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.”
Professor Fefferman, Herbert E. Jones, Jr. ’43 University Professor of Mathematics at Princeton University (New Jersey, 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. Fefferman “has introduced groundbreaking techniques to study the detailed structure of functions and the behavior of solutions to partial differential equations, including those arising in fluid dynamics,” in the words of the citation.
Le Gall 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. The citation singles out his “seminal contributions to the theory of stochastic processes by establishing key results about the fine properties of Brownian motion, and by deriving the scaling limits of discrete random geometric objects as their sizes diverge to infinity.”
“I feel as if the problems pick me”
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 every full professor. In his long career, he has maintained strong links with Spain, particularly with the mathematics school of the Universidad Autónoma de Madrid (UAM); a relationship forged when the Spanish mathematician Antonio Córdoba, currently Professor Emeritus of Mathematical Analysis at UAM, moved to Chicago to take up a place as Fefferman’s first PhD student. The two have remained close, with Fefferman reporting important mathematical results with Córdoba’s son Diego.
Fefferman “stands out as an all-rounder,” in the view of Prof. Córdoba, one of the five nominators behind his candidacy for this award. “The normal thing is for a mathematician to make fundamental contributions in one or two areas; Fefferman has made them in harmonic analysis, partial differential equations, problems in quantum mechanics, and in the field of fluid mechanics, where he came up with a result that suggested a whole new tack for understanding turbulence.”
Other outcomes of his research touch on questions in computation, financial mathematics, neural networks and solid state physics. “It is this broad sweep that makes Fefferman such an exceptional mathematician,” Córdoba adds.
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.”
Fefferman has spent long research periods in Spain, and has supervised the doctoral theses of seven Spanish mathematicians as well as collaborating with a dozen more. With Diego Córdoba’s group at the Institute of Mathematical Sciences (ICMAT) in Madrid, he has managed to mathematically describe how waves break, demonstrating that, just as expected (and as any observer could note, since waves do break), the movement of the fluid drives phenomena known as singularities, corresponding to the splash. This is important, because it proves that the model used by physicists to describe the phenomenon is indeed correct. “One of the jobs of a mathematician is to be like a notary public, certifying that scientific models are correctly specified,” Córdoba explains.
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, making more tools available to tackle different problems. 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.”
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 manoeuvers 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.
For Córdoba, “Fefferman’s research style is to open up new paths and perspectives that others can work on over many years, and quickly switch to a new theme.”
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 sphere, 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 had 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 thought about what could be calculated and made idealized machines. Then, motivated I think by World War II, came the idea of actually building these machines, and this led to the development of the first computers, that were devised by mathematicians.” The IT revolution is accordingly 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.”
