The search for knowledge is an end in itself

Interview with Nalini Anantharaman

A mathematician specializing in chaotic dynamical systems,Nalini Anantharaman is interested in the influence of object geometry on wave propagation. In 2012, she was awarded the Prix Henri-Poincaré and, since 2019, she has been an elected member of the Académie des Sciences. She will hold the Spectral Geometry chair at the Collège de France in 2022.

When did you realize that you wanted to dedicate your life to mathematics ?

Nalini Anantharaman : As I grew up in a family of mathematicians, I was immersed in a mathematical culture from an early age. I had books at my fingertips, and was aware that research was active in this field, whereas public opinion tends to think that this discipline is stuck somewhere between the ^18th and ^19th centuries. Throughout my studies, I always preferred scientific subjects, such as physics or biology, but without any particular distinction at the time. Then, during my university studies, I started by studying both physics and mathematics. During an experimental internship, I realized that abstract reasoning, with its intense and sustained intellectual activity, interested me more. Experimental research often requires us to spend time solving practical rather than conceptual problems, and at times the intellectual activity can be less intense when it comes to setting experimental parameters.

What obstacles might an aspiring mathematician encounter along the way?

Most of the obstacles I encountered had their origin in myself. They can be personal difficulties to overcome or particularly complex intellectual challenges. Having been fortunate enough to grow up with access to information about the various scientific streams, I knew how to orientate myself in my studies. I went through the preparatory classes, where the initial shock is quite harsh, as the amount of work to be done is much greater than in high school, which, for me as for many, was the source of difficulties in the first few months. But I always reasoned that it was up to myself to find the resources to overcome these problems. In short, I never felt that the obstacles came from outside.

You wrote yourthesis on the theory ofchaotic dynamic systems.How are thesesystemscharacterized ?

At the end of the ^19th century, the French mathematician Henri Poincaré tried to show that the solar system had a regular, almost periodic behavior, but he made a mistake. In realizing this, he saw, on the contrary, that the solar system is unpredictable under certain conditions. It was from this observation that the theory of chaotic dynamical systems developed throughout the ^20th century. The term " chaotic " comes in handy, as it's quite meaningful, but on the other hand, it's not very well defined mathematically. Three characteristic elements stand out. First , what defines a chaotic dynamic system is its high sensitivity to initial conditions. If we are interested in the movement of a point, or a planet, and we make a very small error, even an infinitesimal one, this error will increase exponentially over time. As a result, over a fairly long timescale, we will have completely lost the ability to predict the behavior of this point. In a chaotic system, it makes no sense to study only the motion of a point, so we work instead with clouds of points. In a second time, if we consider two very close points, their destinies will very quickly become independent. Knowing what has become of one of the two points gives no information about the fate of the other. In other words, their trajectories are decorrelated. The third aspect is that even physical models described by deterministic equations ^[1] have evolutions that appear random. For example, the solar system is described by Newton's laws of gravitation, i.e. differential equations of degree two. They are deterministic in the sense that, if we know the position and velocity of all the planets, we can in principle calculate all their trajectories. However, in reality, on extremely long time scales, the motion may appear random. This theory was born out of the study of celestial mechanics, and a priori it doesn't apply to wave propagation. So I had to adapt certain ideas to describe the disorder that can occur in wave propagation.

How did you come tofocuson the link between chaos theory and wave mechanics ?

The link between chaos theory and wave mechanics came to me almost miraculously. As a young researcher, I took part in a conference, and at dinner found myself face to face with a researcher I didn't know at all, Leonid Polterovich. He was polite enough to ask me what I was researching, so I told him about my work on the Hamilton-Jacobi equation, which describes geometrically how wavefronts propagate. He pointed out that it would be interesting to look at the applications of these notions to the field of quantum chaos, and in particular to try and solve a famous conjecture : the unique quantum ergodicity. This comment popped into my head, so I started asking around, both by reading and by asking questions of researchers I didn't know. We're using the fashionable word quantum here, but what interests me more generally is the propagation of waves and, in nature, there are a wide variety of them, be they seismic, electromagnetic, acoustic... In quantum mechanics, there's this notion of wave-particle duality, the idea that any object has two natures ; it can be both a wave and a particle. This is the case for electrons, for example, which it may be more interesting to conceive of as one or the other. In the end, physics always translates into mathematical terms : all these types of waves are described by equations that are very similar. This led me to take an abstract interest in how geometry affects wave behavior. When a wave propagates in a room, the way it bounces off the walls depends completely on their shape - whether they are flat, curved, form corners... I like to give concrete examples, but in reality, my day-to-day research is fundamental ; I study this kind of question from a purely mathematical point of view by establishing theorems.

These interactions are referred to as Spectral Geometry. What does " spectrum " mean in mathematics, and how was the link established between the geometry of an object and the waves it emits ?

In physics, ever since Newton's work, we've known that white light is actually made up of a superposition of colors. The visible color spectrum extends from violet to red, passing through blue, green, yellow... Gradually, in the ^19th century, physicists noticed discontinuities in the light emitted by the Sun : its spectrum contains dark lines, known as Fraunhofer lines. By heating chemical elements, scientists also saw the appearance of discontinuous spectra in which only certain colors were present. These findings were astonishing, since we had always thought of the world as continuous. The challenge was to understand why these discontinuities were occurring. In the years 1920, the founders of quantum mechanics, Heisenberg and Schrödinger, understood something very important : to calculate the missing frequencies that appear in the observed spectra, we need to calculate numbers that correspond to a notion already known for seventy years in mathematical circles : the eigenvalues of matrices. It was a miraculous moment, and from then on, the notion of calculating eigenvalues came to be known as spectral theory. This was a crucial period, because on the one hand, mathematics was enriched by the questions raised by physics, and on the other, quantum mechanics became extremely mathematical. This also made it more difficult for the uninitiated to understand, as spectral theory called upon rather abstract notions, such as infinite-dimensional spaces. By studying the geometry of an object, we can try to determine whether we're going to obtain a continuous or discontinuous spectrum (known in mathematics as a " discrete "). The hydrogen atom, for example, has both types of spectrum, but the one we observe is the discrete spectrum. This is the case for most physical objects...

As a pure mathematician, are you motivated by the potential applications of your work, or is the pursuit of fundamental knowledge an end in itself ?

I've never really worked with the aim of finding applications or answering concrete questions posed by experimentalists. I'd say my main motivation is knowledge in itself. That said, the existence of a link with the real world, however remote, is important to me. I think it would be hard for me to work on subjects that are totally disconnected from questions that are rooted in the real world. This very fundamental approach brings with it great freedom and intense intellectual activity, but can sometimes be frustrating. In particular, when I try to explain the nature of my work to a lay mathematician, I get the feeling that he or she doesn't fully grasp what I'm talking about, because certain notions are far too far removed from the tangible. Incidentally, in my field, we often have discussions with physicists, at conferences where we compare our work. Even if we soon realize that we don't speak exactly the same language, we still seek out this discussion, this interdisciplinary fine-tuning, which is very enriching.

The discipline of mathematics has a reputation for having a small number of women in its ranks.Why do you think this is the case ?

A few years ago, I brought my daughter to work one day when she was ill and couldn't go to school. At five years old, she attended a seminar on probability theory. As she entered the room, she remarked to me, astonished, that I was the only woman in the room. I wasn't sure what to say. Then she told me that, in the future, she would refuse to work in a profession where she was the only woman. When young girls who are interested in science see that it's a male-dominated field, it can be very off-putting. Personally, I didn't find it that way. As I remember it, I had a few classmates who were interested in maths, but they quickly went into teaching. I think parents play a very important role in this orientation. If a boy has decent grades in science, he's immediately encouraged to go into scientific preparatory classes and engineering studies. On the other hand, if a girl has good grades in science, she is not necessarily steered towards the same fields. The message is not the same.

What advice would you give to young female students consideringmathematicalstudies andcareers ?

I'd like to say that these mathematical research careers are very compatible with personal and family life, contrary to what some people imagine. You're very free to organize your life and make your own choices. Unfortunately, it's true that for the last ten years or so, the first years of a career in France have been more precarious. It used to be that you could start out very quickly with a stable career, after long years of difficult studies, but the situation has deteriorated quite rapidly in recent years. Once you're in the business, you realize that it is indeed a very male-dominated environment, but not very hierarchical, and I've always found the working atmosphere very benevolent. What's more, since academia allows us to be fairly free in our choice of research subjects, you have to know how to judge yourself, criticize as well as congratulate yourself, and have confidence in yourself.

Interview by William Rowe-Pirra

Glossary

[1]Deterministic equation : an equation that always behaves in the same way when faced with a given event.