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# Bringing a new light on old problems – Interview with Laure Saint-Raymond

*Laure Saint-Raymond is a French mathematician working in partial differential equations, fluid mechanics, and statistical mechanics. She is a professor at École Normale Supérieure de Lyon. In 2008 she was awarded the EMS prize and in 2013, when she was 38 years old, she became the youngest member of the French Academy of Sciences. Roberto Natalini, Chair of the Raising Public Awareness Committee of the EMS, took an interview with her, which was first published on EMS Newsletter, No. 102, Septeber 2017, pp. 23-25. It is reprinted here with the permission of the journal. Here the original version.
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*Roberto: Let me start with a very trivial question: when did you become interested in mathematics?*

Laure: Actually it was quite late. In high school I was a good student, but somehow I was more interested in music. But, being good in maths, as it was the habit in France, I entered in the so called “Classes préparatoires” (which prepare to the entrance selection for the “Grandes Écoles”), and then in the École Normale Supérieure (ENS) in Paris. Here I found very enthusiastic teachers and so my interest for mathematics started.

*R.: How were your parents involved in your interest in mathematics? Did you have an important teacher before the university?*

L.: I had a maths teacher during the “Classes préparatoires” with a strong passion in mathematics and in particular in logic. However, even though my father is a mathematician, I was not really pushed by my parents to go in this direction. I was quite free to make my choice.

In the ENS I found many inspirational professors, like the physicist Yves Pomeau, who used to introduce baby models to catch important physical phenomena such as the growth of trees. On the mathematical side, I should mention Yann Brenier, with his very original way to see all things, and Henry Berestycki. And finally it was with *François* Golse that I really discovered the connection between mathematics and physics, or I could better say, how to couple the rigor of maths with the inspiration arising from physics.

*R.: What are your main fields of interest in mathematics and how and why did you start to work on them?*

L.: I started my research in plasma theory, looking at the qualitative behaviour of beams of charged particles in strong magnetic fields. The approach was driven by kinetic theory methods, with a deep interplay of mathematics and physics. In collaboration with my PhD advisor *François* Golse, we solved one part of Hilbert’s sixth problem. This problem consists in developing mathematically “the limiting processes [merely indicated in Boltzmann’s work] which lead from the atomistic view to the laws of motion of continua”.

What we established is the rigorous transition from the Boltzmann kinetic description, where the gas is considered as a collection interacting particles described statistically, to a fluid description, given by the Navier-Stokes equations, where the flow is described only through macroscopic quantities such as the average speed or the pressure of the fluid.

*R.: What have been your main original ideas in proving the limit from Boltzmann to Navier-Stokes equations?*

L.: Actually, I have contributed both to collect and organize in an original way many existing techniques, and to develop some new mathematical tools, as the so called L^1 velocity averaging lemma related to dispersion and mixing. The Boltzmann equation describes the state of a gas using a distribution function that depends on space, velocity and time. It expresses a balance between two mechanisms, the transport and the collisions. This equation has no regularizing effect, and so, if we have a singularity in the solution, we keep it forever. And this is a problem when you study the fast relaxation limit (i.e. the asymptotic behaviour when the relaxation to local equilibrium due to collisions is much faster than the transport which correlates close positions) because you need some compactness.

It was noticed by Golse, Lions, Perthame and Sentis, that observables, which are obtained by taking averages with respect to the velocity variable, are more regular than the solution itself. We were able to combine this result with hypoelliptic properties of the transport to prove that if you gain some nice behaviour in the velocity, then you can gain something also in the space variable. This was one of the main tools to prove our convergence result.

*R.: What about some other problems you have considered?*

L.: The other part of my work is concerned with large scale geophysical flows where the Coriolis force is dominant, taking into account the dominating influence of the earth rotation. Classical methods for linear singular perturbation problems fail when the oscillations cannot be described explicitly because one does not even know whether the waves will be captured or dispersed. For instance, close to the equator, the spatial variations of the Coriolis acceleration cannot be neglected. The spectral structure of the propagator is completely modified and one can prove that fast oscillations are trapped in a thin band of latitudes.

Another challenging problem is to understand the interaction with the boundaries, which is responsible for most energy exchanges (forcing and dissipation), even though it is concentrated on very thin layers close to the bottom and the surface.

Now I try to understand the propagation of internal and inertial waves in ocean, in regions with a variable topography. I collaborate with physicists to understand how to separate the different time and space scales, neglecting the very complex dynamics at small scales but keeping the qualitative behaviour of the solutions.

*R.: Are you still working on the Hilbert’s Sixth problem?*

L.: Yes, of course! More recently, mainly in collaboration with Isabelle Gallagher and Thierry Bodineau, I worked on the full problem, namely, to make a rigorous derivation of fluid models from particle models, which I think is a much more difficult problem. A very challenging question is to explain the appearance of irreversibility at the macroscopic level. At this stage there is no general theory, but some special results have been obtained. For instance, we were able, under some specific scaling assumptions, to obtain the Stokes equations directly as the limit of particle models. It is not the optimal result, but it is the first rigorous derivation of fluid equations from Newton’s mechanics. Our starting point is solely the deterministic collisions of hard spheres, coupled with a suitable entropy bound. However it is quite clear that we cannot hope to obtain the full result, to say the convergence to the Navies-Stokes equations, using the same ideas. So, we are looking around for some new ideas.

*R.: You have been awarded with many prizes. Which one is the most important for you?*

L.: First, I have to say that, when you receive a prize, then you receive a lot of them, which does not mean that you have more merit. Of course prizes come as some recognition from the mathematical community, and I am very proud of the EMS prize I received in 2008. But I think that prizes should be overall understood as an encouragement to go further and maybe to take more risks and more responsibilities.

*R.: Speaking of responsibility, I remember your intervention in 2015 about publications, during the event for the 25th anniversary of the EMS at the Institut Poincaré in Paris.*

L.: Yes, I am really concerned by this point. I believe that, as mathematical community, we publish really too much and that senior people with an accomplished career should be more careful and selective when submitting papers. Most of the time nobody reads these papers, and it is even difficult to find somebody to do a good peer review. On my own I adopted as a rule to refer each year at least twice the number of the papers I publish. It is crucial to properly review the papers, and also to read and discuss articles from other researchers. This is the only way to be a scientific community.

I believe that science is a common project, not an isolated enterprise. On the other side, unfortunately, we are faced to all these national and international rankings, which are very often quite meaningless and based on quantitative metrics. Nobody is interested in what people are really doing, and I think it is bad for mathematics.

*R.: How much in your work is intuition and how much it is just hard work?*

L.: The starting point of each of my papers is to try to bring a new light on a problem. Unfortunately, many of my papers are a mess of technical details, but still we try to explain one or two new ideas. In this sense my works are not only technical, but there is always some intuition to be made rigorous. You have an idea, then you try to work the details and you struggle with some problems. And to solve these problems, you have to understand something that you missed before. You don’t fully understand until you have a complete proof. This is, in my opinion, the essence of the mathematical work.

*R.: How do you organize your work? Do you follow a routine, or it varies a lot according to external conditions?*

L.: I work most of the time with the same collaborators, since it takes a lot of time to share the same language, the same feelings on the topics and so on. I’m not this kind of person who goes to a conference, meets some people and immediately starts a new collaboration.

Two years ago, I spent a sabbatical in the US, where I had a lot of time and no duties. It was really quite and I had a great time to work with no constraint, but somehow it was not long enough to develop new collaborations.

*R.: How has been important for you to be in Paris for many years?*

L.: For a very long time we didn’t leave Paris to stay close to our parents who helped a lot with the children, and I have to say that I didn’t quite realize the great opportunity I had. Actually, in Paris it is possible to discuss and collaborate with a lot of people with different backgrounds and ideas.

Out of Paris you are maybe not exposed to such a large mathematical community, but somehow it gives you more opportunities to meet people doing something really different and to go into new research directions. I have now moved to Lyon where I am very happy.

*R.: In France, women in mathematics are not so common, even if in recent years something changed. Could you explain the difficulties that sometimes women can experience in having a satisfactory career in mathematics?*

L.: Actually, I have to say that in my experience I didn’t feel any discrimination against women. My impression is that somehow the problem is more in our society. One reason why women are not following scientific careers is maybe the French system of education based on selection and competition, which can discourage women to follow this path.

Also there is the dominant model of family, where men are choosing their jobs and women are following their husbands. In the academic careers very often it is hard to stay together.

*R.: And how did you manage to face these problems? You have a large family, with six children. How is it possible to work so hard with a lot of children and commitments? *

L.: My husband is just great and makes everything at home (she smiles). Also, for many years our parents helped us taking care of the kids very often. Besides, the French school system (starting at the age of 3) is helpful in this regard.

But nevertheless, for a long time I needed to be at home at 5 pm almost every day…. and I wrote less papers than most of my colleagues!

*R.: What do you do outside math? Have you hobbies? What do like to do?*

L.: I do plenty of things, like hiking and skiing, and this is also one of the reasons why I like very much to be in Lyon. Also I enjoy music, playing the cello. Sometimes I even play chamber music with some colleagues.

*R.: A last question. What is your bedtime reading?*

L.: It is hard to say, sometimes I just sleep (laughing). But, for instance I like very much Eric-Emmanuel Schmitt, for his positive attitude about life. More generally, I look for books where I find a supplement of energy to live, something helping to find the positive sides in our life.

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