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Maryna Viazovska, the Ukrainian who solved a centuries-old problem and who is one of the most admired mathematicians

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Maryna Viazovska is the second woman in history to win the Fields Medal, a distinction considered the Nobel Prize for mathematics.

Since they began to be granted, in 1936, only one woman had obtained it: the Iranian Maryam Mirzakhani, in 2014.

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“It’s a brilliant math,” Christian Blohmann told BBC Mundo days before. “I admire her because her solution to the sphere packing problem is very beautiful and extremely unexpected.”

The researcher at the Max Planck Institute for Mathematics, in Germany, refers to the fact that in 2016, Viazovska solved two cases of the famous geometric problem that had been proposed, in the 17th century, by the great German scientist Johannes Kepler.

For this feat, he has received several distinctions, but his contribution has not stopped there.

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“As a result of the Viazovska result, in the last five years, lines of research have been opening in different parts of the world,” Pablo Hidalgo, a researcher at the Institute of Mathematical Sciences of the Higher Council for Scientific Research of Spain, tells BBC Mundo. .

The expert in number theory was distinguished at the International Congress of Mathematicians, in a ceremony in Finland.

The other three winners of the award given to mathematicians under 40every four years, were: the French Hugo Duminil-Copin, the American June Huh and the British James Maynard.

Viazovska’s name had been sounding loud to win this award, even before the Congress that was held in 2018. At BBC Mundo, we tell you why.

Euclid’s daughter

Albert Einstein said it: “If Euclid failed to ignite your youthful enthusiasm, you were not born to be a scientific thinker.”

The Greek mathematician is precisely one of Viazovska’s heroes, who says he admires the extraordinary figures who were able to “change mathematics or the way one thinks about it.”

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This is how he told it in an interview made by the organizers of the Prize New Horizons in Mathematicsdistinction that was awarded in 2018.

Viazovska was born in Kyiv and has been fascinated by mathematics since she was a child, so when it came time to decide on her university career, it didn’t take long.

Something that he likes about this science is that it is possible to determine where the “truth” is, to distinguish what is wrong from what is right.

After graduating from Taras Shevchenko National University, he went to Germany for his postgraduate studies.

During his postdoc in Berlin, one of the problems he included in his research proposal was that of the spheres that Kepler formulated in 1611.

He focused on it for about two years and the “magic” moment came to find the solution.

“It turned out to be easier than I thought.”

And although in that interview she reveals her teaching skills by simplifying the problem into a question: “How many balls can you put in a very big box?”, the truth is that the mathematics he used to arrive at his answer is of immense complexity.

thinking of oranges

For Hidalgo, this problem “has a certain importance for the real world in the sense that people without mathematical studies can understand what it is about” and could even have faced it at some point:

What is the most optimal way to occupy a space with a certain number of spheres, for example oranges?

Kepler posed the problem in three dimensions.

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“Surely the greengrocers had already realized that the best way to organize the oranges was in a pyramid shape,” says the Spanish researcher.

“But there is a substantial difference between: ‘it seems that this form occupies the space well’ and being certain that ‘really this form is unbeatable for occupying space’”.

Kepler could not prove it and he was not the only one, extraordinary mathematicians did not succeed either.

It was in the late 1990s, when American mathematician Thomas Hales did the proof for three dimensions.

But the fascinating thing about this conjecture is that it can be carried to circles (two dimensions) or to spheres of any dimension.

“What Viazovska achieves in 2016 is to generalize the problem.”

He found the optimal way to pack eight-dimensional spheres.

“It is not that mathematicians have been complicated by inventing a strange way of packing spheres, it is the same problem, but in a dimension that as humans we cannot visualize”, indicates Hidalgo.

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And while such packings of higher-dimensional spheres are difficult to visualize, “they are eminently practical objects,” mathematician Erica Klarreich wrote in 2016 in the journal article how much: Sphere Packing Solved in Higher Dimension.

“They are closely related to the error-correcting codes that cell phones, space probes and the Internet use to send signals through noisy channels.”

25 pages

According to Hidalgo, the proof Hales arrived at “was very long and very complicated.”

His result was presented in about 250 pages and required many calculations with computers.

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“It took almost 20 years to verify that those calculations with computers were correct.”

“While Viazovska did, for the problem of dimension eight, a 25-page article.”

“If we remove the introduction, the bibliographic references and other aspects of form, she has 10 or 15 pages of mathematics, nothing more, and with them she demonstrates a problem in a higher dimension, so we could say that it is more difficult than the that Hales demonstrated.

Highlights the “such painstaking work, so exactwhich makes a demonstration easier to understand than the previous one, which occupied dozens of pages”.

“That doesn’t want your math pages to be simple, they’re complex,” he says. But for experts, it’s 10 pages of pure math.

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From Switzerland, Özlem Imamoglu, professor at the Department of Mathematics at the Federal Polytechnic School of Zürich (ETH Zürich), notes that the solution that Viazovska arrived at “through the construction of so-called magic functions It was a spectacular achievement.”

“The existence of such functions had been conjectured by (Henry) Cohn and (Noam) Elkies in 2003, but it remained elusive despite the efforts of many brilliant mathematicians,” he tells BBC Mundo.

The simplicity and elegance of his demonstration is astonishing and admirable”.

And it would also be surprising that, after solving the problem of the packing of spheres in dimension eight, only a week later – this time with other colleagues – he solved the problem in dimension 24.

His first demonstration is considered a masterpiece, which allowed his classmates to “understand the problem well and generalize it to solve a similar problem, although even more difficult,” says Hidalgo.

It clarifies that the problem of optimal packings in high dimensions is still open, since only the configurations for dimension eight and 24 have been found.

The bridges

Experts point out that the beauty of Viazovska’s solution is that it interconnects different areas of mathematics.

His result of the packing of spheres has a lot to do with signal analysis or Fourier analysis, a French mathematician and physicist of the 19th century.

GETTY IMAGES The Fourier transform is a mathematical operation that is used in different disciplines.

“All the power of Viazovska’s result arises from joining, in ways that were not known, two areas of mathematics: number theory and Fourier analysis,” explains Hidalgo.

And therein, in his opinion, lies the strength of current mathematics.

There are areas that have evolved separately and “the difficult and really interesting in recent decades is to build bridges between them”.

“It can be extremely fruitful if someone is able to establish a robust bridge between two different areas of mathematics and that is precisely what Viazovska did.”

“You need a lot of knowledge and understanding of what the important properties of each area are to really be able to put them together. From that union is that its result arose”.

“Thanks to the fact that he established contact between the two areas, it is already understood where the relations are going.”

“It has opened up new mathematics that are still being explored and yielding results, and that will surely continue to happen in the future.”

Indeed, Imamoglu notes that although Viazovska is “most famous” for her solution to the sphere-packing problem, “her work on Fourier interpolation formulas and energy minimization questions,” which she has done alongside other distinguished mathematicians , “they deserve so much recognition”.

“Collaboration”

when he received the award newhorizons, Viazovska thanked her professors, colleagues and co-authors, “because without them none of my research would be possible.”

CARL DE SOUZA/AFP VIA GETTY IMAGES The Fields Medal recognizes outstanding achievement by mathematicians under the age of 40. For many, it is the highest distinction that can be obtained in that field.

“Science is a collaborative effort, and rapid progress is possible when people openly share their knowledge and ideas”, he indicated.

She is currently a professor at the prestigious Federal Polytechnic School of Lausanne (EPFL), Switzerland.

Blohmann met her when he was a doctoral student in Germany.

“Maryna is an extremely kind and modest person. The recognitions and positions that she has conquered they haven’t changed it not at all,” he says.

On March 16, the mathematics department of the iconic ETH Zürich, where Einstein studied, offered the first of the Alice Roth Lectures, which were created in honor of the great Swiss mathematician.

The goal with these sessions is to honor women who have made outstanding achievements in mathematics.

Viazovska was the guest and her presentation was titled: “Fourier interpolation pairs and their applications”.

Before delving into the mathematics of her presentation, she recalled a colleague and compatriot.

“We will rebuild peace”

“Three weeks ago my life changed forever in a very dramatic way that I could never have imagined. Preparing for this presentation was very difficult for me,” she said.

COURTESY: ETH ZÜRICH The title of Viazovska’s lecture, which was organized by the mathematics department of ETH Zürich, was entitled: Fourier interpolation pairs and their applications.

“Today I would like to celebrate the life and achievements of Alice Roth, but there is also another math that I would like to remember and I hope you will join me.

I also want to dedicate my lecture to Yulia Zdanovska, a mathematician and computer scientist 21 years old, whose life tragically ended on March 8 in the city of Kharkif.

Zdanovska stayed to “defend” the city after the Russian invasion, but “unfortunately she was killed in a missile attack”.

“Ukrainians are paying the highest price for our beliefs and for our freedom.”

He thanked the support received in these “moments of darkness”.

“I think we’ll somehow come out and we’ll rebuild peace, we’ll rebuild our world, and of course science and creative thinking will play a big role in that.”

Later, he delved into the magic of his mathematics.

Source: Eluniverso

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