The BBVA Fronteiras Foundation for Knowledge in Basic Sciences Award in its 14th edition recognized the fundamental contributions of professors Charles Feffermangives Princeton University (USA), and Jean Francois Le Gallgives Paris-Saclay University (France), to two areas of mathematics that had numerous ramifications with applications in various fields.
According to the jury’s minutes, the winners opened “new perspectives in the Math analysis and the probability theory, with an enormous influence on a whole generation of mathematicians”. The jury also stressed that both “introduced powerful analysis techniques to solve mathematical problems with a long history, some of which are motivated by fundamental questions of theoretical physics”.
Professor Fefferman, professor of mathematics at Princeton University, is considered one of the most versatile mathematicians of our time, author of results in areas apparently as far away as the mathematical description of the fluid behaviorthe analysis of the laws of quantum mechanics or the properties of graphene and other two-dimensional materials. “It introduced new techniques that allow the study of the detailed structure of functions and the behavior of solutions of differential equations in partial derivatives, including those that appear in fluid dynamics”, highlights the jury’s minutes.
In turn, Le Gall investigates in probability theoryand an important part of his work comes from physical models that try to explain the quantum world on the atomic scale and at the time of the origin of the universe, with the development of a quantum theory of gravity.
A very versatile mathematician
Fefferman (Washington, 1949) entered the University of Maryland (USA) when he was just 14 years old and published his first mathematical work a year later. In 1971, at age 22, he became the youngest teacher in the United States. Part of his extensive career has a close relationship with Spain and, specifically, with the school of mathematics at the Autonomous University of Madrid (UAM), began when the Spanish mathematician Antonio Córdoba, currently Emeritus Professor of Mathematical Analysis at UAM, moved to Chicago to be its first doctoral student. Both researchers maintained a close relationship, and Fefferman also obtained important mathematical results in collaboration with Cordoba’s son Diego.
Fefferman “stands out for its versatility,” says Córdoba. “It is normal for a mathematician to make fundamental contributions in one or two areas, but he has made them in harmonic analysisin equations in partial derivativesin trouble quantum mechanics and also in the area of fluid mechanicswhere he found the key to an outcome that paved the way for understanding the turmoil”.
Other results of his work have to do with computing, financial mathematics, neural networks and solid-state physics. “This diversity of areas is what makes Fefferman an exceptional mathematician”, highlights the Spanish mathematician.
Interviewed after hearing the jury’s decision, Fefferman explained that, for him, the jump between areas is natural: “I have the feeling that I don’t choose problems, they choose me. I’ve heard about some problem and it’s so fascinating that I can’t stop thinking about it. And if it’s from an area that I haven’t studied before, but I think I have some possibility of being able to contribute something to solve it, I try”.
However, he doesn’t feel like an expert in many areas: “When people talk to me about what they are doing in the world of mathematics, I sometimes feel very ignorant because so many things are happening that approaching each new topic takes a lot of effort. previous job”.
Fefferman carried out extensive research stays in Spain, supervised the doctoral thesis of seven mathematicians in our country and collaborates with a dozen of them. Your research with the Diego Cordoba at the Institute of Mathematical Sciences (ICMAT) from Madrid, managed to describe mathematically how waves break, and thus demonstrated that, as expected, phenomena called singularities -which correspond to the splash of the wave. The result is important because it attests that, in fact, the model used by physicists to describe the phenomenon is correct. “One of the functions of mathematicians is to act as notaries, attesting that the models of science are well proposed”, explains Córdoba.
I have the feeling that I don’t choose problems, they choose me. I hear about a problem and it’s so fascinating I can’t stop thinking about it
Fefferman counts “several dozen” problems he has solved throughout his career. He chose as one of his favorites the so-called duality theoremas it is a result that connects two very different concepts, providing a functional tool that has opened new perspectives in the Harmonic Analysis. He likes it in part because he is one of the ones that took the least time to solve: “only a few weeks”, compared to others that took “up to 20 years”.
At 73, he is still investigating. He is now working to mathematically define the curious physical properties of the new two-dimensional materials, with problems such as the behavior of electrons on the edge of a graphene sheet. also in a problem control theory: how to control a system whose behavior is not known, the equivalent of what a pilot achieves when “the plane is seriously damaged for some reason and he learns to control it and manages to land. It’s a big problem, but we’re making progress”, guarantees the winner.
In Córdoba’s words, “Fefferman has a habit of research opening new paths and perspectives, leaving the work to others for many years and quickly moving on to another subject”.
The geometry of random movements
As for Jean-François Le Gall (Morlaix, 1959) he “profoundly transformed the field of probability theory”, writes Emmanuel Royerdeputy scientific director of the National Institute of Mathematical Sciences and their Interactions (National Center for Scientific Research, CNRSFrance).
For the professor at the Faculty of Mathematics at University of Barcelona, Marta Sanz-Solealso a probability researcher and a great connoisseur of Le Gall’s work, her contributions are “really crucial, because, in turn, they generated new research around her results and by the impetus of connections with mathematical physics”.
Many of the problems Le Gall works on stem from the physicist, although he describes himself – as he stated in an interview after learning about the decision – as a “theoretical mathematician who works with interesting mathematical objects in themselves, without thinking about the applications”. Mathematics advances, he says, “for an aesthetic motivation”.
His first work focused on mathematical brownian motion. This is an area that goes back to Albert Einstein, who was able to explain the random motion of pollen grains floating in water as a result of the vibration of fluid molecules, thus proving that atoms and molecules really exist.
Le Gall investigated the geometry that results from the trajectories of particles in Brownian motion: “I worked a lot on the study of Brownian motion, which describes the random motion of a particle subject to continuous changes of directionand presented several important objects related to Brownian motion.”
In the last fifteen years his research has created a new branch of probability theory based on the study of so-called “brownian spheres”. They’re not really spheres, but “mathematical objects” – explains the winner – with an uneven surface that arises when tens of thousands of tiny triangles randomly come together.
Physicists invented these spheres as a model for quantum gravity theory”, he says, “my contribution has been to make this model rigorous”. The field now attracts a great deal of mathematical activity and “has opened up new avenues of research”.
One of the results that Le Gall places among his favorites is from nine years ago and refers to these Brownian spheres; specifically, it demonstrates its “uniqueness” in the mathematical sense: “It was a key question, a problem that had been open for about eight years,” he explains. “It’s important because if you can’t demonstrate the uniqueness of your model, you can’t know if it really works.”
The transformative power of mathematics
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 gadgets that we use every day,” says Fefferman, “depends on math, and in order for a device to be able to do what we want, it first had to solve a math problem.”
The Princeton professor considers that “the main utility of mathematics is its ability to provide great ideas that would never have arisen if it weren’t for them, and that transformed the world. We still don’t know what the big idea that mathematics will bring in the 21st century will be, but in the 20th century it was the computer”.
“Before computers existed, mathematicians were dedicated to thinking about what could be calculated and what it meant to compute something, and they imagined machines that, later, in the context of World War II, led to the development of the first computers, which were invented by mathematicians”. Therefore, Fefferman considers that the computer revolution is the perfect example that reflects how, “from the work of many mathematicians, ideas are born that can transform the world in unpredictable ways”.
Le Gall 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: ” math is the language of science, so it is very important to point out, for example, that physicists, such as 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.”