# a mathematician who won the field medal, in july 2022 for solving the problem of how to pack spheres in the most efficient way in a space with eight dimensions.

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## ‘Mathematics is an unknown land’: meet Fields Medal winner Maryna Viazovska

Nature talks to the Ukrainian mathematician about her research and how the field is changing.

NEWS Q&A 15 July 2022

## ‘Mathematics is an unknown land’: meet Fields Medal winner Maryna Viazovska

talks to the Ukrainian mathematician about her research and how the field is changing.

Davide Castelvecchi Twitter Facebook Email Download PDF

Number theorist Maryna Viazovska was awarded the Fields Medal for her work on sphere packing.Credit: EPFL/Fred Merz

On 5 July, Ukrainian mathematician Maryna Viazovska became the second woman in history to earn a Fields Medal, one of the top prizes in the field. Viazovska, who is based at the Swiss Federal Institute of Technology in Lausanne (EPFL), is most famous for solving the problem of how to pack spheres in the most efficient way in a space with eight dimensions.

spoke to Viazovska about the award’s significance, and her vision for mathematics.

## How does it feel to be one of the winners of the Fields Medal?

Of course, I was extremely happy and honoured, because very few people get this award. I have actually known about it for a while: Carlos Kenig, the president of the International Mathematical Union, contacted me and told me the news in January.

## Why did you become interested in the problem of sphere packing?

It’s a very natural, very nice geometric problem — a problem with a very simple formulation, and often very difficult to solve. There are still many open questions surrounding it. Also, what made me interested in the packing problem in dimensions 8 and 24 was, of course, the work by Henry Cohn and Noam Elkies, where they proposed how to solve it, and came very close to a solution. So it seemed like low-hanging fruit. Even now, after it is solved, we still have infinitely many dimensions for which the problem is still open and the same methods don’t work. There are still so many discoveries to be made.

## Calling it low-hanging fruit sounds very modest, given how your work has been hailed as a big breakthrough.

Yes, but it’s maybe in the nature of mathematics that to make a big breakthrough, you have to chase low-hanging fruit — and some of our low-hanging fruits are still pretty high! In mathematics, when we think of open problems, we don’t think in terms of months and years to solve them — often, we think in terms of decades and centuries.

## Does the prize have a special significance at such a hard time for the Ukrainian nation?

I hope so. Maybe this news made somebody’s day better. But compared to how much we are losing right now, of course it’s not comparable.

I was chosen as a winner before the Russian invasion of Ukraine began in February. I believe that this decision is about mathematics and not about anything else — that’s how it should be.

## And it’s also an important win for women in mathematics?

My dream is that women getting major prizes will be a routine event, not a special occasion. It should not be. Maybe this prize could have a positive effect on young women, but what is much more important is what happens early at school — the hard, everyday work that is done by parents, teachers, university professors.

Mathematics is one of the fields where we can just enjoy our diversity — where it’s not a problem but an advantage.

## How does diversity enrich the mathematical community?

Everything we do is in some very indirect way connected to our everyday experience. Even in a very abstract field, people with different backgrounds might have different working habits, or important core beliefs that are not directly related to mathematics but can influence the way they attack problems.

## How would you describe your style of doing mathematics?

I prefer working on concrete examples, not on big, abstract theories. My view of mathematics is that I am like a pioneer discovering an unknown land. So, I don’t try to build castles, but rather I go into a jungle and follow a path, and hope that this path will lead me to new, undiscovered land.

## How is mathematics research evolving? Do you see any particular trends?

On the one hand, mathematics — at least, pure mathematics — is very conservative and does go its own way. But now we live in this exciting time when technology is changing our lives, and, of course, it is changing mathematicians and mathematics. We get a lot of input not from inside mathematics but from outside.

A topic that is attracting more and more attention is mathematical aspects of machine learning. There are many directions; one that interests me is how I could use some of these new exciting tools in my own research. Another very ambitious and noble goal is to create a mathematical theory of machine learning. When does it fail, and when we can we hope for good results?

Also, the idea of quantum computing comes with a big number of interesting mathematical problems.

## Is there a big role for mathematics in quantum error correction, which is crucial to making quantum computers work?

Studying sphere packing is very close in some ways to the problem of error correction — many approaches and methods translate from one to the other.

As mathematicians, we cannot build a quantum computer, but maybe, motivated by the possibility of its existence, we can prove interesting, meaningful theorems.

## Maryna Viazovska Wins Fields Medal for Cracking Sphere

The SciencesWorld

## Maryna Viazovska Wins Fields Medal for Cracking Sphere-Packing Problem in 8D

06/07/2022

The Wire Staff

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Maryna Viazovska at EPFL Lausanne, June 2022. Photo: Fred Merz/Lundi13

Maryna Viazovska became the second woman to win the Fields Medal, for her work on helping solve the sphere-packing problem in eight dimensions.

The Fields Medal is presented by the International Mathematical Union to up to four mathematicians once every four years, all younger than 40 years.

Viazovska hails from Ukraine and spoke up on helping refugees fleeing the war, brought by Russia, and students leaving Ukraine to study in Europe.

**Bengaluru**: Maryna Viozovska became only the second woman to be awarded the Fields Medal, one of the most prestigious awards given for contributions to mathematics. Viazovska – who hails from Ukraine – was joined by Hugo Duminil-Copin (France), James Maynard (UK) and June Huh (USA).

The Fields Medal is presented by the International Mathematical Union to up to four people once every four years. The laureates are required to be younger than 40 years at the time of awarding and can’t win the medal more than once. The age limit, set up by Canadian mathematician John Charles Fields (1863-1932), was to ensure the medal would be awarded to people who had displayed excellent mathematical skill and who could find encouragement in the award to continue doing good work.

Viazovska (37) was born in Kyiv, the capital of Ukraine. Since 2018, she has been a professor of number theory at the École Polytechnique Fédérale de Lausanne in Switzerland. Before the Fields Medal, Viazovska had received the Clay Research Award in 2017 and the Satter Prize and the Fermat Prize in 2019, among others.

She was awarded the Fields Medal this year for her solution to the sphere-packing problem in eight dimensions, developed in 2016. The sphere-packing problem requires one to prove that the way to arrange the most number of spheres of the same size in a given volume is to stack them in a pyramid – like seen in grocery stores around the world. This arrangement is technically called a face-centred cubic packing.

Thomas Hales first solved the problem in 1998 – in a 250-page paper and involving more than 50,000 lines of computer code. Peer-reviewers of the journal Annals of Mathematics – one of the most ‘prestigious’ in mathematics – needed four years just to be able to say Hales’s solution was too complicated to admit a full verification. The paper went on to set the stage for discussions on how mathematicians of the future could verify computer-aided solutions to proofs.

However, it’s notable that in three dimensions – the case that Hales considered – the sphere-packing problem is very difficult. In dimensions eight and 24, there are two arrangements that pack spheres in a highly symmetric way. These arrangements are called the E8 lattice and the Leech lattice, respectively. Their existence is counterintuitive because adding more dimensions should make the problem more complex.

(The number of dimensions of a space is equal to the minimum number of coordinates required to locate a point in that space.)

Eight- and 24-dimensional sphere-packing has a variety of applications as well because the solutions seem to exist at the “nexus of [different] areas of mathematics” (source). So solving them is lucrative.

In March 2016, Maryna Viazovska proved that the E8 lattice is the best solution to the sphere-packing problem in eight dimensions. More specifically, Viazovska’s paper unearthed the function that would describe the highest possible packing density in eight dimensions – the missing piece at the time to solve the problem in eight dimensions.

Hales told Quanta at the time that he believed finding the function “would take a Ramanujan to find” – referring to the Indian mathematician whose ability to deduce solutions to intricate mathematical problems have been the stuff of legend.

Viazovska’s paper was only 23 pages long. And using this work plus the help of some other mathematicians (notably Henry Cohn, among others), the group cracked the problem in 24 dimensions as well. The latter paper appeared less than a week later and was only 12 pages long – a reflection of the importance and cruciality of Viazovska’s findings.

The only other woman to have won a Fields Medal was the Iranian mathematician Maryam Mirzakhani, in 2014. According to the citation, she received the medal for “outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces”.

Similarly, the full citation for Viazovska read: “For the proof that the E8 lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis.”

The 2022 awards ceremony was originally slated to be held in St Petersburg in Russia and included President Vladimir Putin among its guests. After Russia’s invasion of Ukraine and several letters of protest, the International Mathematical Union moved the venue to Helsinki, Finland.

## Fields medal 2022: Work on prime numbers and spheres wins maths prize

Mathematicians who have studied the most efficient way to pack spheres in eight-dimensional space and the spacing of prime numbers are among this year's recipients of the highest award in mathematics, the Fields medal

## Fields medal 2022: Work on prime numbers and spheres wins maths prize

Mathematicians who have studied the most efficient way to pack spheres in eight-dimensional space and the spacing of prime numbers are among this year's recipients of the highest award in mathematics, the Fields medal

MATHEMATICS 5 July 2022

By Matthew Sparkes

The four Fields medal winners, clockwise from top left: Maryna Viazovska, James Maynard, June Huh and Hugo Duminil-Copin

Mattero Fieni/Ryan Cowan/Lance Murphy

Mathematicians who have studied the most efficient way to pack spheres in eight-dimensional space and the spacing of prime numbers are among this year’s recipients of the highest award in mathematics, the Fields medal.

The winners for 2022 are James Maynard at the University of Oxford; Maryna Viazovska at the Swiss Federal Institute of Technology in Lausanne (EPFL); Hugo Duminil-Copin at the University of Geneva, Switzerland; and June Huh at Princeton University in New Jersey.

Kyiv-born Viazovska is only the second female recipient among the 64 mathematicians to have received the award.

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“Sphere packing is a very natural geometric problem. You have a big box, and you have an infinite collection of equal balls, and you’re trying to put as many balls into the box as you can,” says Viazovska. Her contribution was to provide an explicit formula to prove the most efficient stacking pattern for spheres in eight dimensions – a problem she says took 13 years to solve.

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Maynard’s work involved understanding the gaps between prime numbers, while Duminil-Copin’s contribution was in the theory of phase transitions – such as water turning to ice, or evaporating into steam – in statistical physics.

June Huh, who dropped out of high school aged 16 to become a poet, was recognised for a range of work including the innovative use of geometry in the field of combinatorics, the mathematics of counting and arranging.

The medal, which is considered to be as prestigious as the Nobel prize, is given to two, three or four mathematicians under the age of 40 every four years.

The awards were first given out in 1936 and are named in honour of Canadian mathematician John Charles Fields. This year’s awards were due to be presented at the International Congress of Mathematicians in Saint Petersburg, Russia, but the ceremony was relocated to Helsinki, Finland.

More on these topics:

MATHEMATICSFIELDS MEDALS

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