Fields Medal 2022: Ukrainian Maryna Viazovska is the 2nd woman winner of the “Nobel Prize” in mathematics

In 2014, like every four years since 1950, the International Congress of Mathematicians (ICM, International Congress of Mathematicians) awarded four laureates Fields Medal which is often considered the Nobel Prize in Mathematics. He created the surprise by attributing it for the first time to a woman, the mathematician of Iranian origin Maryam Mirzakhani, unfortunately deceased since.

At that time and since the first awards of the Medal from 1936 to mathematicians who must be under the age of 40, the more than 50 past recipients had all been men. We therefore learn with pleasure that one of the new Fields medals awarded in 2022 at the International Mathematical Congress which opened in Helsinki, Finland, goes to the mathematician of Ukrainian origin Maryna Viazovska. Aged 37, born in kyiv, Ukraine on December 2, 1984,she therefore joins the club of more than 60 mathematicians who have been awarded the mythical medal to date.

The event obviously has a particular resonance with current events in Ukraine, but it should be noted that the awarding of the Fields medal had been served on Maryna Viazovska in February 2022, before the invasion of Ukraine launched by Vladimir Putin . However, the mathematician cannot fail to mention this fact in the video produced on the occasion of the presentation of this award and where she also gives some explanations concerning the discovery she made and which she made public in 2016 with an article posted on arXiv.

A generalization of Kepler’s problem in higher dimensions

Technically, and in the jargon familiar to mathematicians, Maryna Viazovska has realized and demonstrated that an elegant combination of mathematical concepts at the intersection of several major areas of research, namely the theory of Lie groups and algebras, the theory of forms modules that we find closely linked to the theory of so-called elliptical analytical functions and conformal transformations of the complex plane, and finally to what is called harmonic analysis with Fourier transforms and the theory of distributions (which the ‘we owe another Fields Medal winner, Frenchman Laurent Schwartz) made it possible to solve in dimensions 8 and 24 a problem whose solution in dimension 3 had been conjectured in 1611 by theastronomer and mathematician Johannes Kepler. It was a matter of finding the optimal stacking of spheres with a minimum of void between them.

The problem is less trivial than it seems because, as Futura explained in a previous article, we may need to solve it in part to understand the structure of the matterin particular when it is crystallized, from the atomic theory by considering a simple model of the atoms in the form of compact spheres. Historically, it also seems like this issue also stems from how to store the most cannonballs in a volume given. mathematicians and physicists have therefore sought for centuries to determine how to stack spheres in the most efficient way possible and it is in the writings of Kepler that we find, for the first time, the conjecture bearing his name and which was only demonstrated in 1998 thanks to the mathematician Thomas Hales.

According to Kepler, the most efficient stacking was the one giving a face-centered cubic structure that is precisely known today in crystal lattice theory. Such a “tiling” of a volume by spheres makes it possible to occupy approximately 74% of a given volume.

Such considerations are useful for explaining the density of a crystal, for example, and also for predicting the extent to which atoms of a different type and occupying a smaller spherical volume than those having initially served to constitute this crystal can be introduced into it. The design ofalloys with given properties benefits from research on these issues.

From Kyiv to the Federal Polytechnic School of Lausanne

You can find on the page of theInternational Mathematical Union several links that allow you to learn more about the career of the Ukrainian mathematician and the work that earned her the Fields medal.

So there is an interview which started on February 18, 2022 and continued on June 6, 2022, led by two Russian mathematicians Andrei Okounkov and Andrei Konyaev. The former was also a Fields Medal winner, but in 2006. Futura had devoted an article to his work at that time.

In this interview, Maryna Viazovska explains that she comes from a family of chemists and that she had a passion for mathematics very early on. ” We don’t have mathematicians in the family; my mother, my father, my grandmother and my grandfather are all chemists. It seems to me that my story is quite banal. When I was 12, I entered a school specializing in physical and math and started participating in math olympiads. Then I understood that I wanted to be a mathematician. »

In another excerpt from her interview, she explains other elements of her journey: ” I remember having a book in which the theorem fundamental of algebra was proved by topological methods. Among the Olympiad problems, I preferred those of combinatorics, especially the older ones. They may require only one idea, but a very beautiful one. In more recent problems, you might have to stack several ideas together – like making a sandwich. I also remember – maybe it has nothing to do with the math – that we had a neighbor who died a long time ago. He was an old man who had fought in the Great War and then worked with my grandfather at the same university. And he had a huge – really huge! – collection of different popular physics and math home books. And then at some point, I think his grandchildren didn’t engage in scientific studies, he gave them to me. The whole collection, the whole pile. And there I found an astronomy book that really impressed me. She would later cite among the geometry books she read one of those of the Ukrainian mathematician Aleksei Pogorelov. You can find many books remarkably pedagogical works of mathematicians from the former USSR translated into English which are archived.

As for his discovery of a generalization of the solution to Kepler’s problem in dimensions 8 and 24, it can be understood at least in outline with an L3 background in mathematics and with the help of the presentation that in fact for the cultivated public of this level again Andrei Okounkov.

We learn that it has to do with the mathematical equivalent of crystal lattices and the symmetry groups associated with the geometry of these lattices. However, we know that in physics we can analyze the crystal lattices of atoms with the theory of Fourier transformations using diffraction experiments. We are therefore ultimately not very surprised that generalized solutions in dimensions greater than 3 of Kepler’s problem involve harmonic analysis in relation to group theory. We also know that symmetries related to crystal lattices can be connected to the theory of functions of the complex variable and more precisely with those of these functions that are called elliptical functions and related to Riemann’s surface theory as well as with mythical works of the French mathematician Henri Poincaré concerning what are called functions automorphs.

Remarkably, in dimension 8, one of the key ingredients of Maryna Viazovska’s demonstration is related to the famous exceptional Lie group of superstring theory: E8.

Maryna Viazovska holds the Chair of Arithmetic at EPFL, the Swiss Federal Institute of Technology in Lausanne.