Maryna Viazovska is the second woman in history to receive the Fields Medal for a “simple,” “elegant,” and “unexpected” solution to the bead packaging problem.
Maryna Viazovska is the second woman in history to receive the Fields Medal, considered the Nobel Prize in mathematics. Until then, only one woman, Iranian Maryam Mirzakhani, had received this medal, which had been awarded for the world’s greatest mathematical achievements since 1936.
“Viazovska are brilliant mathematicians,” Christian Blohmann told BBC News Mundo, the BBC’s Spanishlanguage service. “I admire her because her solution to the bead packing problem is very beautiful and extremely unexpected.”
The researcher at the Max Planck Institute for Mathematics in Germany points out that in 2016 Viazovska solved two cases of the famous geometric problem proposed by the great German scientist Johannes Kepler in the 17th century.
She received several awards for this achievement, but her contribution did not stop there.
“As a result of the Viazovska Resolution, research lines have been opened in different parts of the world in the last five years,” said Pablo Hidalgo, a researcher at the Institute of Mathematical Sciences of the Superior Council for Scientific Research in Spain.
Ukrainian mathematicians received the medal at the International Congress of Mathematicians at a ceremony in Finland. The other three winners of the prize, which is awarded every four years to mathematicians under the age of 40, were Frenchman Hugo DuminilCopin, American June Huh and Briton James Maynard.
Viazovska’s name was heavily in favor of victory. BBC News World explains why.
daughter of Euclid
Albert Einstein (18791955) said, “If Euclid failed to ignite your youthful enthusiasm, you were not born a scientific thinker.”
The Greek mathematician is precisely one of Viazovska’s heroes, who says he admires the extraordinary personalities who managed to “change mathematics or the way of thinking about it”.
Einstein admired Euclid
Image: Science Photo Library
Born in Kyiv, the capital of Ukraine, Viazovska has been fascinated by mathematics since childhood. When it came to deciding to study at university, she had no doubts.
The Ukrainian likes that in mathematics it is possible to determine where “the truth” lies, to distinguish wrong from right. After graduating from Taras Shevchenko National University, she went to Germany for postgraduate studies.
During his habilitation in Berlin, he included the spheres formulated by Kepler in 1611 in his research proposal.
This is what Viazovska focused on for about two years, until the “magic” moment came to find the solution. “It turned out to be easier than I thought.” The problem she solved can be simplified to this question: How many balls can you put in a very large box? But the math used to arrive at the answer is immensely complex.
think of oranges
For Hidalgo, this problem has “a certain realworld relevance in the sense that even people without a mathematical degree can understand what it is about” and may even have had to face this question at some point.
What is the ideal way to fill a space with a certain number of balls, for example oranges?
Kepler posed the problem in three dimensions.
Image: Getty Images
“The greengrocers had certainly already realized that the oranges are best arranged in the form of a pyramid,” says the Spanish researcher.
“But there’s a key difference between ‘this shape looks like it occupies space well’ and knowing that ‘this shape is really unbeatable at occupying space optimally'”.
Kepler couldn’t prove it and he wasn’t the only one, nor did extraordinary mathematicians. In the late 1990s, the American mathematician Thomas Hales provided the proof for threedimensionality.
But the fascinating thing about this conjecture is that it can be transported in circles (two dimensions) or in spheres of any dimension. “What Viazovska achieved in 2016 was to generalize the problem.”
She found the ideal way to wrap eightdimensional spheres.
What is the best way to organize oranges?
Image: Getty Images
“It’s not that mathematicians complicated themselves by inventing a weird way of packing balls, it’s the same problem but on a scale that we as humans can’t imagine,” says Hidalgo.
And while this packing of higherdimensional spheres is hard to imagine, “they are extremely practical objects,” mathematician Erica Klarreich wrote in a 2016 Quanta article.
“They are closely related to the errorcorrecting codes that cell phones, space probes and the internet use to send signals over noisy channels.”
25 pages
According to Hidalgo, the mathematical solution Hales found was “very long and very complicated”. His result was presented on about 250 pages and required a lot of computing work with computers.
“It took almost 20 years to verify that these calculations with computers were correct. Viazovska, on the other hand, has written a 25page paper for the eightdimensional problem,” Hidalgo points out.
“If we take the introduction, the bibliography, and other aspects of the form, it has 10 or 15 pages of math, no more, and so it shows a problem at a higher dimension, so we could say it’s more difficult than what Hales is demonstrating Has.”
He points out that the Ukrainian’s work was “so meticulous, so precise, that it’s a more easily understood demonstration than the previous one, which took up dozens of pages.” “That’s not to say your math pages are simple, they’re complex,” he notes. But for experts, it’s 10 pages of pure math.
Özlem Imamoglu, a professor in the Department of Mathematics at the Swiss Federal Institute of Technology Zurich (ETH Zurich) notes that the solution Viazovska arrived at “was a spectacular feat by building socalled magic functions”:
“The existence of such functions was conjectured by (Henry) Cohn and (Noam) Elkies in 2003, but has remained elusive despite the efforts of many brilliant mathematicians,” he told BBC News World.
“The simplicity and elegance of his demonstration is incredible and admirable.”
To top it off, after solving the problem of packing the eightdimensional spheres, just a week later this time with other colleagues she solved the problem in dimension 24.
Viazovska’s first demonstration is considered a masterpiece that allowed his colleagues “to get a good understanding of the problem and to generalize it to solve a similar but even more difficult equation,” Hidalgo said.
He clarifies that the problem of ideal packaging remains open in many dimensions since they have only found configurations for dimensions 8 and 24.
Bridges to other areas of mathematics
Experts point out that the beauty of the solution Viazovska arrived at is that it connects to different areas of mathematics. The result of packing the balls has a lot to do with signal analysis or Fourier analysis, a 19thcentury French mathematician and physicist.
Fourier calculations can be applied in various disciplines and were useful in Viazovska’s discovery
Image: Getty Images
“The full power of Viazovska’s result comes from combining two fields of mathematics, number theory and Fourier analysis, in a way that has never been seen before,” explains Hidalgo.
And therein, in his opinion, lies the strength of current mathematics. There are areas that have developed separately and “what has been difficult and really interesting over the past few decades is bridging between them.”
“It can be very helpful when you can build a robust bridge between two different areas of mathematics, and that’s exactly what Viazovska did.”
“It takes a lot of knowledge and understanding of the important characteristics of each area to really put them together. From this union emerged the result of Viazovska’s calculation.”
“Thanks to the contact between the two areas, it is already known where the relations between them are going. This has opened up a new mathematics that is still being explored and producing results, and will certainly continue to do so in the “future”.
In fact, Imamoglu notes that while Viazovska is “famous” for her solution to the sphere packing problem, “her work on Fourier interpolation formulas and energy minimization issues” that she has done with other mathematicians “deserves equal credit.”
cooperation
When she received the New Horizons Award, Viazovska thanked her professors, colleagues and coauthors, “because none of my research would be possible without them.”
The Fields Medal is considered the Nobel Prize for mathematics.
Image: CARL DE SOUZA/AFP via Getty Images
“Science is a collaborative effort, and rapid progress is possible when people openly share their knowledge and ideas,” she said.
The Ukrainian is currently a professor at the renowned Federal Polytechnic School in Lausanne (EPFL), Switzerland. Blohmann met her as a doctoral student in Germany.
“Maryna is an extremely kind and humble person. The recognitions and positions that she has received have not changed her at all,” she says.
On March 16, the Department of Mathematics at ETH Zurich, where Einstein studied, offered the first of the Alice Roth Lectures, created in honor of the great Swiss mathematician.
The aim of these sessions is to honor women who have made outstanding achievements in mathematics.
Viazovska was a guest and her lecture was entitled: “Fourier interpolation pairs and their applications”. Before delving into the mathematics of her presentation, she recalled a colleague and compatriot.
“We will restore the peace”
“It’s been three weeks since my life changed forever in a very dramatic way that I could never have imagined. It was very difficult to prepare for this performance,” he said.
“Today I want to celebrate the life and achievements of Alice Roth, but there is other math to remember as well. I would like to dedicate my talk to Yulia Zdanovka, a 21yearold mathematician and computer scientist whose life tragically ended on March 8 in the city of Járkif.”
Zdanovka stayed to defend the city from the Russian invasion, but died “unfortunately in a missile attack”. “Ukrainians are paying the highest possible price for our faith and freedom,” said Maryna Viazovska.
She also thanked for the support she received during these “dark moments”. “I believe we will restore peace, we will restore our world, and certainly science and creative thinking will play an important role in this.”
Then he began to uncover the magic of his mathematics.
Originally published text at https://www.bbc.com/portuguese/geral62069614