The every four years Fields Medal awarded by the International Mathematical Union, which is endowed with 15,000 Canadian dollars (about 11,000 euros), is also considered the “Nobel prize in mathematics”. This year, in addition to three male prizewinners, there is also a woman, and only for the second time: the Ukrainian number theorist Maryna Viazovska from the EPF Lausanne. Other laureates include University of Geneva stochasticist Hugo Duminil-Copin, Princeton University’s June Huh, who works in algebraic geometry, and number theorist James Maynard of the University of Oxford.
The two to four Fields Medals are only awarded to mathematicians who are under the age of 40 at the beginning of the year of the award, so you have to hurry. Viazovska, a professor at the École Polytechnique Fédérale in Lausanne, is only 37. She received the award for her work on sphere packing.
It’s a problem that Johannes Kepler dealt with centuries ago: How can you stack balls in such a way that as many as possible can be accommodated in a small space? Kepler had surmised that the best option is the pyramid pack, the way apples or oranges are sometimes stacked in the market. This was not proven until 1998 by Thomas Haleswith massive computer support.
But as so often in mathematics, there are half a dozen more general variants to a problem. So you can consider spheres not only in three, but in any dimension – a sphere’s surface is simply the area in each dimension in which every point has the same distance to the center. What part of a volume can be filled with such spheres? In two dimensions, for example, it is about arranging circles on a surface. If you distribute them in such a way that six circles are grouped around one in the middle (“hexagonal”), you can cover 90.7 percent of the area. In three dimensions, the pyramid pack still achieves around 74 percent. But what about higher dimensions? Not much was known about that for a long time.
Maryna Viazovska was born in Kyiv. She liked mathematics at school and studied the subject at the University of Kyiv. She later went to Germany to do her master’s degree at the TU Kaiserslautern. Then she did her doctorate with Don Zagier, a well-known number theorist at the University of Bonn. On the phone Tuesday morning, Zagier is surprised by the news: The Fields Medal for his former graduate student? “It’s fantastic!” he says. “She did a lot of nice things.”
The young woman is distinguished by quiet perseverance
The first story that comes to mind about Viazovska says something about her quiet perseverance. When she was still working for him, she once casually mentioned to colleagues that she was going to work for the Annals of Mathematics had submitted – the absolute top publication in mathematics, in which only the most important works are published. “Secretly I thought, well, that’s probably too much,” says Zagier. But then he was on the committee for a mathematical conference series, and someone wanted to host a conference on a new, groundbreaking result – it was the work of Maryna Viazovska and two colleagues. “She did it very quietly alongside her doctoral thesis, I didn’t know anything about it,” says Zagier. The work then really appeared in the annals.
Soon after, Viazovska began to turn to higher-dimensional packings of spheres. There are upper limits to the maximum density of such packing. Two cases were also known that came very close to these absolute limits: in eight dimensions, an arrangement in the so-called E8 lattice is close to the optimum, in 24 dimensions it is the Leech lattice. But is that really the optimal arrangement, or is there a better way?
One thing tyrants can’t do, explains the Ukrainian: keep people from mathematics
In 2016, Maryna Viazovska won with a amazingly straightforward solution acclaimed by colleagues show that the E8 lattice is indeed optimal for eight dimensions. A little later, based on this, she and four colleagues were able to prove it for the 24-dimensional case. However, the volume yield is modest with so many dimensions: in the Leech lattice, each sphere has 196,560 neighbors, and the volume covered is less than one percent of the space.
Today Viazovska lives in Lausanne with her husband and two children. She left Ukraine for a long time. After the outbreak of war, her two younger sisters, a niece and a nephew, fled according to a report on Viazovska im Quanta Magazine to her in Switzerland. But the grandmother did not want to leave Ukraine, and Viazovska’s parents cannot leave the old woman.
But at least, she told the magazine, “bullies can’t stop us from doing math. That’s something they can’t take away from us.”