The world of mathematics is neat and tidy, but the real world is messy and seemingly chaotic. To make order out of all this, researchers rely on different types of equations. A type of these equations, called “partial differential equations” (or PDE), is especially common in modeling physical processes. These equations model how specific variables change with respect to each other. Luis A. Caffarelli is among the leading figures in this field, and for his contribution, he has been awarded the Abel Prize.
Mathematics doesn’t have a Nobel Prize. Why Nobel didn’t establish this prize has remained a subject of controversy, but whatever the reason was, it left mathematics, which is essential to virtually all fields of science, woefully unrecognized. But the field of mathematics has compensated for this with different awards: one is the Fields Medal, which is awarded every four years to leading researchers under 40; the other is the Abel Prize.
Named after Norwegian mathematician Niels Henrik Abel, the Abel Prize is awarded annually by the King of Norway to one or more outstanding mathematicians. It’s modeled after the Nobel Prize. This year, the recipient is Luis A. Caffarelli, who was celebrated for his “seminal contributions to regularity theory for nonlinear partial differential equations.”
Differential equations sound very complex (and they can be), but in principle, they measure change — how much one thing changes in regard to another. In pure mathematics, differential equations relate one or more unknown functions and their derivatives, but in practical applications, the functions generally represent physical quantities and the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
Movement and flux are some examples of when these equations come into play. Speed is the first derivative of distance with respect to time, so you can use this in a differential equation. Flow is also commonly described with differential equations. In fact, the notion of flow is critical to the study of ordinary differential equations. But the applications can get very complex very fast, and straightforward techniques don’t always work.
This is where Cafarelli’s work comes in. He introduced ingenious new techniques and produced seminal results that progressed our understanding of how differential equations can be used in multiple different areas of mathematics and physics.
Cafarelli also worked on characterizing singularities, mathematical points at which a given object is either not defined or ceases to be “well-behaved.” For physics, particularly, these are very important because they can represent areas where the behavior of the physical system is hard to characterize, and are surprisingly common. This also ties into another area where Cafarelli worked for decades: free-boundary problems.
As the name implies, free boundary problems happen at “boundaries,” which refers to limits where one thing turns into another thing. For instance, the boundary at which water turns into ice, or a liquid turns into a crystal. But boundary problems also appear in economics, where they play a key role.
Caffarelli worked on a specific type of boundary problem called an obstacle problem, where the challenge is to find the equilibrium position under specific given circumstances. This is particularly useful in gas and liquid flows in porous media and financial mathematics.
Caffarelli, who was born and grew up in Buenos Aires but mostly worked at US universities such as the University of Minnesota, the University of Chicago, New York University, and Princeton, is also remarkably prolific. As if revolutionizing a key field of mathematics is not enough, he published very often, gathering a whopping 320 papers, with over 130 collaborators. His papers have generally been very well received in the community, gathering over 19,000 citations. He also had over 30 Ph.D. students, including one Alessio Figalli, who was awarded the Fields Medal in 2018.
Now, at 74, Caffarelli is still very active and publishes several papers a year, continuing to work with graduate students and other collaborators.
“Few other living mathematicians have contributed more to our understanding of PDEs than the Argentinian-American Luis A. Caffarelli,” the Abel Press release notes.