In late February, just weeks after Maryna Viazovska learned she had won a Fields Medal—the highest honor for a mathematician—Russian tanks and war planes began their assault on Ukraine, her homeland, and Kyiv, her hometown. Viazovska no longer lived in Ukraine, but her family was still there. Her two sisters, a 9-year-old niece, and an 8-year-old nephew set out for Switzerland, where Viazovska now lives.
They first had to wait two days for the traffic to let up; even then the drive west was painfully slow. After spending several days in a stranger’s home, awaiting their turn as war refugees, the four walked across the border one night into Slovakia, went on to Budapest with help from the Red Cross, then boarded a flight to Geneva. On March 4, they arrived in Lausanne, where they stayed with Viazovska, her husband, her 13-year-old son and her 2-year-old daughter.
Original story reprinted with permission from Quanta Magazine , an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences. Viazovska’s parents, grandmother, and other family members remained in Kyiv. As Russian tanks drew ever closer to her parents’ home, Viazovska tried every day to convince them to leave.
But her 85-year-old grandmother, who had experienced war and occupation as a child during World War II, refused, and her parents would not leave her behind. Her grandmother “could not imagine she will not die in Ukraine,” Viazovska said, “because she spent all her life there. ” In March, a Russian airstrike leveled the Antonov airplane factory where her father had worked in the waning years of the Soviet era; Viazovska had attended kindergarten nearby.
Fortunately for Viazovska’s family and other Kyiv residents, Russia shifted the focus of its war effort to the Donbas region in eastern Ukraine later that month. But the war is not over. Viazovska’s sisters spoke of friends who have had to fight, some of whom have died.
Viazovska said in May that even though the war and mathematics exist in different parts of her mind, she hadn’t gotten much research done in recent months. “I cannot work when I’m in conflict with somebody or there is some emotionally difficult thing going on,” she said. On July 5, Viazovska accepted her Fields Medal at the International Congress of Mathematicians in Helsinki, Finland.
The conference, organized by the International Mathematical Union every four years in concert with the Fields Medal announcements, had been set to take place in St. Petersburg, Russia, despite concerns over the host country’s human rights record, which prompted a boycott petition signed by over 400 mathematicians. But when Russia invaded Ukraine in February, the IMU pivoted to a virtual ICM and moved the in-person award ceremony to Finland.
At the ceremony , the IMU cited Viazovska’s many mathematical accomplishments, in particular her proof that an arrangement called the E 8 lattice is the densest packing of spheres in eight dimensions. She is just the second woman to receive this honor in the medal’s 86-year history. ( Maryam Mirzakhani was the first, in 2014.
) Like other Fields medalists, Viazovska “manages to do things that are completely non-obvious that lots of people tried and failed to do,” said the mathematician Henry Cohn , who was asked to give the official ICM talk celebrating her work. Unlike others, he said, “she does them by uncovering very simple, natural, profound structures, things that nobody expected and that nobody else had been able to find. ” The precise whereabouts of the École Polytechnique Fédérale de Lausanne is far from obvious outside the EPFL metro station on a rainy May afternoon.
Known in English as the Swiss Federal Institute of Technology Lausanne—and in any language as a leading research university in math, physics, and engineering—it’s sometimes referred to as the MIT of Europe. At the end of a dual-use lane for bicycles and pedestrians that ducks under a small highway, the idyllic signs of campus life come into view: giant two-tier racks packed with bicycles, modular architecture befitting a sci-fi cityscape, and a central square lined with classrooms, eateries, and upbeat student posters. Beyond the square sits a modern library and student center that rises and falls in three-dimensional curves, allowing students inside and out to walk under and over each other.
From below, the sky is visible through cylindrical shafts punched through the topology like Swiss cheese. A short distance away, inside one of those modular structures, a professor with a security access card opens the orange double doors leading to the inner sanctum of the Math Department. Just past the portraits of Noether, Gauss, Klein, Dirichlet, Poincaré, Kovalevski, and Hilbert stands a green door simply labeled “Prof.
Maryna Viazovska, Chaire d’Arithmétique. ” Inside, the office is spare, pragmatic: just a computer, printer, chalkboard, papers, and books, with few personal effects. The place where the magic happens seems not so much a physical location in spacetime as a higher-dimensional world of abstractions in Viazovska’s mind.
Across the small table in her office, the world’s preeminent sphere-packing number theorist begins to recount her story in her usual matter-of-fact manner. Gradually, she breaks form and smiles, her eyes light up and lift upward, and she grows ever more animated while summoning memories from the past. The earliest memory is of walking with her grandmother as a 3-year-old from her family’s utilitarian Khrushchyovka apartment building (named after the former Soviet leader Nikita Khrushchev), down a wide boulevard to a monument for the geochemist Vladimir Vernadsky, where her grandmother lifted her up and tossed her into the air.
The late 1980s were a difficult time in the Soviet Union, said Viazovska, now 37. “It took people many, many hours to buy even basic things. ” When a shop was low on goods like butter or meat, her mother felt bad about taking more for her three children and worried that others waiting in the long line would get angry at her.
Her family didn’t have much, because there wasn’t much to have, but her parents made sure she and her sisters never went hungry or without heat. No stores carried nice clothes, but workers were sometimes offered a chance to win a stylish pair of shoes made in Czechoslovakia as an incentive for doing good work. The shoes might not fit, her mother explained to her, but if you won a pair, you could trade with someone who had won a pair in your size.
“The Soviet Union fell apart when I was 6,” Viazovska said. Her family was excited to live in a free and independent Ukraine, but hyperinflation only worsened their economic plight. In the Soviet Union, there was money but no goods to spend it on.
In the early years of Ukraine’s independence, there were goods but not enough money to buy them. Her mother worked as an engineer until 1995, and in that last year on the job, she told her daughter, her monthly salary couldn’t pay for a metro ticket. Describing her father as a former chemist who is “extremely energetic,” with “entrepreneurship spirits,” Viazovska recalled how he left his job and embraced the new reality by starting one small business after another.
That new reality was chaotic and unpredictable, she said. “One day, you don’t have much. Then there is another opportunity, and you have a lot.
” Still, both Viazovska and her husband, Daniil Evtushinsky, a physicist at EPFL, remember the hopeful exuberance Ukrainians felt at the prospect of economic growth. “In the economy, what matters is the derivative and not the absolute value,” said Evtushinsky, referring to the importance of the rate of growth over one’s current assets. Given how low that absolute value was at times, Viazovska replied with a laugh: “Maybe the second derivative.
” As a first grader, Viazovska realized that she liked math better than language arts: “In reading, I was too slow. In writing, I was too messy. But with math, I was kind of quick.
” It’s not that she didn’t like reading. She read Alexandre Dumas, Jules Verne, and the assorted pirate adventure books her parents gave her. Later, she discovered science fiction and fell in love with the genre.
“Flowers for Algernon,” the Hugo Award-winning short story about a mentally disabled man and a lab mouse who undergo an experimental procedure to boost their intelligence, was particularly memorable, she said, because it’s “actually about us”—the human condition, not fantastical technology. She also devoured the science fiction stories written by the Russian brothers Arkady and Boris Strugatsky. While their early work was overly optimistic and naïve about communism, she said, their writing grew increasingly dark and “much smarter and much deeper.
” Evtushinsky recalls first meeting Viazovska at an after-school physics circle when they were around 12. Even then, she approached math problems in her own way. One problem, he remembered, involved a physical system with seven elements.
“Maryna made a conjecture that seven is almost infinity,” he said. The extraordinary approximation “worked very well and simplified the problem drastically,” he said. “No one else could suggest that.
” Viazovska’s younger sisters, Natalie and Tetiana, recall how talented and committed she was, even as a child. “When everyone goes to sleep, she has her notepad and she draws some formulas,” Natalie said, adding that their parents were afraid she studied too much instead of playing like the other kids. Natalie did not look forward to getting the same math teacher as her older sister.
“Her math teacher became my math teacher,” Natalie said. “I heard very often that Maryna is a brilliant student. ” Viazovska attended a specialized lyceum (equivalent to high school in the US), where she was invigorated by the advanced math and physics classes, and by the exceptional teachers who were genuinely enthusiastic about explaining difficult concepts and making students put in the work to master them.
There, she fell deeper into the competitive world of math Olympiads, which she had loved for years. It didn’t always love her back. “It teaches you how to lose and how to win,” Viazovska said.
“In my case, I was not as successful as I dreamed about. ” In her last year at the lyceum, her dream was to represent Ukraine in the International Mathematical Olympiad. At the national competition, only the top 12 competitors are invited to a training camp where six national team members are selected.
Viazovska placed 13th. She had worked hard, she said, but “apparently not hard enough. ” Bogdan Rublyov , the head of Ukraine’s math Olympiad program and a math professor at Kyiv University, remembered meeting Viazovska that year.
He called it a “great surprise” that she has become such a prominent mathematician, but he is “very happy about this,” he said, “because she is a very good person. ” She went on to win many university math contests and, he said, served on the jury helping to grade Olympiad competitions in Kyiv. Now the Olympiad team is training in Poland because of the war, Rublyov said, while he is legally bound to stay in Ukraine as a 58-year-old reservist.
In March, the war exacted a far greater toll on Ukraine’s math community, when a Russian air strike in Kharkiv killed the 21-year-old mathematician Yulia Zdanovskaya. Five years ago, Zdanovskaya won a silver medal at the European Girls’ Mathematical Olympiad, which Rublyov helps to organize. “I knew her well,” he said.
“It is a catastrophe for our country that such young and talented people are dying. ” In May, some weeks before the Fields Medals were to be announced, Rublyov was convinced a Ukrainian like Viazovska could not win math’s top prize, given Russia’s influence on the world stage. “It is a pity that she was not given the Fields prize,” he lamented at the time, “because she deserves it.
” Viazovska’s first big moment as a mathematician arrived in 2005 when she collaborated on her first original research result as a senior at Kyiv University. While it wasn’t a major open problem, she realized that it was one she could solve. The joy came, she said, from “feeling that an argument comes together and it does work.
” The result buoyed her confidence. Viazovska had been encouraged to pursue the problem by Igor Shevchuk , a math professor at Kyiv University who helped organize some of the university math competitions she had participated in. Shevchuk discussed the problem with a few people, she said, including her and a master’s student named Andrii Bondarenko .
The paper she and Bondarenko produced together kick-started a fruitful period of collaboration between the two. Later, when Bondarenko was teaching at Kyiv University, he began working with a strong student named Danylo Radchenko . The three young Ukrainian mathematicians teamed up.
In 2011, Viazovska, together with Bondarenko and Radchenko, submitted a paper to the journal Annals of Mathematics on a subject called spherical designs. “ Annals ,” as mathematicians call it, is perhaps the most prestigious journal in mathematics—“the pinnacle of the pinnacle,” according to Don Zagier , who was Viazovska’s and Radchenko’s doctoral adviser at the time. When Radchenko told Zagier of the trio’s aims, Zagier thought to himself, “Dream on … you’re beginners.
” But the paper was accepted, and soon mathematicians were organizing entire conferences to discuss it. “Wow, what a fantastic paper,” thought Cohn, of Microsoft Research and the Massachusetts Institute of Technology, upon reading it. The paper examines the classical problem of analyzing the behavior of a function by looking at its values at some sprinkling of points.
In the version the trio tackled, the function is a polynomial—say, something like 4 xy 2 z 5 + 3 x 4 —and we can think of each possible input to the polynomial as a point living in the space whose dimension matches the number of variables (so for the above polynomial, each input would be a point in three-dimensional space, with its x -, y – and z -axes). In the problem Viazovska and her collaborators studied, we’re interested in the polynomial’s average value on a sphere. We could approximate this average by choosing several points on the sphere and averaging the values of the polynomial at those points.
If we’re really lucky—or if we choose the points carefully—we might even get the exact answer instead of an approximation. Mathematicians have long known that for every polynomial, you can pick some finite set of points that gives the exact answer. What’s more, you can pick a single set of points that will work for all the polynomials up to some given “degree” (the highest sum of exponents in any of the polynomial’s terms).
For example, if you’re working in three-dimensional space, you could embed a regular icosahedron in the s
