Marina Vyazowska is the second woman to win the Fields Medal

Marina Vyazovska, a Ukrainian who is now a professor at the Swiss Federal Institute of Technology in Lausanne, is known for her proofs of high-dimensional equations for stacking spheres of equal size. She is also the second woman to ever win a Fields Medal.

Of the 60 mathematicians who won Fields medals before this year, 59 were men. In 2014, a mathematician from Stanford University, Maryam MirzakhaniShe was the first, and so far only, woman to have one.

“I feel sad that I am only the second woman,” Dr. Vyazowska said. “But why is that? I don’t know. I hope it changes in the future.”

Dr. Vyazowska’s work is a variant of Johannes Kepler’s conjecture over 400 years ago. Kepler is best known for his realization that the planets move around the sun in elliptical orbits, but he also considered cannonball stacking, asserting that regular pyramid stacking was the densest way they could be arranged, filling just under 75 percent of the available. outer space.

However, Kepler could not prove this statement. No one else, until Thomas Hals, then at the University of Michigan, could, He succeeded in 1998 With a 250-page, controversial proof, With the help of a computer program.

Proving something similar to packing equal-sized spheres with dimensions higher than three has so far been impossible – with exceptions.

In 2016, Dr. Vyazovska found The answer is in eight dimensions, showing that the particularly symmetrical filling structure known as E8 worked the best, filling about a quarter of the volume. Within a week, she and four other mathematicians showed that a different arrangement known as a leech network was Best possible packing in 24 . dimensions. In high dimensions, the filled volume is not too full, the Leech 24-dimensional ball grid occupies about 0.2 percent of the volume.

What distinguishes the dimensions of eight and 24?

“I think that’s a mystery,” Dr. Vyazowska said. “Only in these dimensions do certain things happen that do not happen in other dimensions.”

She said that a method that generally gives an upper bound on packing density is the exact solution in these cases.

High-dimensional spherical packages are related to error-correction techniques used to fix interference in information transmission.

She said the Russian invasion of Ukraine had a negative impact on her family. “It’s very difficult,” she said.

Dr. Vyazovska said her parents still live near Kyiv, while her sisters, nephew and niece have left and joined her in Switzerland.