A second Fields Medal to a woman

The Fields Medal is awarded to recognize outstanding mathematical achievement for existing work and for the promise of future achievement. Out of a total of about 60 people, only one woman had so far managed to receive these highest honors for spectacular new mathematical discoveries. This was Maryam Mirzakhani, who received this award in 2014 (and died of cancer in 2017). Now a second woman has received this highest mathematical award (Nobel prizes do not exist for mathematics, after all, the founder Alfred Nobel thought at the end of the 19th century that mathematics was not so important for practical purposes, something he would hardly have believed 25 years later. There are also rumors that he was angry with the mathematician Magnus Gösta Mittag-Leffler for having an affair with the great mathematician Sofja Kowalevskaja – which is not true).

The woman receiving the second Fields Medal is the 37-year-old Ukrainian Maryna Viazovska, holder of the Chair for Number Theory at the Swiss EPFL. Number theory is considered by mathematicians to be the crowning field of their discipline. She was rewarded for solving a problem that can be explained quite clearly at first sight: It is the so-called sphere-packing problem that had preoccupied mathematicians for more than four centuries: how to pack spheres like oranges as tightly as possible (using the smallest volume)? In 1611, Johannes Kepler proposed the hypothesis that the densest type is the pyramid-shaped one stacked on top of the other – but had to leave this hypothesis unproven. This hypothesis was not finally proven before 1998 (for which a computer had to be used, which did not please all colleagues who would have liked to have a „real proof“). The volume is then a little more than 74 per cent filled with spheres – more is not possible. Now the mathematicians, including Maryna Viazovska, wondered whether this statement could also be proven for higher 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 special dimensions is remarkably symmetrical. More than a decade ago, her colleagues Henry Cohn (MIT/Microsoft Research) and Noam Elkies (Harvard) found that particularly efficient sphere packing could be constructed in certain configurations of space, in the so-called E8 lattice for eight-dimensional space, and the so-called Leech lattice, a 24-dimensional lattice. However, they were unable to develop a proof that these are minimal packages. Brilliant work by Viazovska provided the missing ingredient and showed that these lattices are indeed the densest possible packing patterns in their respective dimensions.

Back in 2016, Maryna Viazovska proved in a paper of only 23 pages that the E8 lattice realizes the densest sphere packing in eight-dimensional space (R8) when spheres of equal radii are placed around each of the lattice points. Her proof was considerably simpler than that of Kepler’s corresponding 3-dimensional problem. For this, she used a certain function composed of modular forms, in the search for which she drew on other areas of mathematics – a fact that makes her proof particularly elegant and original, according to experts. Driven by creativity and intuition, Viazovska oriented herself towards the focus of her dissertation: modular forms, a type of mathematical function with a particularly high degree of symmetry. After two years of work, she had found the right function for eight dimensions. Only one week later, Viazovska (with colleagues Cohn, Kumar, Miller, Radchenko) then proved with similar methods as in the eight-dimensional case in a 17-page paper that the Leech lattice provides the optimal sphere packing in the 24-dimensional space R24. This finally confirmed the significance of their original proof for 8 dimensions. This proof was cheered by the mathematical community and earned Viazovska a number of prestigious awards. “This is actually a very useful tool used in many areas of technology,” she told The Associated Press.

Unfortunately, the methods used cannot be generalized to other dimensions, so many mathematicians seem to believe that there will not be such elegant solutions in other dimensions. Nevertheless, mathematicians saw this as an epochal breakthrough and gave enthusiastic descriptions of it (as can be seen, for example, in a 2019 Quanta Magazine article: https://www.quantamagazine.org/universal-math-solutions-in-dimensions-8-and-24-20190513/).

More recently, Viazovska was able to prove the „universal optimality“ of the E8 and Leech lattices, showing that they are the best possible configurations for a continuous set of natural problems. Although Viazovska’s latest research results are very abstract, they could help solve everyday problems. For example, it has long been known that sphere packing (packing everything into spheres) plays a key role in information theory and in the theory of error-correcting codes. Viazovska’s latest research may play a role in this.

Viazovska was born on Dec. 2, 1984, in Kiev, Ukraine. Because she developed a passion for mathematics at a young age, her path to the subject was relatively easy. “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 earning her Bachelor degree at the National Taras Shevchenko University in Kiev, Viazovska went to Germany to earn a Master degree at the Technical University of Kaiserslautern (2007) before moving to the University of Bonn, where she completed her doctorate on modular forms in 2013. During her postdoctoral research at the Berlin Mathematical School and Humboldt University of Berlin, Viazovska tackled and solved the sphere packing problem in 8 and 24 dimensions. Then, in December 2016, she accepted an offer from EPFL to become an assistant professor with tenure-track status. Just one year later, at the age of 33, she was promoted to full professor.

What drives Viazovska is solving problems, which she describes as „doing a jigsaw puzzle”, and understanding abstract concepts „so I can link them to other problems and find practical applications.“ She is not, however, independent of politics. For example, she also says that Russia’s invasion of Ukraine on Feb. 24 profoundly changed her life and that of all Ukrainians.

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