Every four years, mathematicians from all over the world meet at the International Congress of Mathematicians (ICM). This meeting will be punctuated by plenary conferences that will take stock of the main outstanding issues in this area. It is also the occasion of the much-anticipated Fields Medals, which recognize four researchers under the age of 40 for the exceptional quality of their work.
The 2022 edition of the ICM has a special political note. The city where the ICM takes place will be chosen during the previous edition. And the choice fell on Saint Petersburg in Russia. But in response to Russia’s invasion of Ukraine, the Executive Committee of the International Mathematical Union abandoned holding the congress in Saint Petersburg and opted to make it a virtual event, with an awards ceremony at Aalto University in Helsinki, Finland July 5, 2022.
The Ukrainian mathematician Maryna Viazovskaat the EPFL (Eidgenössische Polytechnische Schule Lausanne), Switzerland, was awarded the Fields Medal 2022 for his work on the most compact stacks in high dimensions and for other contributions to extremal problems and interpolation problems in Fourier analysis.
The problem of stacking identical spheres is an old problem…in three dimensions. Kepler conjectured in 1611 that the closest configuration is that found in Primeur: pyramidal stacking. Thomas Hales, then at the University of Pittsburgh, demonstrated this in 1998. His proof relied on computer-aided calculations, which at the time raised many questions about the reliability of such a proof, which was then confirmed with a proof assistant.
The next step was to find the optimal configuration when increasing the number of room dimensions. These questions have direct applications to telecommunications and error-correcting codes. The generalized pyramid structure works well in 4, 5, 6, and 7 dimensions. But in 8 dimensions the situation changes and the network says “E8th » seems denser. Maryna Viazovska and her colleagues proved this conjecture in 2016. They also proved another special case, the 24-dimensional one.
The American June aha, from the Institute for Advanced Studies in Princeton, receives the 2022 Fields Medal for his work in combinatorial geometry, which combines geometry and combinatorics (the science of counting objects that extends to graph theory). In particular, he is credited for demonstrating the Dowling-Wilson conjecture for geometric networks, proving the Heron-Rota-Welsh conjecture for matroids, developing Lorentzian polynomials, and proving Mason’s strong conjecture.
James Maynardfrom the University of Oxford, receives the Fields Medal 2022 for his contributions to number theory, which have led to significant advances in understanding the structure of prime numbers and Diophantine approximations (which consist of approximating real numbers by rational numbers).
In particular, he worked on twin primes. The conjecture about those pairs of primes that are two units apart (like 11 and 13) suggests that there would be infinitely many of those pairs. In 2013, Yitang Zhang showed a weak version of this conjecture: There are infinitely many pairs of primes that are less than 70,000,000 apart. Many mathematicians at the time were working to close this gap. Terence Tao then started a collaboration on the participatory project Polymath, which quickly narrowed the gap to 4680. James Maynard improved this limit to 600 with a new method. Combining his efforts with the Polymath team, the gap narrowed to 246. The conjecture of twin primes remains open.
In 2019, with Dimitris Koukoulopoulos, he demonstrated the Duffin-Schaeffer conjecture, which states how well a real number can be approximated by an infinite series of rational numbers.
Finally, Hugo Duminil Copin, Professor at the University of Geneva and at the IHES (Institut des Hautes Etudes Scientifiques) in Bures-sur-Yvette, completes the list of Fields Medal winners. Expert in probabilities, he works on the mathematical aspect of statistical physics. He is interested in the probabilistic modeling of phase transitions, i.e. changes in the state and properties of matter, such as the transition of water from the liquid to the gaseous state or the magnetization of a crystal. In particular, this allows the study of the porosity of materials (through the theory of percolation), ferromagnetism (through the Ising model) and of polymers (through the study of self-avoiding walks). Hugo Duminil-Copin obtained many results by studying random phenomena in 3 and 4 dimensions, especially according to the Ising model.