П'ятниця, 15 липня 2022 09:34

In Times of Scarcity, War and Peace, a Ukrainian Finds the Magic in Math

The Ukrainian mathematician Maryna Viazovska at the École polytechnique fédérale de Lausanne (EPFL) in Switzerland in May. The Ukrainian mathematician Maryna Viazovska at the École polytechnique fédérale de Lausanne (EPFL) in Switzerland in May. Thomas Lin/Quanta Magazine

With her homeland mired in war, the sphere-packing number theorist Maryna Viazovska has become the second woman to win a Fields Medal in the award’s 86-year history.

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, waiting 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. 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, including some who 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. Today, Viazovska is accepting her Fields Medal at the International Congress of Mathematicians in Helsinki, Finland. This 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 today, the IMU cited Viazovska’s many mathematical accomplishments, in particular her proof that an arrangement called the E8 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 Second Derivative

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. ”

Maryna Viazovska seated at her office desk using her computer to videoconference with students.

Viazovska videoconferencing with students in her EPFL office.
Thomas Lin/Quanta Magazine

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 space-time 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 of 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. ”

Almost Infinity

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 U. S. ), 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. ”

A mug with a picture of Maryna Viazovska and Bogdan Rublyov at the 2019 European Girls’ Mathematical Olympiad in Kyiv, Ukraine.
A mug with a picture of Maryna Viazovska and Bogdan Rublyov at the 2019 European Girls’ Mathematical Olympiad in Kyiv, Ukraine.
Thomas Lin/Quanta Magazine

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. ”

Doing It Right

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.

Viazovska at EPFL’s avant-garde learning center.
Thomas Lin/Quanta Magazine

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 4xy2z5 + 3x4 — 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 sphere and use its 12 corners as your sampling points, and you’re guaranteed to get the exact answer for all polynomials of degree up to 5. A set such as these 12 points is called a spherical design. Since the 1970s, mathematicians have wondered: As you look at polynomials of higher and higher degree, how does the number of points in a spherical design grow? That’s the question Viazovska, Bondarenko and Radchenko answered. “It takes something that a lot of people have thought about a long time, and after a long series of suboptimal constructions, this paper comes along and says, ‘Well, gee, why don’t you do it this way, then you get exactly the right bound, QED,’” Cohn said. “It’s not like they jumped through all sorts of elaborate hoops to get this — they just do it right. ”

Magic Functions

As an undergraduate, Viazovska lived what she called a “double life,” dividing her studies between the seemingly disparate fields of algebra and analysis (a generalization of calculus). But then she went to Bonn for her doctoral studies and started studying modular forms, functions with special symmetries related to the ones that appear in the circular tilings of the artist M. C. Escher. Modular forms involve a lot of analysis, but their symmetries bring algebra into the picture as well. “I realized this is where my two passions meet,” she said. Together with Bondarenko and Radchenko, she started exploring whether modular forms could illuminate a centuries-old question the three had been trying to crack for a while: how to pack spheres together in the densest possible way. Mathematicians already knew that the densest way to pack circles in the plane is in a honeycomb pattern, and the densest way to pack spheres in three-dimensional space is the familiar pyramidal piling you see in stacks of oranges at the grocer. But the question can also be posed in higher dimensions, where it has important applications to error-correcting codes. No one knew what the densest sphere packings were in dimensions higher than three. But two special dimensions — eight and 24 — had strong candidates. In those two dimensions there exist highly symmetric arrangements, called E8 and the Leech lattice, respectively, that pack spheres much more densely than any other arrangements mathematicians could find. Cohn and Noam Elkies of Harvard University had developed a method that uses certain functions to compute upper bounds on how dense a sphere packing can be. In dimensions eight and 24, these upper bounds were an almost perfect match for the densities of E8 and the Leech lattice. Mathematicians felt certain that in each of these two dimensions, there must be a “magic” function whose bound matches E8 or the Leech lattice perfectly, thereby proving them to be the densest packings. But researchers had no idea where to find these magic functi

Viazovska teaches students about modular forms using a book co-written by her former doctoral adviser, Don Zagier.
Thomas Lin/Quanta Magazine

Bondarenko, Viazovska and Radchenko looked to modular forms to try to construct a magic function, but for a long time they made little progress. Eventually Bondarenko and Radchenko turned their attention to other problems. Viazovska, though, couldn’t stop thinking about sphere packing. The problem somehow felt as if it belonged to her, she later told Quanta. After pondering the problem for several years, in 2016 she managed to pinpoint the magic function for dimension eight. The answer, she found, lay not in a modular form but in a certain “quasimodular” form, something with errors in its symmetries. She posted an “absolutely stunning” paper, said Peter Sarnak of the Institute for Advanced Study. It’s “one of these papers you pick up, [and] you don’t put down before you’ve read the whole thing. ”Within hours of the paper’s appearance, news of her result was spreading. That evening, Akshay Venkatesh, a mathematician at the Institute for Advanced Study — himself a 2018 Fields medalist — emailed Cohn a link to the paper, with “Wow!” in the subject line. Cohn devoured the proof. “My initial reaction was, ‘What on earth is this? It looks like nothing anybody has tried to do for constructing these functions,’” he said. To Cohn, the quasimodular form Viazovska used had always seemed “just a defective version of modular forms,” he said. But “there was this whole remarkable rich theory hiding below the surface. ” Feeling convinced that Viazovska’s approach should also apply to dimension 24, he emailed her to propose a collaboration. Viazovska wanted nothing more than to take a break. But she agreed to plunge into the 24-dimensional problem, and over a single intense week she and Cohn, together with Radchenko and two other mathematicians, managed to prove that the Leech lattice is the densest 24-dimensional sphere packing. It was “probably the craziest week of my life,” Radchenko recalled.

A Bold Conjecture

Viazovska and her collaborators emerged from the sphere-packing work with a higher ambition. Mathematicians had long suspected that E8 and the Leech lattice are much more than just the best way to pack spheres. These two lattices, mathematicians hypothesized, are “universally optimal,” meaning that they are the best arrangements according to a host of criteria — for example, the lowest-energy way to position mutually repelling electrons in space or twisty polymers in a solution.

To prove that E8 and the Leech lattice minimize energy in all these different contexts, the team had to come up with magic functions for each different notion of energy — infinitely many magic functions. But they only had partial information about how such a magic function must behave (if it exists). They knew the value of the function at some points, and at other points they knew the value of its Fourier transform, which measures the function’s natural frequencies. They also knew how quickly the function and its Fourier transform were changing at particular points. The question was: Is this information enough to reconstruct the function?Viazovska made a bold conjecture: This information the team had was exactly the right amount to nail down the magic function. Any less, and there would be many functions that fit. Any more, and the function would be too constrained to exist. Cohn had his doubts. What Viazovska was proposing was so simple and fundamental that “if this were true, surely humanity would already know it,” he thought at the time. He also knew that Viazovska didn’t make conjectures frivolously. “I still thought, ‘This is kind of pushing her luck here. ’”Viazovska and Radchenko first managed to prove a simplified version of her conjecture, in which the information is limited to the values of the function and its Fourier transform, not the speed at which they are changing. Then, together with their sphere-packing collaborators, they figured out how to prove the full conjecture — exactly what was needed to show that E8 and the Leech lattice are universally optimal. It seems, Cohn said, that in the process of trying to understand these lattices, “Maryna was also pushing the state of the art in Fourier analysis. ”

In front of EPFL’s central administration building, which sits opposite the math building.
Thomas Lin/Quanta Magazine

The resulting paper, said Sylvia Serfaty of New York University, is on a par with the great breakthroughs of the 19th century, when mathematicians solved many of the problems that had confounded their predecessors for centuries. “This paper is really a great advancement of science,” she told Quanta at the time. “To know that the human brain is able to produce a proof of something like that, to me it’s a really remarkable fact. ”

War and Peace

If Viazovska sometimes seems to inhabit another plane or a different dimension when doing math, it’s probably because, as her teenage son Michael has learned, she’s in her own world. “Sometimes my mom has loops in the ear and does not react when you talk to her,” he said. He remembers being the last kid in his kindergarten class to be picked up when the family lived in Berlin (and Viazovska was working on the E8 proof). He was aware that his mother had won a lot of math awards but was surprised to hear about the Fields Medal, saying, “Now I understand why she worked so much. ”At their apartment in Lausanne in early May, a 20-minute walk from the EPFL campus, an extra bed was tucked into the alcove of the living area to accommodate Natalie and Tetiana, and Tetiana’s daughter Oleksandra and son Maksym. This spring, Oleksandra celebrated her 10th birthday not at home in Kyiv but at her aunt Maryna’s place in Lausanne. On one wall of the apartment hangs a large drawing Viazovska made of a nearby view of Lake Geneva. Outside of math, art has been her principal escape since childhood. Some of her favorite drawings, like the one she made of a Klein bottle containing an Escher-esque fish pattern, incorporate themes from math and science. (It’s hard to study math without taking an interest in Klein bottles and M. C. Escher, she explained. ) She sometimes draws pictures to help visualize geometric ideas in her work, but she is acutely aware that when dealing with higher dimensions, “our two-dimensional and three-dimensional intuition is often misleading. ”

Maryna Viazovska inside her Lausanne apartment with her 13-year-old son, Michael, and her 2-year-old daughter, Sophie.
Viazovska at home with her 13-year-old son, Michael, and her 2-year-old daughter, Sophie.
Thomas Lin/Quanta Magazine

Viazovska walks to work, both for the exercise and because neither she nor her husband drives — a fact the couple affectionately rib each other about. “Maryna has a driver’s license, but in our three-dimensional world, it’s very difficult [for her] to drive,” Evtushinsky joked. “Ha ha,” Viazovska deadpanned. When Evtushinsky explained how he’s in the process of getting his license, she described it as “a long, slowly-going process. ”

“We are probably the only parents who don’t have a car,” Evtushinsky said. “I don’t know why it is so difficult for us. ”As the conversation drifted inevitably back to the conflict in Ukraine, Viazovska shared a dark joke that has become a morbid refrain among friends back home: “Do you remember those old good times of the coronavirus?”Viazovska’s grandmother, who still has no plans to leave Ukraine, told her that even though she’s old and it’s almost her time, she doesn’t want to die before the war ends, because “I want to see the peace and I want to know that somehow everything is going to be OK. ”Viazovska is proud of her country but feels terrible that her compatriots have had to acclimate to the air raid sirens, the shelling, the war. After enduring the first days of the invasion, her nephew Maksym started sleepwalking at night. “This is not for free,” Viazovska said. “This will have some consequences in the future, this kind of extreme stress, extreme fear. ”At least, she said, “tyrants cannot stop us from doing mathematics. There is at least something they cannot take away from us. ”