In 2010, you were awarded the Fields Medal, seen as the ‘Nobel Prize for mathematics’. Can you tell us a bit about this honor and your work?
Cédric Villani: Even when it was established, the Nobel Prize was not designed to recognize all the sciences, without exception. So in 1924, at the International Congress of the International Mathematical Union (IMU), Canadian mathematician John C. Fields – bemoaning the lack of a Nobel Prize for mathematics – suggested awarding a number of medals to recognize notable advances in the discipline. Between two and four Fields Medals are awarded every four years to researchers (not more than 40 years old) at the opening ceremony of the IMU International Congress.
I myself specialize in mathematical analysis, and more specifically in partial differential equations (PDE), which are at the heart of all of physics. In particular, I have been working on the kinetic theory of gases, which aims to characterize gases using the statistical profile of the positions and velocities of the particles of which they are formed. We use two very well-known equations to do this: the Boltzmann equation for low density gases and the Vlasov equation for plasmas. I’m also interested in transport theory, which aims to determine optimal resource allocation by minimizing travel. Both these subjects have a link with reality, but I approach them from their theoretical side.
Computer simulation seems to be mainly use for practical applications. In what ways can it be useful for more abstract research?
CV: Nowadays, in every area of science, digital simulation has really become the third pillar of research, alongside theory and experimentation. Doing evaluations or verifying things using computers is nothing new, but thanks to the computing power now available to us, we can often go much further than you can using experimentation alone in terms of accuracy, scale… What’s more, it is usually cheaper and easier to implement.
In my area of work, computer simulation is an adjunct to thought: using computing, for example, you can get an idea of the far-reaching behavior of an equation, and so derive possible lines for exploration. For instance, in our work on Landau damping – which describes the evolution of the electrical field in plasma –digital simulation has confirmed my intuitive hypothesis. Other times the reverse has been true; digital simulation has disproved my initial intuition and led me to radically change my opinions.
But computer simulation will never be a substitute for theoretical deliberation…
CV: No, that’s right. Supercomputers open up extraordinary possibilities for science, and we can only be fascinated by the speed of their progress. But you have to have very clear ideas right from the start; digital simulation is only there in a supporting role. Without a powerful algorithm, without the study of synthetic models that capture the essence of complex phenomena, the machine will be running on empty. A supercomputer is an amazing tool, but for optimum use it needs to bring together four different kinds of expertise: in modeling, to translate reality into equations; in numerical analysis, to determine the computable quantities behind these equations; in algorithms, to find a rapid way of performing these calculations; and in programming, to build the code capable of running the algorithm as efficiently as possible. We have had fantastic improvements in algorithmic performance, in the order of a million times in the space of few years: 1,000-fold because of supercomputer architectures which are doubling in power every year thanks to technology progress, but the other 1,000-fold is due to improvements in algorithms. There always has to be intelligence behind the computer. You can change the world with one algorithm!
So digital simulation demands complementary skills. And in France there is an internationally-renowned specialist mathematical university and, with Tera 100, built by Bull, the world’s most efficient supercomputer and the most powerful in Europe, would you say that we have everything we need to build this kind of essential ecosystem which brings together research and technology?
CV: Yes, I believe we do. It’s a major industrial and economic challenge. The race to ‘Exascale’ computing – to a billion billion operations a second – has begun. France is very well positioned in that race, with Bull, which is the European leader in this area. Given the importance of computer simulation to scientific research, encouraging the advent of these kinds of collaborations is a major issue for France and for Europe more widely. And theoretical reflection must not be forgotten. There is highly advanced mathematics and physics behind miniaturization of systems, signal transmission… Without realizing it, we live in a world built around Shannon’s theories of information and certain key algorithms, like Metropolis and Fast Fourier Transform (FTT). Other models, other algorithms will come along. But when? Which ones? Who will discover them? These are vast areas of thought and these new possibilities for multi-disciplinary collaborations are very exciting for academics and young people who want to become researchers.
Cédric Villani: biography
Born in 1973, Cédric Villani is a French mathematician who is currently Professor at Lyon University and Director of the Institute Henri-Poincaré: a national mathematical institute founded in 1928, supported by the CNRS[1] and UPMC[2], which acts like a vast ‘project hotel’ in all disciplines relating to mathematics.
Having studied at the école Normale Supérieure de Paris and taken a PhD at Paris-Dauphine University, Cédric Villani spent ten years teaching at the école Normale Supérieure de Lyon. His research interests lie at the crossroads where analysis, probability, statistical physics and differential geometry meet. A specialist in mathematical analysis, he is particularly interested in the kinetic theory of gases and the theory of optimal transport. With his colleague and former student Clément Mouhot, Cédric Villani has contributed to clarifying a controversy amongst mathematicians and physicists relating to plasmas, by explaining the spontaneous relaxation of their electrical fields under certain conditions.
In 2010, Cédric Villani was awarded the Fields Medal, the world’s highest distinction in mathematics. This honor was preceded by numerous others, including the Henri Poincaré Prize awarded by the International Association of Mathematical Physics, the Fermat Prize, the European Mathematical Society Prize, the Jacques Herbrand Prize from the French Academy of Sciences, the Peccot-Vimont and Cours Peccot Prize from the Collège de France and the Louis Armand Prize.
[1] CNRS: Centre National de la Recherche Scientifique – the French National Center for Scientific Research
[2] UPMC: Université Pierre et Marie Curie – Pierre and Marie Curie University, Paris
