A Fields medal, a sort of Nobel in mathematics, is once again awarded to a Frenchman this year as part of the International Congress of Mathematicians taking place in Helsinki. She returns to Hugo Duminil-Copin who is a permanent professor at the Institut des Hautes Etudes Scientifiques (IHES)I’Institute for Advanced Study of French Princeton, where a mathematics legend taught and worked Alexandre Grothendieckand who also gives of his time as a full professor at the Mathematics Section of the Faculty of Sciences of the University of Geneva.
Hugo Duminil-Copin’s career is quite classic for many of the French Fields medal winners he has just joined, such as Alain Connes Where Cedric Villani. Born on August 26, 1985, he joined the École Normale Supérieure on rue d’Ulm after completing two years of preparatory classes at Lycée Louis-le-Grand, also in Paris. He described in several videos his trajectory which in no way suggested in college or high school that he would later become a world-class professional mathematician. It contrasts with that of the Ukrainian Maryna Viazovska, also winner of the Fields Medal 2022from a family of chemists and participating from college in the Mathematics Olympiad.
Hugo Duminil-Copin, permanent professor at IHES, tells us about his career and his research. © Paris-Saclay University
Two Fields Medals as teachers
Hugo Duminil-Copin’s parents – his mother was a dancer before becoming a teacher and his father a sports teacher – although exposing him to several activities did not particularly push him to study science nor put the pressure to excel in school. In addition to these videos, there is a very complete article dating from December 2020 which can be read online on the University of Geneva website. where we learn that, in his youth, his parents ” notice that their eldest son is gifted. They are proud of him, but not to the point of pushing him to make the most of his academic talents or to compete with his classmates. Both teachers, they probably had the time to notice the damage that this kind of education can cause on the balance of children. And then, the father himself had skipped a class in his youth and had experienced it as a trauma. No question that his son lives the same ordeal ” and ” rather identifying with tortoise that to the hare of the fable, he congratulates himself on having given time to time in his youth and having taken advantage of it to exercise other activities such as music and sport “.
We also learn that at Louis-le-Grand, barely arrived ” he eats his “first big slap” there. He finds himself with students who are all superior to him. He is immediately in trouble. He has no choice but to get to work. He does it well and slowly climbs the slope. At the end of the year, he is even the best in his class in math. As a reward, he is sent to a special class that brings together all the gifted among the gifted. And there he collects his “second big slap”. He is penultimate in the class at the first math test. Once again, he toils and finishes first at the end of the year “.
A graduate of a master’s degree from Paris-Sud University, now Paris-Saclay University, he was noticed by one of his professors, Wendelin Werner, a winner of the Fields Medal in 2006 which Futura had already mentioned in a previous article. This one will prefer to direct it towards a doctorate and a post-doctorate at the University of Geneva under the supervision of Stanislas Smirnovhimself a winner of the Fields Medal in 2010, and whom he considers to be the man of the moment for the research theme that interests Hugo Duminil-Copin.
These are questions at the intersection of the theory of phase transitions and in particular in magnetic materials, from the random walk theory such as those for the movement Brownian (in order to model for example the behavior of polymers as’DNAimmersed in a solvent) and in connection with percolation patterns (Latin percolare, flow through). A beginning of explanation of his work on magnetic materials can be found in the video below, and more developed explanations and with the notion of percolating in the following video, which we owe to the Simons Foundation, which publishes the famous Quanta Magazine.
Hugo Duminil-Copin explains that his work relates to the ising models of magnetic materials and that they come under what is called statistical mechanics. © Paris-Saclay University
Probabilities and magnetism, very French traditions
The work of Hugo Duminil-Copin is part of a long French tradition at the interface of mathematics and physical and are based on the theory of probability and the modelization from magnetism of the material.
Indeed, although sketched by the Italian mathematician and physician Jérôme Cardan (well known for his method of resolution general of equations of the third degree and the discovery of complex numbers) in his book Liber de ludo aleae (The book on games of chance), the theory of the calculation of probabilities will only really develop from the contributions of Blaise Pascal and Pierre de Fermat via their epistolary correspondence. It will then receive considerable impetus from the hands of Pierre Simon de Laplace.
It is now a major branch of thetree organic structure of mathematics which comes in several possible definitions of the notion of probability (classical, frequential, Bayesian) and which received in the XXe century of other determining impulses in particular of the Russian Andrei Kolmogorovbut we could also cite the English Harold Jeffreys.
Let us now pass to the study of the magnetism of matter. It was Charles Coulomb who, in addition to discovering the law of force electrostatic between two electrical charges, introduces the notion of magnetic moment/dipole and shows that a law similar to that of electrostatics exists between the poles of a magnet. He’s still a Frenchman Andre-Marie Ampere who, a few decades later, will propose the existence of microscopic current loops in matter to generate the magnetic field materials constituting magnets. It’s finally Pierre Curiewhich we no longer present, which will then show that by heating magnets they lose their magnetization beyond a certain critical temperature. This is a phase transition analogous but not identical (second order) to that passing from a liquid has a gas (first order).
Theory of the magnetism of matter and the theory of probability are found combined in the most general framework of what is called physics and science. thermodynamics statistics and which has been established by the mathematician, physicist and American chemist Josiah Willard Gibbs, generalizing the work of James Clerk Maxwell and Ludwig Boltzmann (the young Einstein had found, independently of Gibbs, the same theory or almost).
Matter being made up of a very large number of particles, much more than billions of billions in a single liter of air, it is impossible to describe its behavior by solving the equations of motion of these particles and we must s rely on estimates of probability laws with averages on the characteristics of microscopic physical quantities, averages giving macroscopic quantities such as the pressure of a gas or the magnetization of a block of iron.
On the theory of the magnetism of matter and in particular of ferromagnetism magnets, we can consult the famous physics course of Richard Feynman and as regards an introduction to statistical physics there is the famous course of Frederick Reif, Statistical Physics, and for those who would like to know much more, his treatise Fundamentals of Thermal and Statistical Physics.
Further explanations from Hugo Duminil-Copin regarding his work on the Ising model and its connections to percolation. To obtain a fairly accurate French translation, click on the white rectangle at the bottom right. The English subtitles should then appear. Then click on the nut to the right of the rectangle, then on “Subtitles” and finally on “Translate automatically”. Choose “French”. © Simons Foundation
Ising and Onsager, the pioneers
As sketched in the video below, the behavior of magnetic materials can be understood from theories generalizing a model proposed in the 1920s by Ernst Ising in his thesis. It is about a simple model in which one can consider particles on a line which would be horizontal like small magnets whose magnetization would always be vertical. In fact, we introduce the notion of spin with two possible states, “high” and “low”. Like two magnets, these particles are in magnetic interaction, but it is assumed that each particle is only sensitive to the influence of its two neighbours.
The question Ising asked was whether spontaneous magnetization could occur with a majority of spins oriented in one direction. His answer was negative in the one-dimensional case and he thought that it must be the same for a 3-dimensional object, such as a crystal whose atoms would behave similarly. He was wrong.
Although he had given up research to devote himself to teaching, he was unaware for decades that his model had caught the attention of Heisenberg in his own attempts to explain the ferromagnetism magnets and that theoreticians like Rudolph Peierls and especially Lars Onsager had extended his calculations to the two-dimensional case. In the late 1940s and early 1950s, Onsager and Chen Ning Yang concluded that a phase transition explaining the spontaneous magnetization below the Curie temperature of a magnet did indeed appear in a two-dimensional Ising model. Since then, countless articles have been published using the Ising model to describe phase transition phenomena in statistical physics.
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