# Interview with a Young Mathematician: Emmanuel Trélat

#### Emmanuel Trélat Talks About His Career as a Mathematician

DIANOIA is publishing a series of interviews with young applied mathematicians. This time, Roberto Natalini interviews Emmanuel Trélat. Trélat is a professor at Université Pierre et Marie Curie (Paris 6), Laboratoire Jacques-Louis Lions, and is the director of the Fondation Sciences Mathématiques de Paris.

**Q: How did you decide to become a mathematician?**

**A:**When I was a student I hesitated between mathematics and physics, and I was maybe slightly more attracted towards physics, actually. But then I studied mathematics at Ecole Normale Superieure de Cachan and I discovered so many fascinating subjects. . .many of which were motivated by physics!

*Q: Could you mention some people who have been important for your education?*

**A:** My mother, who was a schoolteacher, undoubtedly developed my taste for mathematics when I was a child. Much later on, it was my teachers at ENS who opened many issues, and in particular Jean-Michel Coron, who introduced me to control theory. Then, my PhD advisor Bernard Bonnard taught me much and has become a close friend.

*Q: What is your main focus in mathematics, the main direction in your research?*

**A:** I have a special interest in control theory. This branch of mathematics is concerned with the analysis of systems on which one can act by means of a command, a control. Control problems emerge in any situation where one would like to perform some guidance procedure, steering a system from some initial configuration to some final one: vehicle, satellite, chemical reaction, biological planning, economical issue, etc. Optimal control theory is concerned with control problems in which one wishes more- over to minimize some cost functional. Many mathematical problems have also a strong intersection with control theory, such as Riemannian or sub-Riemannian geometry, optimal transport, mean field theories, etc.

*Q: Could you single out your best achievement in mathematics?*

**A: **I am proud of having pointed out the important role of the so-called singular trajectories in optimal control theory. In various contexts, I proved that the absence of singular minimizers is the main assumption allowing one to derive theoretical results (like smooth regularity for the value function, solution of the Hamilton-Jacobi equation), but also to prove convergence of numerical algorithms in optimal control, and of combined homotopy methods, in relation to conjugate point theory. I have studied extensively the relevance of such an assumption, proving that it is generic, or valid for some large classes of systems.

*Q: You have a strong focus on applications. Why are you interested in this direction and, also, is there a sort of neat line between applied and industrial math?*

**A: **It has always been a kind of challenge for me to try to go at the extreme point of the scientific chain of a given problem of mathematics motivated by a concrete application. Solving the Ariane launchers problem and implementing the real-life code was a personal challenge that I would like to solve, at least once in my life. I am not sure that now I would still have time to develop the whole chain as I did at that time. This is my personal taste: to be, in general, motivated by applications. But I also strongly and proudly support mathematics that does not seem to have an immediate application – note that this word “application” is in itself very difficult to define. . .and a bit dangerous to use. There is room for all mathematicians, for all tastes. Myself, I have recently deeply invested in the fascinating area of investigating spectral and geometric properties of sub-laplacians. Should it have some industrial application? Who knows. . .to answer your question about applied and industrial mathematics: I would rather speak of relationships between mathematics (at large) and industrial issues. For sure, they certainly feed each other. Not only should new mathematics be transferred to an industrial context, in view of applications and of creating innovation, but conversely, industrial problems raise difficult challenges that open new interesting mathematical issues. This is a two-way interaction that is very fruitful.

*Q: In 2012 you received the Felix Klein prize from the European Mathematical Society. This prize is awarded every four years to a young scientist for using sophisticated methods to give an outstanding solution, which meets with the complete satisfaction of industry, to a concrete and difficult industrial problem. In the motivation for your prize, we can read that it was awarded to you “for combining truly impressive and beautiful contributions in fine fundamental mathematics to understand and solve new problems in control of PDEs and ODEs” What kind of problems have you solved and how was your contribution important to them?*

**A:** I developed for Airbus Defence & Space (formerly, EADS Astrium Space Transportation) software that computed automatically and instantaneously the optimal trajectories for the problem of minimal consumption for the last stage of an Ariane launcher, and did this for any chosen terminal conditions, and any configuration of launcher. The main challenge of this problem was in the word “instantaneously,” and moreover error is not allowed. It was of course already known how to provide a fair solution to the problem, but computing only one flight required at least several hours, and even several days. The code I designed is able to provide the optimal solution within one second. This work, realized in collaboration with Thomas Haberkorn, took me around five years. The approach is based on a combination of the classical Pontryagin Maximum Principle in optimal control with numerical continuation methods and with a refined geometric analysis of the extremal flow, using quite recent results that are of a geometric nature (this is part of the so-called geometric control theory). Successfully integrated to the global optimization tools of Airbus, this real-time algorithm brought a big improvement for Ariane 5 trajectory planning. It also permits the design of new strategies for the forthcoming Ariane 6 launchers. I am proud of this success story which shows that mathematics plays a crucial role in technological innovation.

*Q: Are you able to directly interact with your industrial partners, or do you need some intermediate collaborators to translate math in practical implementations?*

**A: **I have experimented both ways.

- With Airbus I implemented by myself the code in its final form, which they now use. As I said before, this was a personal challenge to achieve the complete chain, at least once. Frankly, I would not have the time now to do that again.
- I have also experimented, with CEA (the French atomic agency) or with CNES (the French space agency), collaborations where an engineer turns the mathematical part into a final code. Both ways are fine; after all what is important is the final success of the overall scientific issue. In the case of my work with Airbus the second solution would have been impossible because some code issues (parameter tuning) were too closely related to fine mathematical aspects.

When there is a collaboration with someone doing the practical implementation, what is also very important is that anyone in the chain finds his/her own interest and has pleasure in performing the work; that should be as rewarding as possible.

*Q: Recently you have been appointed as Director of the Fondation Sciences Mathématiques de Paris. Could you explain succinctly the role of this foundation and your main goal as director?*

**
A:** The Fondation Sciences Mathématiques de Paris (FSMP), hosted at (and partner of) Institut Henri Poincaré, is a network of excellence covering all mathematics and theoretical computer science departments centred in Paris, several INRIA teams and departments in engineering schools – around 1500 permanent people. This is the largest concentration of mathematicians in the world. The role of FSMP, as a gathering structure, is to guarantee the visibility of the community of mathematicians, and to search for collective funds (region, country, Europe, industries) in order to implement all pro- grams with these financial supports: funding for master students, PhD students, postdoctoral positions, and invited researchers that hold specific chairs. The FSMP animates many other programs and, thanks to its flexible structure, is able to organize actions that would not be possible in the universities because of administrative constraints. My role as a director is to animate all programs and direct the committees, and search and attract new funding sources to ensure durability of our main actions. I am fortunate to have at FSMP a fantastic team, very devoted and efficient. The mathematics community is a wonderful one; it is very well organized and has been able to make emerging structures like FSMP in order to ensure better cohesion and international visibility. In the very complex Paris scientific landscape (there are around 20 universities in Paris), FSMP ensures a unification of mathematical sciences around big programs.

*Q: How do you spend your time when you are not working?*

**A: **Well, I love cutting the grass and bushes, tilling the ground in my garden and watching tomatoes grow, and after that, drinking a beer on my balcony! Before quickly coming back to mathematics, of course.

*Q: Have you other interests or hobbies? Who are your favorite writers?*

**A:** A great pleasure is to run with my wife, with our three children accompanying us by bike. My favorite writer is Jacques-Louis Lions. I’m joking. . .I am not much versed in literature but before sleeping I like reading mystery novels. For instance, I enjoyed several books by Harlan Coben. I also read a lot of French history, actually any local historical issues or anecdotes attract my attention. I live in Orléans, in the Loire Valley, which is an incredibly rich country from a historical and architectural point of view. I like looking at things, buildings, places with, you know, different eyes, various points of view. You travel so much when you simply look at everyday things with different eyes.

*Q: Finally, a last general question. What do you wish for mathematics in 2017?*

**A:** We are certainly attending the advent of data science. We can expect that in the next years there will be a real explosion of jobs related to connected objects, may it be at home, in our (soon autonomous) cars, or at work. Mathematics has a great role to play, in order to provide and develop powerful tools to calibrate and face the enormous challenge brought by big data, massive information and measurements. “Prediction” is likely to be the guideline of many forthcoming studies and projects. I believe that “data science” will soon emerge as a new very visible branch of mathematics, full of exciting open issues, and raising new questions combining probability and statistics, PDE analysis, modeling and model reduction, linear algebra, computing issues, etc. This will be the advent of the data science age. So many challenges are about to be solved, for instance in the domain of Math-Health. Just think of all we can do with a simple smartphone. . .I wish for mathematics to proudly carry the torch of data science in 2017. Why not a forthcoming year of data science?

*This post is being republished from the July issue of Dianoia, the ICIAM newsletter.*

Roberto Natalini has been director of the Istituto per le Applicazioni del Calcolo “Mauro Picone” of the National Research Council of Italy since 2014. His research interests include fluid dynamics, road traffic, semiconductors, chemical damage of monuments, and biomathematics. He is on the board of the Italian Society of Industrial and Applied Mathematics and is chair of the Raising Awareness Committee of the European Mathematical Society. |