Of the 60 mathematicians who won Fields Medals before this year, 59 were men. In 2014, a Stanford mathematician, Maryam Mirzakhani, was the first and, so far, only woman to receive one. “I feel sad that I am only the second woman,” said Dr. Viazovska. “But why is that? I do not know. I hope that will change in the future.” Dr. Viazovska’s work is a variation of a conjecture by Johannes Kepler more than 400 years ago. Kepler is best known for realizing that the planets move around the sun in elliptical orbits, but he also looked at the stacking of spheres, arguing that the usual pyramid stacking was the densest way they could arrange them, filling just over 75 percent hundred of available space. However, Kepler could not prove this statement. Nor could anyone else until Thomas Hales, then at the University of Michigan, succeeded in 1998 with a 250-page proof and, controversially, the help of a computer program. Proving something similar for the packing of equal-sized spheres in dimensions greater than three has so far been impossible — with a few exceptions. In 2016, Dr Viazovska found the answer in eight dimensions, showing that a highly symmetric packing structure known as E8 was the best possible, filling about a quarter of the volume. Within a week, she and four other mathematicians showed that a different arrangement known as a Leech lattice was the best possible packing in 24 dimensions. At high dimensions, the filled volume is not very crowded, with the Leech lattice of 24-dimensional spheres occupying about 0.2 percent of the volume. What is special about dimensions eight and 24? “I think this is a mystery,” said Dr. Viazovska. “It’s only in these dimensions that certain things happen that don’t happen in other dimensions.” He said a method that generally gives an upper limit on packing density turns out to be the exact solution in these cases. High dimensional sphere packings are related to the error correction techniques used to correct distortions in information transmission. She said the Russian invasion of Ukraine had taken its toll on her family. “It’s very difficult,” he said. Her parents still live near Kyiv, Dr Viazovska said, while her sisters, nephew and niece left and joined her in Switzerland.
