The two winners have opened “new perspectives in mathematical analysis and probability theory”, according to the jury
Mathematicians Charles Fefferman, from Princeton University (USA), and Jean-François Le Gall, from Paris-Saclay University, were recognized this Thursday with the BBVA Foundation Frontiers of Knowledge Award in Basic Sciences. According to the minutes of the jury, these researchers have contributed to the development of different fields, such as mathematical analysis and probability theory, “with an enormous influence on a whole generation of mathematicians.” Furthermore, both “have introduced powerful analysis techniques to solve mathematical problems with a long history, some of which are motivated by fundamental questions of theoretical physics.”
Fefferman (Washington DC, United States, 1949), who entered university at the age of 14 and published his first mathematical work a year later, is considering one of today’s most versatile mathematicians, the author of results in areas as apparently distant as the mathematical description of the behavior of fluids, the analysis of the laws of quantum mechanics or the properties of graphene and other two- dimensional materials.
For his part, Le Gall (Morlaix, France, 1959) researches probability theory, and an important part of his work comes from physical models that attempt to explain the quantum world at the atomic scale and at the time of the origin of the universe, with the development of a quantum theory of gravity.
“I feel that problems trap me”
In 1971, at the age of 22, Fefferman became the youngest professor in the United States. Part of his extensive career has a close relationship with Spain, specifically with the mathematical school of the Autonomous University of Madrid (UAM),
Fefferman “stands out for his versatility,” says Professor Antonio Córdoba, currently Emeritus Professor of Mathematical Analysis at the UAM. «It is normal for a mathematician to make fundamental contributions in one or two areas; Fefferman has done them in harmonic analysis, in partial differential equations, in quantum mechanical problems, and also in the area of fluid mechanics, where he hit on the key to a result that opened a path to understanding turbulence».
Other results of his work have to do with computation, financial mathematics, neural networks, and solid-state physics. “This diversity of areas is what makes Fefferman an exceptional mathematician,” adds Córdoba.
After hearing the jury’s decision, Fefferman explained that for him, the jump between areas is natural: «I have the feeling that I do not choose the problems, but that they choose me. I hear about some problem and it’s so fascinating that I can’t stop thinking about it. And if it belongs to a field that I haven’t studied before, but I think I have some possibility of being able to contribute something to solve it, I try it».
Fefferman has carried out long research stays in Spain, has supervised the doctoral thesis of seven mathematicians from our country, and collaborates with a dozen of them. His research with Diego Córdoba’s group at the Institute of Mathematical Sciences (ICMAT), in Madrid, managed to mathematically describe how waves break, thus demonstrating that, as expected −and as anyone can observe because waves do indeed break−, in the movement of fluids phenomena called singularities are produced −which correspond to the splash of the wave−. The result is important because it certifies that the model used by physicists to describe the phenomenon is indeed correct.
Fefferman counts in “several dozen” the problems he has solved throughout his career. Some have been resolved in “just a couple of weeks”, compared to others that have taken “up to 20 years”. At 73 years old, he continues to investigate. He is now working on mathematically defining the curious physical properties of new two-dimensional materials, with problems such as the behavior of electrons at the edge of a graphene sheet. Also, on a problem in control theory: how to control a system whose behavior is unknown, the equivalent of what a pilot achieves when “the plane is badly damaged for some reason and he learns to control it and lands. It is a huge problem, but we are making progress, ”says the winner.
Random moves
For his part, Jean-François Le Gall has “profoundly transformed the area of probability theory”, according to Emmanuel Royer, deputy scientific director of the National Institute of Mathematical Sciences and their Interactions (National Center for Scientific Research, CNRS, France).
Much of the problems Le Gall works on come from physics, although he describes himself as a “theoretical mathematician who works on interesting mathematical objects in themselves, without thinking about the applications.” Mathematics advances, he asserts, “for aesthetic reasons.”
His early work focused on mathematical Brownian motion. This is an area that goes back to Albert Einstein, who succeeded in explaining the random motion of pollen grains floating in the water as the result of fluid molecules vibrating, thereby proving that atoms and molecules really do exist. Le Gall has investigated the geometry that results from the trajectories of particles in Brownian motion.
In the last fifteen years, his research has created a new branch in probability theory based on the investigation of the so-called ‘Brownian spheres’. They are not really spheres, but “mathematical objects” -explains the winner- with an irregular surface that arises when tens of thousands of tiny triangles randomly stick together.
Big ideas
Both winners defend the crucial importance of mathematics in today’s world, both to promote the advancement of knowledge in all fields of science and to lay the foundations for technological development. “The operation of any of the gadgets that we use every day – Fefferman points out – depends on mathematics, and for a gadget to be able to do what we want, it has first had to solve a mathematical problem”.
The Princeton professor considers that “the main utility of mathematics is its ability to provide great ideas that would never have emerged were it not for them, and that has transformed the world. We still don’t know what will be great idea mathematics will bring in the 21st century, but in the 20th century, it was the computer. Before there were computers, mathematicians dedicated themselves to thinking about what could be calculated and what it meant to compute something, and imagined machines that, later, in the context of World War II, led to the development of the first computers, which were devised by mathematicians. For this reason, Fefferman considers that the computer revolution is the perfect example that reflects how, “based on the work of many mathematicians,
Le Gall, for his part, highlights not only the fundamental role of mathematics in the technologies we use in our daily lives, “such as GPS, which is based on advanced mathematical analysis”, but also its indispensable contribution to the advancement of knowledge in all fields: «Mathematics is the language of science, so it is very important to highlight, for example, that physicists, like chemists or biologists, use mathematics to understand nature. Quantum mechanics, for example, or relativity, depend on deep mathematics. It is essential for science to have good mathematical models.”