine, and in return to learn things in those fields. In a specialized conference, we know that we will make progress on the questions we address and that we will be able to have fairly detailed discussions. In a generalist conference, I will not necessarily listen to the presentations that are most naturally related to my discipline. One may be pleasantly surprised to be confronted with questions and interactions that one would not have expected at all, which will eventually allow one to address a problem in an unexpected way.
7- What role does collaborative work play in your work?
Collaborative work has become progressively dominant and essential through the encounters I have had and through mentoring students - my list of publications attests to this. Recently, the lockdown related to the covid19 health crisis has made me realize that as far as I am concerned, mathematics is a social activity. I need to meet my colleagues for real! I am quite puzzled by what the video-conferences during the lockdown have given us to believe about a possible generalization of this way of working.
8- Mathematics, even when it is called applied, is a fundamental science. What does this mean to you?
"There is no watertight boundary between pure and applied mathematics" is the title of an interview I gave to CNRS le Journal1 a few years ago, which I could repeat word for word. The distinction between pure and applied mathematics seems artificial to me. I am in a department that does application-oriented mathematics, but it is essential for applications to also do mathematics that is not application-oriented. Without taking the field, without offering the possibility of moving to an abstract framework that goes beyond the application, that does not aim at it, one deprives oneself of a treasure for both fundamental and applied advances. I would add that mathematics is a profoundly human science in the sense that human beings in all their generality do mathematics from the data that their environment offers them: in daily life, each and every one of us optimizes, linearizes, extrapolates, interpolates, classifies, generalizes, filters... In my opinion, it is the mathematical language rather than the mathematical reasoning that is the cause of the rejection of mathematics by a part of the population. It is at the moment when it is necessary to use this language that the cleavage occurs because it requires an initiation.Access to mathematical language is not immediate and it is understandable that it is divisive in society. It is like playing jazz or tasting wine, there is always a principle of pleasure at the heart of the initiation, but it is also a language to be learned, and which is learned with more or less ease!
Download the text of the interview in French
