Understanding how spheres can be packed in multiple dimensions is a unique talent of the mathematician Maryna Viazovska.| Image: Catherine Leutenegger

How did you feel when you were awarded the Fields Medal?

A great joy, of course. It is an honour to be one of the few people to receive this prize. But I noticed a paradox: it rewards our research in mathematics but then pushes us to do a lot of simplification, in which we don’t address the technical aspects that make our work so valuable.

Too much simplification for your taste?

I struggle a bit to spend time alone with mathematics. But talking to people who don’t know anything about my subject is also a very interesting experience, because it is completely new to me. Scientists must be curious, right? So I ask myself: why have these people come to listen to me? What do they know about mathematics? What are they interested in?

In September 2022, you were invited to the Federal Parliament together with Hugo Duminil-Copin, the other Swiss Fields Medallist of 2022. What were your impressions of the meeting?

Maybe I’m naive, but I felt a positive energy and a sincere interest. Maybe because my work is neutral and does not concern politics. We also discussed the importance of international collaboration and basic research, and the demands scientists can put to politicians. I don’t like to justify basic research with reference to future utility. I like the fact that it allows us to slow down a bit and think about what we are doing and what we want. For me, many problems persist not because we lack the tools to realise our vision, but because we have the wrong vision.

Your speciality is the problem of stacking spheres. It was only solved in three dimensions in 1998. Why is it so difficult?

It is an optimisation problem with an infinite number of degrees of freedom. Moreover, the solution is not unique: the best three-dimensional approach is a stack of hexagonal layers, but they can be shifted in an infinite number of ways without destroying the arrangement. From the 1950s onwards, ways were found to reformulate the problem with a finite number of parameters, which nevertheless turns out to be a gigantic number.

When did you tackle the problem of eight-dimensional spheres?

In 2014 – during my postdoc at the Humboldt University in Berlin. It was a kind of gamble, because in mathematics you never know whether or not you will succeed. If you know how to solve a problem, it is in fact already solved.

You have also solved the problem in 24 dimensions. Why eight and 24?

Because these dimensions have special symmetries. In 2001 the mathematicians Cohn and Elkies from Harvard University estimated the density of the stack of spheres in eight and 24 dimensions very precisely by computer, but without being able to provide an explicit form to the auxiliary function they had used. My contribution was to find this function. I used what are called ‘modular forms’, which I had worked on during my PhD. Luck played a role.

Luck? Experts write that your approach is very elegant, very creative.

My job is like digging for gold. You get some tools, you go to Alaska, you start quarrying, there’s a lot of dust, and if you’re lucky, you find some little nuggets: formulas, results, some theorems. In mathematics, we work with ideas, and most of them die, just like that.

I see my approach as a bit naive. I work on problems that are simple enough for me to understand but which raise a question that seems fundamental to me. Then I tackle them with all the tools I have and develop any I lack. You need to try and gain a global vision of the issue. I am more of a problem-solver than a theorist.

Any upcoming projects?

I will continue with geometric optimisation. It is a fascinating field with an infinite number of problems. But I don’t like to say too much about it, it’s bad luck.

Is it superstition? Or are you afraid that the competition will steal your subject?

A bit of both, I think. A mathematical problem is also something intimate that I want to keep to myself. But I can say that I will work on the stacking of spheres in spaces with a high number of dimensions. We know that the density tends towards zero, or, in other words, there is more and more void between the spheres when the number of dimensions tends towards infinity, but without knowing the speed of this decrease. And that the best arrangement is probably random, not structured as in small-dimensional spaces. It’s a major open question: does randomness win over structure in high-dimensional spaces?

Could this be of practical use?

Yes, it is linked to information theory. How can we best densify the messages we send, and how can we correct them in case of transmission errors? Claude Shannon’s work shows that the number of bits contained in each message must be as large as possible – and this corresponds to the size of the spheres.

You are the second woman to have won a Fields Medal, out of 64 people. Do you like the fact that we are talking about this?

I hope that one day I won’t be asked this question, not for myself but for mathematics! Diversity is important, because scientists always bring their personality to their research. The situation is not so bad in my institute, with seven female professors out of 30.

You grew up in Kiev in the former USSR. What are your memories of that time?

Not much, I was six years old when the Soviet Union broke up... It is difficult to know whether a memory is real or reconstructed when you hear a lot about the past.

How are you coping with the war in Ukraine?

My sisters and their children left Kyiv at the beginning of the invasion and we took them in for a few months – they now have their own home. My grandparents don’t want to leave Ukraine and we have done everything we could to help them, including heating.

Do you work with Ukrainian institutions?

Yes, this summer I visited my alma mater, Taras Shevchenko University in Kyiv, where I have been teaching an online course for three years.

Could your international recognition inspire students?

I don’t know to what extent I am a role model for the people there. I’ve never experienced a situation like theirs. And anyway, the scientists who teach there are very inspiring people. But I think it’s important to feel support from outside. I hope mathematics allows them to forget about their everyday lives for a while.