## Between maths and leadership: an interview with Sylvie Benzoni*

### Interview by Francesca Arici and Anna Maria Cherubini

- Born inFrance
- Studied inFrance
- Lives inFrance

### Interview

**Sylvie Benzoni is the Director of the Institut Henri Poincaré.**

When have you realised you liked mathematics and that you wanted to become a mathematician?

I think, as far as I can remember, I always liked mathematics.

**So you were supported by people telling you that you could do research. And how did you end up working on your field of research?**

Yes, I was supported by the studies director, Jean Giraud. His advice was actually more precise: he knew I was from Lyon and the ENS was relocating to Lyon, so he suggested I should move back from Paris to Lyon, where one of the professors, Denis Serre was looking for a student to supervise. I accepted to go to Lyon to start the DEA and to prepare for the * agrégation.* It’s a competitive examination to get a degree as a teacher but everyone did it at that time. After that I also had an idea of using my PhD to work in industry, and in fact I found a subject related to industry. An agreement was concluded between the ENS Lyon and the Institut Français du Pétrole (IFP) and I had an advisor from IFP, besides my academic advisor.

**For your PhD you worked on applications of fluid dynamics. What are you working on now?**

I started on fluid mechanics equations and I more or less stayed on that. The first problems I worked on were concerning equations for time dependent phenomena, at the time it was about two-phase flows in pipelines. It was not so clear which where the good equations.

Since then, I’ve worked with equations well agreed on when they arrived on the scene, like the Euler equations in gas dynamics. We don’t know how to solve them, but we are interested in finding special solutions and study them, in particular their stability.

It’s rather easy to identify some special solutions. For instance, if we think of the equations for the motion of the air around us, small perturbations give the modeling of sound, and we can find solutions which give the speed of sound. This is the starting point, then we can think of more complicated motions, and then we can consider shockwaves.

I was trained at shockwaves – hyperbolic systems of partial differential equations – during my master degree. These are the kind of waves that we can see and hear, for example from the sonic booms of fighter jets. They are nonlinear waves in the sense that their speed depends on their shape. But we can also consider phenomena in which the wave speeds depend on their wavelength, this is called dispersion. In the last two decades I’ve been interested in nonlinear dispersive waves and in their stability, because to make them observable, it’s better if they are stable. My ultimate goal is to really understand dispersive shockwaves: this is an oscillatory version of shock waves. Sharp shock waves are reasonably well understood in nondispersive media. But if you have dispersion, then the pattern is not just a step function, it’s an oscillatory pattern. And it’s not steady, it expands as time goes on. This problem has been studied since the 70s by physicists, but results are not really rigorous: I’m trying to understand rigorously how this works. For instance, we have tidal waves on rivers going into the ocean. In France we call them Mascaret. When the tidal wave arrives, there is a difference of level in the water: the higher level tends to go up the river and behind it there are some oscillations. This really is the physical manifestation of nonlinear dispersive waves.

The building blocks of nonlinear dispersive waves are typically solitons and periodic waves. I’ve been studying the stability of periodic waves, surprisingly enough, it’s a rather recent topic in mathematics.

Studying these equations is a way of tackling stability, it’s called modulational stability.

**Physicists have worked on periodic waves: have they tried to solve the equations?**

There are some completely integrable equations involving this kind of waves, for example the Korteweg-de Vries equation, which they studied with algebraic computation, using inverse scattering.

But in fact, I’ve been interested in a more general class of nonlinear dispersive equations, which is nice because you have some structure and you can apply them to several fields of physics such as water waves, capillary fluids dynamics, nonlinear optics or quantum dynamics. I prefer to see the general structure, otherwise on specific cases calculations are a nightmare, from the algebraic point of view.

**Are there any results you are very proud of?**

Well, I could say there are several ones. I think there are at least three topics I think I can be proud of, not only for my results, but also because this led other people to work on problems on which almost nobody was working when I started. On the way I started collaborations with great people.

The first topic is about the stability of shockwaves, but not the usual ones, the ones you associate with phase transitions. In my case it was a two-phase problem. In a mixture of liquid and vapour, you can study the dynamics of liquid-vapour interfaces. Using some assumptions in the modelling that are not completely clear even to physicists, I studied the stability of liquid-vapour interfaces by means of techniques known for classical shocks, and I found some surface waves which are similar to seismic waves, where you have elasticity: there are some waves which propagate down into the earth and other waves which are roughly localized on the surface. It was not that easy to find them.

I’m not sure they are important in the physics of liquid-vapour mixtures because it’s an idealised model, but my work encouraged people – from China to Germany and the USA – to work on the stability of phase transitions with various other assumptions.

Another topic is discretization of shock waves. If you use a numerical scheme to simulate shock waves, then it’s not so easy to understand how the shock waves are discretised by the scheme. I started to work on this issue and I showed results on existence and stability for semi-discrete shock waves, which arise when you discretize the space variable only. I got stuck on fully discrete shock waves and I stopped working on them. But I still like this topic very much. It’s about lattice dynamical systems, it’s full of open problems, and people continue to work on them.

The next topic comes again from liquid-vapour problems. I considered some equations which were those of the gases, but with a dispersion effect because it was a way of deriving my idealised model. I called them Euler-Korteweg equations (and now everybody calls them in this way) because they are Euler equations with additional terms coming from Korteweg’s theory of capillarity. I started to work on them and I met a physicist working on two-phase flows for nuclear plants using exactly the same equations. So I realised that the equations were interesting for physicists, and I should continue working on them. And I got some results, for example on the existence and regularity of solutions, really classical topics in Partial Differential Equation analysis. But what is most interesting to me is understanding the qualitative properties of special solutions like solitary or periodic waves, so I continued on that path. But I know that many of the people continue to get more existence and regularity results.

**What is your experience as director of IHP?**

To be a director, you need an entirely different type of skills: it’s not easy but it’s really a great adventure, humanly speaking.

It’s not easy but it’s really a great adventure, humanly speaking. When I arrived I struggled a bit to step in as director after the previous one (Cédric Villani, editor’s note), but the atmosphere is much better now that people really work together in a collaborative and cooperative manner. I can concentrate more on scientific aspects, even if it’s science communication more than research. I try to save one day a week to do some maths for my research.

**How did you manage the situation in the last year?**

As all similar institutes in the world we’ve had to shut down for some long periods. When we reopened, we set up hybrid events. Even when we don’t have any visitors we have online activities and we also have plenty of projects, in particular the permanent exhibition project for the Maison Poincaré. The IHP team has coped very well with the situation.

**Can you tell us something about the exhibition?**

At IHP, we are preparing a permanent maths exhibition on the ground floor of Perrin’s building, just opposite to Borel’s where the institute has been located since 1928. The project was initiated by Cédric Villani. We have been working on it since 2018 with researchers, teachers and science communicators. We want to show maths ‘alive’.

We want to show maths ‘alive’.

In Paris, we have a science museum, Le Palais de la découverte, which is closed now but hopefully will reopen, but we want to do something different. We are a research institute and we have researchers visiting on a regular basis, so we really want to bridge the gap between mathematicians and the public, and to show that maths is a very wide field of research and there are plenty of surprising things to learn.

**When did you get interested in popularisation of maths?**

In fact I really got interested in outreach when I saw Cédric Villani doing his first public lectures. And I said to myself: “OK, I could do this and I would like it”.

So I started to make some public lectures. I went on radio too, it was at the time of the Mathematics for Planet Earth year, and I got involved in that. I was also responsible for a few years of a monthly press review for the website Images des mathématiques. It was journalistic work more than popularisation of maths and it was really fun because we were a team and I always valued team work. It was also very interesting because it gave me more opportunity to interact with many mathematicians, both ‘pure’ and ‘applied’. You know, I was a researcher in applied maths, and it’s not alway easy to speak to people doing different types of maths.

I also joined the ‘Raising Public Awareness Committee’ of the European Mathematical Society, and I find very interesting to share experiences at European level.

So I have a lot to do and write, both for the exhibition and for other media. I really enjoy it. Of course it’s not research, but it’s rewarding and I’m proud of my work in this regard too.

**What do you think are the big challenges in communication of science?**

I think it’s difficult to communicate science, but the gap between laymen and mathematicians is so big that any mathematician could explain something, even in areas far from their field of specialisation.

It’s difficult to communicate science, but the gap between laymen and mathematicians is so big that any mathematician could explain something.

But it’s not that easy, and many mathematicians don’t want to do it, and often my job is trying to make them aware that it’s important to make maths more accessible. I think also that beside doing exhibitions, videos, documentary films, it’s also important to meet people, in particular young people in high schools. So I’m meeting students and setting up training for teachers.

It is not limited to high schools. For example, I was contacted by a teacher in a middle school in a disadvantaged suburb of Paris: students are requested to choose some kind of ‘role model’, in particular a female scientist, and they found me in internet and contacted me. I’m going to do some workshops with them, and I think it’s really cool.

**Regarding women mathematicians, does IHP have a policy on gender issues?**

When I arrived I noticed that the proportion of women in programs at IHP was low, most of the organisers and speakers were men. So I worked hard to make people aware that it was important to have more women involved. Now we are monitoring this and we tend not to accept proposals in which there are not enough women involved. Moreover, at Maison Poincaré we also chose to have an equal representation of women and men: we are going to present some mathematicians, and there would be as many women as men.

Gender gap involves many issues, the prominent one from my point of view is that we women don’t feel comfortable enough in the mathematical world as it is, and this definitely has an influence on our career paths and more generally on our lives.

On a more general ground regarding gender gap issues: gender gap involves many issues, the prominent one from my point of view is that we women don’t feel comfortable enough in the mathematical world as it is, and this definitely has an influence on our career paths and more generally on our lives. Even though from this interview it could look like I was always supported by men, this is not completely true. I did suffer from sexism, and I still suffer from it to some extent. Unfortunately, despite all the attention devoted to gender issues, I don’t think that the situation has improved for women entering academic life today. We need new perspective. In my opinion the latest book by Eugenia Cheng *x+y* should be a must-read. She suggests that we should change the world to make it more congressive, as she calls it. I like this idea very much.

* Photo Credit: Camille Cier