Maryna Viazovska, explorer of mathematical dimensions
Summer Series: A calling for research. Maryna Viazovska, who holds the Chair of Number Theory at EPFL, has resolved the problem of sphere packing in 8 and 24 dimensions.
Talking to Maryna Viazovska, you get the sense that her brain operates on a higher level. Mathematical dimensions are her playthings. For Viazovska, the problem of sphere packing in 8 and 24 dimensions might be akin to a 30,000-piece jigsaw puzzle for a normal human brain. She solved that problem, coming up with a brilliant – and elegant – solution. Viazovska is one of the brightest and well-known mathematicians in the field of numbers theory.
The 35-year-old native of Kiev seemed destined for a career in mathematics. “I've liked mathematics since school,” Viazovska says. “It always seemed like the most straightforward subject. And since I liked it, I spent more time on it, and I became better at math than at other subjects. So then I liked it even more, and so on.” As her career developed, she delved ever further into an exploration of the most fascinating dimensions of her chosen field of study. Her aim in pure mathematics – in addition to problem solving – is to understand things that she finds intriguing “so that I can connect them to other problems.”
It was as part of her postdoctoral research at the Berlin Mathematical School and at Humboldt University of Berlin that she took on and resolved a question that had stumped mathematicians for more than four centuries: how to pack spheres 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.
Elegant and optimal
With the third dimension resolved, it was time to move on to other dimensions. “What's hard is that 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.” That’s because the ways the spheres are packed is remarkably symmetrical in these particular dimensions using the E8 and Leech lattices, respectively. When it comes to optimization, this is close to perfection – to one billionth of a percent, according to the findings of Henry Cohn and Noam Elkies, mathematicians in Cambridge, Massachusetts.
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. For this, she drew on modular forms – the focus of her dissertation. Modular forms are mathematical functions with a high level of symmetry. After two years of work, fueled by creativity and intuition, she came up with the right function for 8 dimensions. She published her results in March 2016. Her proof filled 23 pages. Concise, for mathematicians; impenetrable, for the uninitiated. Henry Cohn congratulated her and immediately suggested that she extend her method to the Leech lattice in 24 dimensions. Viazovska, Cohn and two other colleagues posted an analogous theorem for 24 dimensions online one week later.
This proof was cheered by the mathematical community and has earned Viazovska a number of prestigious distinctions that are listed on her Wikipedia page (which even has a short version in Hindi). In December 2016, she accepted EPFL’s offer to become a tenure-track assistant professor. Just one year later, in January 2018, 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.” A good sign that her feet are firmly planted on the ground.
Viazovska continues her work on 8 and 24 dimensions, delving deeper into both sphere-packing and the ‘universal optimality’ of the E8 and Leech lattices. “These configurations appear in other optimization problems and topics in combinatorics – probably because they always work best, not only for packing spheres but also in explaining energy expenditure, for example,” she says, adding: “That’s rather unusual.” Another mystery to solve.