Marianne Akian: From “New Maths” to the Search for New Model

What gave you a taste for mathematics?

The story inevitably began with science. I always excelled at mathematics, but was much less talented in the other subjects. I loved investigating and reasoning, but I also enjoyed formalisation and abstraction. I belonged to the new maths generation, which suited me perfectly. 

Even though I was “destined” to do maths, I found it almost too much “fun”. It wasn't at all sure what you could do with mathematics professionally apart from teaching, and being a teacher didn't appeal to me at all.

My parents were fairground stallholders from a modest background that was far-removed from the world of science. They were great believers in education, no doubt because they regretted being denied the opportunity to study to the best of their abilities because of the war.

How did you become involved in research?

After my final year of high school, I was torn between Medicine and Mathematics. I wasn’t drawn to medicine for clinical practice, but for research – I must have seen examples of it in films and, unlike mathematics, I could clearly see the usefulness of the profession. The pathways into medical research were not very clearly documented. I had learned that it was possible to enter the Ecole Normale Supérieure (through a competitive examination process), after two years of Medicine. But there were only four places! So, I chose to enrol in a scientific preparatory class (Maths-Sup), with the idea that I could switch to medical studies if I felt I had very little chance of being accepted into a Grande École on a research programme.

What path led you to applied mathematics?

In 1982, I joined “Sèvres”, the École Normale Supérieure for young women. I enrolled in the mathematics stream and immediately asked to change to biology. The lecturers told me about a third option that captivated me: mathematics applied to biology. While following the mathematics curriculum, I took biology classes and also did an internship in biology. But I soon realised that it was too early: biologists were not yet embracing mathematics and very few mathematicians were interested in biology. So, I thought I'd wait until I had a stronger mathematical grounding.

That's how I started down the path of applied mathematics for my PhD. As I became increasingly absorbed in mathematics and was unable to find a quick way to apply mathematics to what truly interested me in biology (genetics and how the brain works), I continued along the applied maths pathway.

And how did you end up at Inria?

For my DEA (Master’s 2), I studied with Alain Bensoussan who was a professor at Paris Dauphine University and went on to become President of INRIA*. His teaching and research work combined several fields of applied mathematics: modelling, partial differential equations (PDEs) and probability, all applied to decision-making in uncertain contexts (stochastic control). I asked him to propose a PhD topic and he suggested one in stochastic control with an application in finance. What really appealed to me was the idea of linking several areas of mathematics, along with the potential applications. 

I completed my PhD at INRIA Rocquencourt, where I was jointly supervised by Jean-Pierre Quadrat. Slowly but surely, my PhD work evolved, encompassing numerical analysis, algorithms and code which I inserted into an “expert system” in stochastic control – a form of artificial intelligence before its time. It was during this period that I became a researcher at INRIA.

I then turned to a new field that Jean-Pierre Quadrat was studying at the time: “max-plus” algebra, now known as tropical algebra.

We formed a group of researchers called “Max Plus” and together we published a number of works. In 2003, this collaboration gave rise to an Inria research project team. I then obtained my HDR (accreditation to supervise research) in 2007, and in 2008 the project team joined the Centre de mathématiques appliquées de l'École polytechnique (CMAP), where I had the opportunity to supervise my first doctoral students.

Has being a woman made a difference to your experience in a very male-dominated environment?

Initially, as I have said, I thought I needed to do something useful, probably because I'm a woman and women are often associated with professions that focus on others, such as health, teaching and care. Later, I realised that you could choose other career paths (mathematics for me) purely for the intellectual pleasure they bring.

That's really a message I would like to pass on to young girls: don't be afraid to do a job that you enjoy. You can enjoy maths for maths’ sake, without having to find an immediate use for it. The usefulness often comes later. If you like mathematics, go for it! 

When I was at the ENS, there were still two separate institutions: Sèvres for women and Ulm for men. Co-education was introduced in 1986, and since then the number of girls has dropped sharply.

I've never encountered any direct obstacles, but I've often been the only woman in a conference room. We even used to laugh about it among female colleagues. But the fact remains that there are still very few women in mathematics and computer science research today, and the situation is even worse in the more theoretical fields.

Your current research focuses on tropical algebra and stochastic control. Can you describe them in simple terms?

The aim of stochastic control is to model and optimise situations involving randomness. This is a vast field with many applications: economics, finance, dam management, resource management, modelling of human behaviour, and more.

Tropical algebra is an abstract structure that can be highly useful for solving real-life problems. It is as if you are in another world in which the ordinary operations are replaced; for example the “+” becomes a “max” and multiplication becomes addition. It may sound trivial, but this approach can be used to solve certain complex problems in a different way, particularly through dynamicoptimisation (optimal control). 

Tropical algebra also offers a logarithmic perspective on the world, enabling the formalisation of reasoning about orders of magnitude or asymptotic phenomena. This formalisation can be applied to various fields of mathematics, including modelling, probability, optimisation, dynamic systems and PDEs. 

In pure mathematics, it has led to the emergence of a new branch of mathematics called “tropical geometry”, linked to the study of polynomial equations, which can itself prove highly useful in applications.

What I particularly enjoy is establishing links between very different areas of mathematics, working on several fields of application, and simultaneously contributing to advances in both fundamental and applied mathematics.

And what are your current research challenges?

I am continuing to explore the links between tropical algebra or tropical geometry and stochastic control or stochastic games, especially with a view to designing efficient algorithms. Many challenges remain, such as solving high-dimensional problems and information problems in stochastic control – especially in stochastic games.

These challenges are equally important for industrial applications. For example, during two successive PhDs in collaboration with researchers at Orange, we applied the latest tropical geometry tools to mobile data pricing problems, and more recently to spectrum auctions.

What I enjoy most of all, as you will have gathered, is diversity. And that's what mathematics is all about, it's a field in which there's never a dull moment.

*INRIA (Institut national de recherche en informatique et en automatique - National Institute for Research in Computer Science and Automation) became Inria (Institut national des sciences et technologies du numérique - National Institute for Research in Digital Science and Technology) in 2011. Discover the history of the Institute