Questions :
1- What is your research field?
2- Would you tell us more about random matrices?
3- In what other fields have random matrices come to develop?
4- What explains the presence of random matrices in many fields?
5- What do you like about being a mathematician?
6- What would you say to a young woman who would like to embark on a scientific career?
7- What does it mean to you to be invited speaker at ECM?
8- What does it mean to you to be participating in an international congress?
9- What is your relationship to mathematical outreach?
10- How would you describe your role as an academician?
11- Is there a message you would like to pass on to young people?
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1- What is your research field?
My research area is mainly probability with forays into operator algebras and applications in physics. I first became interested in questions arising from statistical physics where one tries to understand macroscopic physical properties from microscopic models of particles, for example the phenomenon of magnetization of a sample of matter from the magnetic magnetization of electrons. In these questions, one studies a large number of particles which often have a random disordered behavior but which interact according to the laws of physics and one seeks to extract a global behavior when this number of particles becomes very large, that it tends towards infinity. This led me to study large random matrices, a beautiful mathematical object has been a major part of my activity. We can think of them as large arrays of data whose coefficients are chosen at random. They are used in many fields of mathematics and physics, from statistics to quantum mechanics. First of all in the analysis of data that we often know in an incomplete way, unknown often modeled by randomness. But also in operator algebras, where they provide particularly rich examples that have been revealed in the relatively recent field of free probabilities. Moreover, they provide very interesting archetypes of problems in strong interaction particle physics. Random matrices are thus particularly rich mathematical objects, at the interface with many fields.
2- Would you tell us more about random matrices?
Random matrices appeared at the beginning of the 20th century in statistics. The Scottish mathematician John Wishart introduced them in the context of data analysis. A random matrix is an array of coefficients: you measure one thing by several different measures. One question you might ask is whether your measurements are correlated. In sports, for example, if you want to see if there is a correlation between being good at swimming and being good at ice hockey, you measure the performance of several athletes in different sports and try to figure out if there is a correlation between them being good at the two chosen sports. The problem is that the data is noisy, in the sense that there may be a counter-performance. We must therefore try to understand the reality of the data beyond the noise. Wishart had introduced this model of random matrices to try to understand this question. These are problems that are still very important in data analysis, especially when you're analyzing large data tables.
They are called random matrices because their coefficients are chosen randomly, with randomness modeling what is unknown. In general, probabilities are used to model what is not known. When you draw a die, you will say that it falls with a probability of one sixth of a six because you don't know much else. A priori you could know more if you knew the impulse given, because everything is subject to the laws of physics. But there are so many things that happen in ways that you can't know or calculate exactly, that you model it by chance. In the same way, for data, there are many things that we do not know exactly and we will model this unknown by randomness. So much for the beginnings of random matrices.
3- In what other fields have random matrices come to develop?
Random matrices are also related to Hamiltonians. A little later in the 20th century, the Hungarian physicist Eugene Wigner became interested in them to model Hamiltonian operators of quantum systems. These Hamiltonians, as there are many electrons, can also be very difficult to study. Wigner's idea was to approximate them by large random matrices, fixing this randomness in such a way that everything we know about physics is satisfied. In these cases it is the eigenvalues, which are the energies of the system, that we will try to understand.
They are also related to other fields of mathematics, for example operator algebras. Matrices are also operators, in the sense that they do not switch. When we multiply a by b, it is not equal to b times a. In everyday life we can think of them as operations, like drying and washing clothes: we will not have the same result by washing the clothes and then drying them as the opposite! This is the birth of the theory of operator algebras. As non-commutative objects, large random matrices produce important examples in operator algebras and have allowed important advances in this field, in particular thanks to the theory of free probabilities.
Large random matrices have also allowed the development of a whole theory of random variables, the strongly interacting variables. In probability, questions are often asked in terms of independent variables. For example, when you draw a die twice in a row, the two draws are assumed to be independent. What you drew on the first throw will not influence what you will draw on the second or third throw. Probability theory has developed around this notion of independence and we have few tools to deal with problems where the random variables are highly correlated. The eigenvalues of random matrices are highly correlated and this provides an archetype for other models, for example random tilings. If you try to pave a certain domain, say with diamonds, you can have many possibilities. But if you don't want to make a hole in that tiling, the tiling positions are very constrained. You can ask yourself what a tiling that you have randomly chosen from all the possibilities looks like. It turns out that mathematically these tessellations are very related to the eigenvalues of random matrices and their study has benefited from the theory of random matrices.
4- What explains the presence of random matrices in many fields?
Probabilities are a fairly recent field, it was only formalized in the 20th century by Kolmogorov, but it has developed in all sciences to model and predict the unknown. Random matrices are even newer mathematical objects, which combine the notions of probabilities and operators. They are very natural objects, and therefore they are used in many different problems and fields. This is what I like about this theory. It offers a mathematical diversity, but also in the applications and contacts. I work with physicists, I turn to computer science and statistics. I am even on a committee on wireless telecommunication.
That random matrices come naturally in many problems of all sciences does not mean that they are easy! But this is why I have devoted a large part of my activity to them.
5- What do you like about being a mathematician?
I like to have time on my hands, to sit and think about a problem, to feel the solution not far away, but not yet there, to turn around it until it appears. That's not the only part of being a mathematician, there are prize committees, recruitment committees. My responsibility as unit director takes up my time... But doing math is the heart and the pleasure of the job.
6- What would you say to a young woman who would like to embark on a scientific career?
Women often don't dare to embark on a scientific career, in my opinion because the society around us doesn't think it's a natural place for them. I would simply tell a young woman to go for it, to do what she loves; that being a researcher is exciting and gives you a freedom that you rarely find elsewhere. It seems to me that there is often a muted, but very real, social pressure on young women not to go into research careers in science, and I have heard several testimonies of this.
7- What does it mean to you to be invited speaker at ECM?
It's a great honor.
8- What does it mean to you to be participating in an international congress?
International congresses are general. Many events are held in parallel. It can be frustrating not to be able to attend everything you would like to. In practice, you run from one presentation to another and it can be difficult to make contacts and discuss. But you also get to listen to extremely good people in different fields and this can lead to unexpected discoveries!
I think that these congresses are a very good opportunity for young people. I think that we should send more young people than we do to these congresses, so that they can see a little more of what is being done in math.
9- What is your relationship to mathematical outreach?
Dissemination is very important, even essential to create vocations and to be understood by the general public. It's a very difficult and time-consuming activity. It is complicated to diffuse well. It takes me more work to do a broadcast presentation than to do a math presentation! And it is an activity that is often not sufficiently recognized.
For me, the goal of dissemination is to succeed in making modern science understood, active today, to make people feel the science that we do nowadays, to eventually make them understand an idea, since the work of the mathematician is to have ideas: this is what we can bequeath. And the challenge is to talk about this starting from a rather small prerequisite. For a broadcasting presentation that I had to do recently, I gave up choosing random matrices. The subject is complex and I haven't found a good approach yet, I don't know where to start talking about it.
At the ENS of Lyon we have a house of mathematics and computer science, the MMI, whose aim is to understand diffusion as a research domain. We can see dissemination as a field of research. Finding good ways to do dissemination is something that requires a separate reflection. We have to think about how to transmit, about tools, about the best way to make young people understand concepts that are not always easy. It is a reflection of research because it requires the creation of new tools.
And then it seems to me that the diffusion must always remain logical and rigorous and transmit a mathematical content. It is a question of trying to make people understand specialized things in a simple way, to transmit the quintessence of things, and to bring the person to whom we address ourselves to make a research process on a current subject.
10- How would you describe your role as an academician?
As an academician, I am expected to participate in the life of the academy. The core of the activity is to be involved in research reports, to award prizes to motivate research, to do dissemination and to represent the academy.
The academy plays an important advisory role to society. Committees of reflection meet on current issues, covid19, global warming... I participate in a committee on open science. The challenge is to bring a reflection that will allow us to propose the best solutions for open science, solutions that are common to all sciences. The committee has members from all sections. It meets regularly and teaches me a lot. Participating in this committee gives me access to a lot of information, much more than I have ever had on the subject. At the end of this reflection, the academy will publish a report on solutions and negotiations with scientific publishers.
The academy is a place where a lot of science and scientists are concentrated, where you can interact with a lot of people. When we meet in section, the mathematics talks allow us to try to see what is being done in mathematics research in France, to discuss prizes. We do our best to boost research in France. It is exciting but also energy consuming... It is also good to be able to meet in front of a math problem!
11- Is there a message you would like to pass on to young people?
If there's one message I'd like to get across, it's pleasure: the pleasure of doing math. Once you get into it, you can't get out of it!
