# An EPFL mathematician is awarded a Fields Medal

Maryna Viazovska, who holds the Chair of Number Theory at EPFL, has been awarded a Fields Medal, widely considered to be the highest accolade in her discipline and described as the Nobel Prize of Mathematics (a field in which the Nobel Foundation does not sponsor an official award). At 37 years of age, Viazovska has thus become only the second female Fields Medalist – after Maryam Mirzakhani in 2014 – and joins a list of over 60 mathematicians to have received the prestigious honor to date. The Fields Medal was created in 1936 and is awarded every four years to one or more mathematicians under the age of 40. The official conferral is to take place at the International Congress of Mathematicians, which opens today in Helsinki.

Viazovska, who specializes in number theory, has been awarded a Fields Medal for solving the sphere-packing problem in 8 and 24 dimensions. In doing so, she resolved a question that had stumped mathematicians for more than four centuries: how to pack spheres – such as oranges stacked in a pyramid – as close together as possible. It was in 1611 that Johannes Kepler posited, without proof, that the best solution for packing spheres in a three-dimensional space was in the shape of a pyramid. That hypothesis was finally proven in 1998.

With the third dimension resolved, it was time for mathematicians to move on to other dimensions. “Formulating the problem in the same way complicates matters because each dimension is different, and the optimal solution depends very much on the dimension,” says Viazovska. Why did she focus on 8 and 24 dimensions? “Because these are special dimensions, and the solutions are particularly elegant.” The way spheres are packed in these particular dimensions is remarkably symmetrical, and uses the E8 and Leech lattices, respectively. More than a decade ago, Henry Cohn (MIT / Microsoft Research) and Noam Elkies (Harvard), found that these lattice patterns were close to perfection – to one billionth of a percent – but were unable develop a proof. Viazovska’s brilliant work provided the missing ingredient, demonstrating that these lattices are the densest possible packing patterns in their respective dimensions.

But Viazovska wanted to prove it, suspecting that an auxiliary function existed that could provide the right answer and match the density of the E8 and Leech lattices. In her quest for the right function, she drew on other areas of mathematics – a fact that, according to experts, makes her proof particularly elegant and original. Fueled by creativity and intuition, Viazovska turned to the focus of her dissertation: modular forms, a type of mathematical function with a high level of symmetry. After two years of work, she came up with the right function for 8 dimensions.

## Universal optimality

She announced her results in March 2016. Her proof filled 23 pages: concise for mathematicians. Cohn subsequently got in touch and suggested that she extend her method to 24 dimensions. One week later, Viazovska, Cohn and two other colleagues posted a theorem online demonstrating that the Leech lattice is the optimal sphere-packing configuration for 24 dimensions, confirming the significance of her initial proof for 8 dimensions. This proof was cheered by the mathematical community and has earned Viazovska a number of prestigious distinctions.

Viazovska continues her work on 8 and 24 dimensions, delving deeper into sphere-packing. “These configurations appear in other optimization problems – not only for packing spheres but also in explaining energy expenditure, for example,” she says. “That’s rather unusual.” Viazovska has recently proved the “universal optimality” of the E8 and Leech lattices, demonstrating that these are the best possible configurations across a continuous set of natural problems. It has long been known that sphere-packing plays a key role in information theory and in the theory of error-correcting codes. Viazovska’s latest research could one day help solve a wide range of other everyday problems.

## An early passion for mathematics

Viazovska was born in Kiev, Ukraine, on 2 December 1984. Having developed a passion for mathematics at a young age, her path into the discipline was relatively simple. “I've liked mathematics since my schooldays,” she says. “It always seemed like the most straightforward subject. And since I liked it, I spent more time on it, and eventually became better at math than other subjects. So then I liked it even more, and so on.”

After obtaining a Bachelor’s degree from the Taras Shevchenko National University of Kyiv, Viazovska moved to Germany to obtain a Master’s degree at the Technical University of Kaiserslautern (2007) before joining the University of Bonn, where she completed her PhD on modular forms in 2013. Since then, she has also started a family. “Studying pure mathematics is bit like reading a book with illustrations,” she says. “The images are linked to the text, but they don’t match the written word exactly.” Viazovska is driven by problem-solving, which she describes as being akin to “doing a jigsaw puzzle,” and by understanding abstract concepts “so I can link them to other problems and find practical applications.”

## Six years at EPFL

It was during her postdoc research at the Berlin Mathematical School and the Humboldt University of Berlin that Viazovska took on and solved the sphere-packing problem in 8 and 24 dimensions. In December 2016, she accepted EPFL’s offer to become a tenure-track assistant professor. Just one year later, at 33 years of age, she was promoted to full professor. “I really like the fact that at EPFL there’s not just pure mathematics, but also a lot of people working in very applied fields,” she says.

According to EPFL President Martin Vetterli, Viazovska is an asset to the School for two reasons: “Maryna was already an outstanding researcher when she joined us six years ago. But it’s been even more impressive to watch her flourish and play an active part of our Institute of Mathematics. EPFL is becoming a center of excellence in this field, not least through the Bernoulli Center, which is forging a global reputation in mathematics, physics and theoretical computer science. I would like to congratulate not only Maryna, but all the professors and researchers who’ve been involved in shaping this dynamic ecosystem.”

EPFL extends its congratulations to Hugo Duminil-Copin, a professor in the Mathematics Section at the University of Geneva (UNIGE), who has also received a Fields Medal. These two accolades further anchor the Lake Geneva region as a center of excellence in the basic sciences.

## Honors and awards won by Maryna Viazovska

Salem Prize (2016)

Clay Research Award (2017)

SASTRA Ramanujan Prize (2017)

European Prize in Combinatorics (2017)

New Horizons in Mathematics Prize (2018)

Fermat Prize (2019)

European Mathematical Society Prize (2020)

Fields Medal (2022)