Fefferman and Le Gall win Frontiers of Knowledge award for solving fundamental mathematical problems

The awardees, says the committee, have opened “new perspectives in mathematical analysis and probability theory, which have had a great influence on a generation of mathematicians.” They have also “introduced powerful analysis techniques to solve longstanding math problems, some of them arising from fundamental questions in theoretical physics.”

Charles Fefferman, a professor at Princeton University in the United States, is considered one of today’s most versatile mathematicians, who has brought new insights to such seemingly disparate fields as the mathematical description of fluid dynamics, analysis of the laws of quantum mechanics or the properties of graphene and other two-dimensional materials.

Le Gall, a professor at Université Paris-Saclay, works in probability theory, and much of his research draws on physics models that attempt to explain the quantum world at the atomic scale and in the early universe, with the construction of a quantum theory of gravity.

Mathematical child prodigy

Fefferman entered the University of Maryland, United States, at just 14 years of age and published his first mathematical paper the following year. In 1971, at the age of 22, he became America’s youngest full professor. In his long career, he has maintained strong links with Spain, particularly with the mathematics school of the Universidad Autónoma de Madrid.

Fefferman reckons that over his career he must have solved “several dozen” problems. Asked about his favorites, he picks the duality theorem, which connects problems from fields far removed from math, providing a functional tool that opens up new vistas in harmonic analysis. He likes it partly because it took the least time to resolve, “just a couple of weeks,” compared to others he has worked on for “up to twenty years.”

In an interview after hearing of the award, Fefferman explained that, for him, jumping between fields is second nature: “I have the feeling that I don’t pick the problems, they pick me. I hear about a problem and it is so fascinating that I cannot stop thinking about it. If it happens to be in a field I have not worked in before but I think I have a chance to get involved and maybe do something, then I try.”

This is not to say that he considers himself an expert in multiple areas: “When people tell me what’s going on in the world of mathematics, I sometimes feel very ignorant, because so much is happening, and it takes a lot of preparation before you can take on the next problem.”

He is still doing research at the age of 73. A current project is to mathematically describe the curious physical properties of new two-dimensional materials, with problems like the behavior of electrons at the edge of a graphene sheet. Another problem occupying him is one of control theory, consisting of how to control a system whose behavior is unknown; the equivalent in math of a pilot’s maneuvers when “the plane is badly damaged for some reason and they manage to gain control and bring it safely to land. This is a daunting problem, but we are making headway,” the new laureate remarks.

The geometry of random movements

Jean-François Le Gall has “profoundly transformed probability theory,” writes Emmanuel Royer, Scientific Director of the National Institute for Mathematical Sciences and their Interactions – INSMI (CNRS, France), which put his name forward for the award.

For Marta Sanz Solé, Professor of Mathematics at the University of Barcelona, who also researches on probability and is a close follower of Le Gall’s work, his contributions are “truly pivotal, in the sense of spurring new research around his results, and strengthening the connections with mathematical physics.”

Many of the problems Le Gall works on come from physics, although he describes himself in the interview granted after hearing of the award as “a theoretical mathematician who works on mathematical objects of inherent interest, without thinking of the applications.” Advances in mathematics, he insists, derive overwhelmingly from an “aesthetic motivation.”

His first object of study was mathematical Brownian motion. This field traces its ideas back to Albert Einstein, who was able to explain the random movement of pollen grains floating in water as the result of the vibration of molecules in the fluid, and thus prove that atoms and molecules really exist. Le Gall has explored the geometry arising from the trajectories of particles in Brownian motion: “I have made an extensive study of this kind of motion, which describes the random movement of a particle that is constantly changing direction, and have introduced several key objects related to Brownian motion.”

In the last fifteen years, his research has birthed a new branch within probability theory based on the study of “Brownian spheres.” These are not in fact spheres but irregularly surfaced “mathematical objects” – the awardee explains – that appear when tens of thousands of minute triangles stick randomly to one another. “Physicists invented these spheres as a model for the theory of quantum gravity,” he explains. “My contribution was to make this model rigorous.” The field is now a hive of mathematical activity and “has opened new perspectives in research.”

A result that Le Gall names among his favorites dates from nine years back and refers precisely to these Brownian spheres, specifically to proving their “uniqueness” in the mathematical sense: “That was a major issue, a problem that had been open for eight years,” he relates. “Because if you aren’t able to prove the uniqueness of your model, you can’t tell if it really works.”

The transformative power of mathematics

Both laureates defend the crucial importance of mathematics in today’s world, both for the advancement of knowledge in all fields of science and for laying the foundations for technological development.

“The operation of any of the gadgets we use every day depends on mathematics,” Fefferman points out. “In order to make the gadget do what you want, you have to first solve a math problem.”

The Princeton professor is convinced that “the main utility of mathematics is its ability to contribute enormous ideas that would never have occurred without studying math, and which completely change the world. What great idea math will bring in the 21st century we still don’t know, but in the 20th century it was the computer. Before there were computers there were studies of what it means to calculate something; mathematicians made idealized machines. Then, World War II led to the development of the first actual computers, which were devised by mathematicians.” So Fefferman argues that the IT revolution is the perfect example of how “from an awful lot of work by mathematicians come a few ideas that fundamentally change the world in ways that could not have been predicted.”

Le Gall, for his part, talks not only about the essential role of mathematics in the tech we use in our daily lives, “like GPS, which is based on advanced mathematical analysis,” but also about its indispensable contribution to advancing knowledge across all domains: “Mathematics is the language of science, so it is important to stress that physicists, like chemists or biologists, use mathematics to understand nature. Quantum mechanics, for instance, or relativity rely on deep mathematics. It is essential for science to start from sound mathematical models.”