The Sciences

Interview: 'If We Want New Revolutions, We Need New Abstract Concepts as Well'

A conversation with the noted mathematical physicist Nalini Anantharaman about the importance of fundamental research, quantum chaos and unpredictability in the Solar System.

Nalini Anantharaman is a French mathematician who teaches at the Institute for Advanced Mathematical Research, Strasbourg. She is noted for her contributions to the study of quantum chaos, ergodic theory, microlocal analysis and the Schrödinger equation. (A more detailed description of her work is available to read here.)

She won the Salem Prize in 2011 for her work on Laplace eigenvalues and the Henri Poincaré Prize for Mathematical Physics in 2012 for, among other things, “a remarkable advance in the problem of quantum unique ergodicity”. She shared the latter with Freeman Dyson, Barry Simon and Sylvia Serfaty. Anantharaman was elected member of the Academia Europaea in 2015, was a plenary speaker at the 2018 International Congress of Mathematicians and won the Infosys Prize for Mathematical Sciences the same year.

She recently visited the International Centre for Theoretical Sciences, Bengaluru, where Anupam Ghosh caught up with her on behalf of The Wire. The questions were prepared by Vasudevan Mukunth (any reference to the first-person in the questions refers to him). It was transcribed by Damini Yadav.

The interview is presented in full, with light edits for clarity.  The editor’s notes are noted as such.

This might seem like an annoying question to begin with, but it’s always interesting to find out how different mathematicians answer it: why is mathematics important?

Why is mathematics important? This might be a difficult question to answer but I think one has to answer it. I hear many colleagues say that mathematics is fun and they do mathematics for fun. I wouldn’t say that. Personally, if I wanted to have fun, I know many other ways than mathematics. Nevertheless, I think it’s very important to me and it’s also a very important subject for humankind.

It’s difficult to explain why but doing mathematics is deeply rooted into human needs and activities. People have done mathematics from the start. I think it is the natural curiosity of man to try to understand what goes on around him and in particular when he sees geometric figures or repetition of patterns, he wants to explain it. And mathematics is one way to understand the world.

This being the case, how do you and your peers around the world convince administrators to fund math departments, especially if funding for research is limited in a country like India?

I don’t think I’m a very good politician personally. In France, like in India, funding for pure mathematics is scarce and we tend to insist, to emphasise the applications, when we ask for funding. Nowadays, mathematics has applications everywhere and some of them are really crucial – in medicine and in environmental issues. So we really emphasise this when we ask for funding. I’m sad to hear that it’s the same in India because I thought in India there was a deeper culture for pure math.

Our authorities also need to know, to be aware, that fundamental research is crucial for developing applications. In the past, all the great scientific revolutions happened because new abstract concepts were developed. If we want new revolutions to happen, we need even more abstract concepts.

This is a bit unpredictable and that’s why it’s difficult to explain it to authorities. For instance, general relativity was made possible because of the invention of non-Euclidean geometry, which is a very abstract thing. The invention of the computer was made possible in part by new foundations of logic in mathematics. So we can’t hope to have applications of mathematics without fundamental research as well.

Could you tell us what motivated you to first examine the quantum mechanical roots of chaos in classical dynamical systems? And, more generally, why mathematical physicists like yourself think this is important to know?

[Editor’s note: Classical chaos is the study of systems that are very sensitive to their initial conditions. For example, a double-rod pendulum will swing in significantly different ways when dropped from only slightly different positions. And chaos theory tries to explain why this is so in the terms of classical mechanics. In this picture, quantum chaos is an attempt to understand if classical chaos has a quantum mechanical counterpart.]

I truly don’t consider myself to be a mathematical physicist. My PhD was in pure math. I was interested in chaos from a very abstract point of view. I think it’s when I received the Henri Poincaré Prize [in 2012] that people started to call me a mathematical physicist. How things happened is that… after my PhD, I happened to be talking about my work with a senior colleague during a dinner and he suggested I look into applications in quantum mechanics. He told me in particular about the quantum unique ergodicity conjecture. He explained it to me over dinner, and I immediately became interested in the subject. I read up on the literature.

In my country, and I think it’s the same in many other countries, mathematics is seen as a ‘selection’ topic.

I really liked the fact that it was related to physics. I’m not a physicist myself. I studied some physics as an undergraduate but it was not my major. But I always had a taste for physics, so when I heard from this senior colleague that some of my ideas may have some relevance to quantum physics, I was immediately excited, and I quickly became convinced that I had something to say.

So basically it wasn’t the application of physics that motivated you. It was more of a mathematical pursuit…

Yeah. Even now, my main motivation is the mathematical pursuit and the abstract questions. I’m glad if from time to time there are relations to physics but this is not my only motivation.

So how are classical chaotic systems characterised? One way is with Lyapunov exponents – could you tell us more about this?

[Editor’s note: The Lyapunov exponent is a number that denotes whether a system is chaotic and, if so, to what extent. It is named for Alexander Lyapunov, a Russian mathematician who worked in the late-19th and early 20th centuries.]

Chaotic systems were discovered by the mathematician and physicist Henri Poincaré towards the end of the 19th century. He had wanted to prove that the Solar System was regular, and predictable, so that you could compute things for an arbitrary long period of time.

[Editor’s note: Isaac Newton’s laws of motion are interesting because they predict how an object that follows them will move at any point of time in the future. You just need to know the initial conditions and plug the numbers into an equation. So Newton figured we’d be able to do the same thing with the Solar System: that if we knew the arrangement of the planets today, we’d be able to able to tell how they’d be arranged, say, a million years from now.]

But in trying to prove this, Poincaré actually realised that it was exactly the opposite – that the Solar System is completely unpredictable. If you want to predict its behaviour for a very long time, and you only know the initial conditions up to some accuracy, then you can’t really do better than when you toss a coin.

[Editor’s note: In other words, you can predict the Solar System’s future configuration as well as you can predict a coin toss.]

This was the birth of the theory of the classical chaotic system. As you said, one way to measure chaos is through Lyapunov exponents. This means that if you just change the initial conditions even by one tiny bit, the trajectory that your system will follow will be very strongly perturbed, and will divert exponentially from the unperturbed one.

One other feature is a technical term, called the Bernoulli property of chaotic systems. It means that even though a classical system is described by deterministic equations – typically the Newtonian equations – and if you don’t know the initial condition very well, then what you can predict won’t be better than the one that you’d make just by tossing a coin. So there is a random behaviour even though the system is deterministic.

So why do mathematical physicists think this is important to know?

This is a question that Albert Einstein asked as early as 1917. At that time, quantum mechanics was still in its prehistory, and people were trying to understand the spectrum of atoms.

[Editor’s note: When an atom is excited to a higher energy, it emits light of certain frequencies to lose that extra energy and return to its original state. The atoms of different elements emit light at different frequencies, and no two atoms have the same spectrum of emission. As a result, an atom can be identified by its emission spectrum. The spectra of hydrogen and iron are shown below.]

The emission spectrum of hydrogen. Credit: Wikimedia Commons

The emission spectrum of hydrogen. Credit: Wikimedia Commons

The emission spectrum of iron. Credit: Wikimedia Commons

The emission spectrum of iron. Credit: Wikimedia Commons

They had a method to understand the spectrum of the hydrogen atom. It had been proposed by Niels Bohr but it was just for one atom, which was very simple – just one electron going around the nucleus.

Einstein’s question was about what you could say about the spectrum of a quantum system that is more complex than that, and in particular has some disorder in it. So if you either have a very large atom or more generally a complicated chaotic system and you consider it from the perspective of quantum mechanics, what can you say about the spectrum of this system?

In 1925, Erwin Schrödinger introduced the notion of a wave function in quantum mechanics. [It accompanied] the idea that an electron, or more generally a physical quantity like a particle, actually behaves like a wave and the wave is driven by a certain physical equation – the Schrödinger equation.

It doesn’t only apply to quantum mechanics. All kinds of waves – seismic waves, acoustic waves, electromagnetic waves – are described by the same equation. And for us mathematicians, the central object is the equation. We tend to forget about the physical meaning of the waves; we focus on the equation. We try to understand how the geometry of the system and the presence of chaos will influence the propagation of the waves [through that equation].

Thank you for that detailed answer, professor.

My next question: I am not very keen on discussing applications when writing about research because knowing something can be its own purpose. But as a mathematician, are you motivated by potential applications for what you’re working on – or do you also feel just knowing something suffices?

In my daily work, I can say I’m absolutely not motivated by applications. When I look for results, my main motivation is that I want to know, I want to understand, and I want to gain knowledge of mathematical objects. So I don’t work with a precise application in mind.

I think that some of my work could have applications. But if they were to be developed, I would have to work in collaboration with someone with a more applied frame of mind. And maybe I will do it one day. I’m not doing it at the moment but I don’t rule out the possibility of doing it one day. But I really don’t have that frame of mind, so I will need some inputs from someone else who already has some experience with applications.

The last question: An AMS survey has found that fewer than 15% of all female mathematicians in the US have tenure-track positions. In 2016, two scientists found that women made up less than 10% of the 13,000 editorship positions in 435 math journals. Why do you think the gender-representation problem is so acute in mathematics? Do you think the field presents any unique barriers to non-male mathematicians?

So in France, as in many other countries, there are very few women who start their career in mathematics. The statistics show that when you go up the hierarchy, there are fewer and fewer women.

In my country, and I think it’s the same in many other countries, mathematics is seen as a ‘selection’ topic. It means that many students who study mathematics don’t do it because they like mathematics but because it is a way to show that you are amongst the best students. To enter the best academic institutions, you have to have good grades in math, so it’s considered to be a very competitive subject.

Our authorities also need to know, to be aware, that fundamental research is crucial for developing applications.

I think that parents as well as teachers tend to encourage boys more than girls to enter competitive curricula. That is one explanation. Another is perhaps the image that society has of mathematicians – it’s not seen as a very feminine subject. And I think that’s a pity because jobs in mathematics are very suited to women and is as compatible with family life as others, if not more. I don’t know what else to say. [laughs] As for why it gets so in the hierarchy, I don’t really know or understand why.

A few years ago, I was invited to a meeting of the Association of Indian Women in Mathematics and we had some discussions about problems encountered by women, especially in India. Some of these problems were specific to India. Some of them were I think universal.

Women in India marry quite early and have children very early. Some of the women said that professors had refused to supervise their PhD work under the pretense that, once they are married and have children, they would stop working on their PhDs. Some professors had explicitly said they didn’t want to take women as students for this reason.

So how’s the situation in France?

I think these situations, where professors don’t want to supervise the PhDs of women, occurred maybe in the 1970s and 1980s. I’ve heard one of my colleagues say that. And it’s less and less the case.

However, according to the statistics, there are very few women doing mathematics. It is worse in pure math than in applied math. There is also maybe a cultural aspect – that women like to know that what they do is useful for society and pure math is not perceived to be as useful to society as applied math. I think that is also a cultural aspect.

Thanks to inputs from Madhusudhan Raman.