Ukrainian Mathematician Maryna Viazovska Wins Fields Medal

Russian tanks and war planes began their assault on Ukraine, Maryna Viazovska's hometown, just weeks after she won the highest honor for a mathematician.

Viazovska's family was still in Ukranian. Viazovska's two sisters, a 9-year-old niece and an 8-year-old nephew left for Switzerland. The drive west was so slow that they had to wait two days for the traffic to stop. After spending several days in a stranger's home, waiting their turn as war refugees, the four walked across the border with help from the Red Cross, then boarded a flight to Geneva. They arrived in Lausanne on March 4 and stayed with Viazovska and her family.

The family of Viazovska stayed in Kyiv. Viazovska tried every day to convince her parents to leave. Her parents would not leave her behind, even though she had experienced war as a child. Viazovska's grandmother spent her entire life in Ukraine.

The factory where Viazovska's father worked in the waning years of the Soviet era was destroyed by a Russian airstrike in March. The focus of Russia's war effort in eastern Ukraine was shifted to the family of Viazovska. The war is still going on. Some of Viazovska's friends have died in fights.

Viazovska said in May that she hadn't gotten a lot of research done in the last few months. She said that she can't work when she's in conflict with someone.

Viazovska is going to accept her Fields medal at the International Congress of Mathematicians today. Over 400 mathematicians signed a petition to boycott the conference due to the host country's human rights record. The IMU pivoted to a virtual ICM after Russia invaded Ukraine.

The IMU cited Viazovska's proof that the E 8 lattice is the densest packing of spheres in eight dimensions at the ceremony. In the 86-year history of the medal, she is the only woman.

Henry Cohn, who was asked to give the official ICM talk celebrating her work, said that Viazovska manages to do things that are completely non-obvious that a lot of people tried and failed to do. She does them by finding things that nobody else had been able to find.

The Second Derivative

On a rainy May afternoon, the location of the cole polytechnique fédérale de Lausanne is hard to see. The Swiss Federal Institute of Technology Lausanne is often referred to as the MIT of Europe. The idyllic signs of campus life can be seen at the end of a dual-use lane for bicycles and pedestrians that ducks under a small highway. The library and student center is three-dimensional and allows students to walk under and over each other. The sky can be seen from below through cylindrical shafts. A professor with a security access card can open the double doors of the math department inside one of those modular structures. The chaire d'Arithmétique is just past the portraits of Noether, Gauss, Klein, Dirichlet, Poincaré, Kovalevski and Hilbert.

There is a computer, printer, chalkboard, papers and books in the office. Viazovska doesn't see the magic happening in a physical location, but in a higher-dimensional world of abstract ideas.

Across the small table in her office she begins to recount her story in her usual matter-of-fact manner. Slowly, she breaks form and smiles, her eyes light up and lift upward, and she grows more animated.

She remembers walking down a wide boulevard with her grandmother as a 3-year-old from her family's Khrushchyovka apartment building. Viazovska said that the late 1980s were hard for the Soviet Union. It took a long time for people to purchase 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 people waiting in the line would get angry at her. She and her sisters never went hungry or without heat because her family didn't have much. Workers were sometimes offered a chance to win a pair of shoes if they did good work, but no stores had nice clothes. Her mother told her that if you won a pair, you could swap with someone who had won a pair.

Viazovska said that the Soviet Union fell apart when she was a young girl. She and her family were excited to live in a free and independent Ukraine, but the hyperinflation only made things worse. There was no money to spend in the soviet union. There were goods but not enough money to purchase them. She was told by her mother that she couldn't afford a metro ticket because of her monthly salary.

Viazovska described her father as an energetic chemist who left his job to start many small businesses. She said that the new reality was disorganized and unpredictable. You don't have a lot one day. You have a lot of opportunities after that.

Viazovska and her husband remember how excited Ukrainians were about economic growth. The rate of growth over one's current assets is important in the economy.

Viazovska laughed and said, "Maybe the second derivative."

Almost Infinity

Viazovska realized that she preferred math over language arts when she was a first graders. I didn't write well because I was too messy. I was quick with mathematics.

She liked to read. Her parents gave her a lot of books. She fell in love with the genre after discovering it. She said that 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 unforgettable because it was actually about us. The Strugatsky brothers wrote science fiction stories. She said that their early work was too optimistic and nave about communism.

You don't put down the paper until you read the whole thing.

They met at an after-school physics circle when they were young. She was able to approach math problems in her own way. A physical system with seven elements was one of the problems he remembered. He said that Maryna thought seven was almost infinite. The approximation worked well and made the problem simpler. Nobody else could suggest that.

Even as a child, Viazovska was talented and committed. Natalie said that their parents were afraid that she studied too much and that she drew formulas when they went to sleep.

Natalie didn't like the idea of having the same math teacher. Natalie's math teacher was her math teacher. Maryna is a great student.

Viazovska was inspired by the advanced math and physics classes and the teachers who were enthusiastic about explaining difficult concepts when she attended a specialized lyceum. She fell in love with the world of math Olympiads, which she had enjoyed for a long time.

Sometimes it didn't love her back. Viazovska said that it taught them how to lose and win. I wasn't as successful as I thought I would be. She wanted to represent her country in the International Mathematical Olympiad. Six national team members are selected to attend a training camp after the national competition. 13th place went to Viazovska. She said she had worked hard but wasn't hard enough.

The head ofUkraine's math Olympiad program and a math professor remembered meeting Viazovska that year. She is a very good person and he is very happy that she has become such a prominent mathematician. He said that she went on to win many university math contests and was on the jury helping to grade Olympiad contests.

Rublyov said that the Olympiad team is training in Poland because of the war in his home country. In March, a Russian air strike in Kharkiv killed a young mathematician. Zdanovskaya won a silver medal at the European Girls' Mathematical Olympiad five years ago, which Rublyov helped to organize. He knew her well. The deaths of young and talented people in our country is a disaster.

Russian influence on the world stage made Rublyov believe that a Ukrainian like Viazovska could not win the Fields medal. She was not given the Fields prize because she deserved it.

Doing It Right

Her first big moment as a mathematician came in 2005 when she collaborated on her first original research result. She realized that she was able to solve the problem. She said that the joy came from feeling that an argument works. She was boosted by the result.

Viazovska was encouraged to take on the problem by a math professor who helped organize some of the university math contests she had participated in. Shevchuk and Andrii Bondarenko talked to a few people about the problem. She and Bondarenko collaborated on a paper. When Bondarenko was teaching at the university, he began to work with a student named Danylo. Three young mathematicians collaborated.

A paper on spherical designs was submitted in 2011. "Annals," as mathematicians call it, is perhaps the most prestigious journal in mathematics. Zagier thought to himself, "You're beginners," when he heard about the trio's aims.

The paper was accepted and mathematicians started organizing conferences to discuss it. After reading the paper, Cohn, of Microsoft Research and the Massachusetts Institute of Technology, exclaimed, "WOW, what a fantastic paper."

The classical problem of analyzing the behavior of a function is examined in the paper. We can think of each input to the function as a point in the space where the number of variables matches the dimensions of the space. Viazovska and her team are interested in the average value on a sphere. If we averaged the values of the polynomial at the points on the sphere, we could get an approximation of the average. We might get the exact answer if we choose the points carefully.

The mathematicians have known for a long time that you can pick a finite set of points that give the answer. You can choose a single set of points that will work for all the polynomials up to a certain degree. If you are working in three-dimensional space, you can put a regular icosahedron in the sphere and use the 12 corners as your sampling points, and you will get the answer you were looking for. A spherical design is a set of twelve points.

Since the 1970s mathematicians have wondered how the number of points in a spherical design grows. The question was answered by Viazovska, Bondarenko and the others.

It takes something that a lot of people have thought about for a paper to come along and say, "Well, gee, why don't you do it this way, then you get the right bound, QED." They don't jump through all sorts of hoops to get this, they just do it right.

Magic Functions

Viazovska lived a double life as an undergrad, dividing her studies between the two fields of math. She studied modular forms, functions with special symmetries related to the ones that appear in the circular tilings of the artist M.C. Escher, while she was at Bonn for her PhD. There is a lot of analysis involved in modular forms. She said that she realized that her two passions met here.

The three of them had been trying to figure out how to pack spheres in the densest possible way. 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 a pyramid. There are important applications to error-correcting codes that can be found in higher dimensions.

They didn't know what the densest sphere packings were. Eight and 24 were strong candidates. The Leech lattice and E 8 are both highly symmetric and pack spheres much denser than any other arrangement mathematicians could find.

There is a method that uses certain functions to compute upper bounds on how dense a sphere packing can be. The densities of E 8 and the Leech lattice were almost perfectly matched by these upper bounds. The densest packings must have a magic function that matches the Leech lattice perfectly in each of the two dimensions. Researchers didn't know where to look for magic functions.

For a long time, Bondarenko, Viazovska and Radchenko tried to build a magic function, but they didn't make much progress. The two men turned their attention to other issues. Viazovska thought about sphere packing. She said the problem felt like it belonged to her.

She was able to find the magic function for dimensions eight and nine. She found that the answer was not in a modular form but in a quasi-modular one. Peter Sarnak of the Institute for Advanced Study said that she posted an amazing paper. You don't put down the paper before you read it.

News of her result spread quickly after the paper was published. A mathematician at the Institute for Advanced Study sent a link to a paper to Cohn with the subject line "WOW!" The proof was devoured by the man. My first thought was, 'What on earth is this?' He said that it looks like nothing has been done for the construction of these functions.

Viazovska's quasimodular form had always seemed like a bad version of modular forms. There was a rich theory hidden below the surface. He thought Viazovska's approach should apply to dimensions 24 and 25.

Viazovska didn't want anything more than a break. The Leech lattice was proved to be the densest 24-dimensional sphere packing by her and two other mathematicians over a single intense week. It was a crazy week.

A Bold Conjecture

Viazovska and her team came out of the sphere-packing work with higher ambitions. The Leech lattice and E 8 are more than just the best way to pack spheres, according to mathematicians. The two lattices are said to be universally optimal, meaning that they are the best arrangements according to a number of criteria.

This will have consequences in the future.

The team had to come up with magic functions for each idea of energy in order to prove that E 8 and the Leech lattice reduce energy. If there is a magic function, they only have partial information. At some points, they knew the value of the function, and at other points, they didn't. They were aware of how fast the function and its transform were changing. Is this information sufficient to reconstruct the function?

Viazovska made a hypothesis that the team had the right amount of information. There are a lot of functions that fit. The function was too limited to exist.

The man had doubts. He thought that if Viazovska's proposal were true, humanity would know it. He was aware that Viazovska didn't make a lot of silly questions. I still believed that this was pushing her luck.

They were able to prove that the information is limited to the values of the function and its Fourier transform and not the speed at which they are changing. They figured out how to prove the full conjecture in order to show that the Leech lattice is optimal. Maryna was pushing the state of the art in Fourier analysis in the process of trying to understand these lattices.

Sylvia Serfaty of New York University said that the paper is similar to the breakthrough of the 19th century, when mathematicians solved many of the problems that had confounded their predecessors for centuries. She said that the paper was an advancement of science. To know that the human brain is capable of producing a proof like that is remarkable.

War and Peace

As her son Michael has learned, Viazovska is in her own world when she does math. He said that his mother has loops in her ear. When the family lived in Berlin, he was the last kid in his class to be picked up. He was aware that his mother had won a lot of math awards but was surprised to learn that she worked so hard.

An extra bed was tucked into the alcove of the living area at their apartment in Lausanne, a 20 minute walk from the EPFL campus. Oleksandra celebrated her tenth birthday at Maryna's place in Lausanne.

A large drawing by Viazovska is on the wall of the apartment. Art has been her main escape since she was a child. One of her favorite drawings is the one she made of a Klein bottle with an Escher-esque fish pattern. She said it was hard to study math without an interest in Klein bottles. She sometimes draws pictures to help visualize geometric ideas in her work, but she is acutely aware that her two-dimensional and three-dimensional intuition is often misleading.

Viazovska walks to work because she and her husband don't drive. It is difficult for Maryna to drive in our three-dimensional world. Viazovska made a joke. She described it as a long, slowly-going process when she heard about it.

The only parents who don't have a car are us. It is difficult for us.

Viazovska shared a dark joke that has become a morbid refrain among friends back home: "Do you remember those good times of the coronaviruses?"

Viazovska was told by her grandmother that she doesn't want to die before the war is over.

Viazovska is proud of her country but feels bad for her countrymen who have had to adjust to the war. Maksym began sleepwalking at night after the first days of the invasion. Viazovska said that this wasn't for free. This will have consequences in the future.

She said that tyrants can't stop us from doing math. Something they can't take away from us.