Mathematics is the language that lets us describe the universe. Galileo Galilei was already convinced of that in the 16th century. But even everyday phenomena such as the melting of an ice cube in a glass of water can lead to equations that are so complex that they overwhelm those with high levels of mathematical expertise. That has not stopped Argentine mathematician Luis Caffarelli from devoting himself to precisely such problems during his research career, however. The Norwegian Academy of Sciences and Letters has now honored Caffarelli with this year’s Abel Prize, the highest honor in mathematics.
Caffarelli was born in Buenos Aires in 1948. Early in his career, he primarily dealt with the properties of polynomials, algebraic expressions with multiple terms, until he completed his doctorate at the University of Buenos Aires. When Caffarelli took a position as a postdoctoral fellow at the University of Minnesota in 1973, he started to devote himself to the broad field of differential equations.
Differential equations are formulas that contain derivatives, which describe properties such as the rate of change in a physical system. Although all this sounds complicated and abstract, it is these kinds of equations that describe the constant physical flux of the world around us. Differential equations explain how certain variables change over time and space. They allow us to glimpse into the future because they predict how a system will change temporally and spatially. Suppose you throw a ball in the air: the parabolic trajectory that it follows can be represented as the solution of a differential equation.
Many natural phenomena—the speed of a river’s flow or the direction of wind—depend on the time and position of a system being observed. This makes differential equations very difficult to solve because they contain both time and space derivatives. Caffarelli began his research by initially devoting himself to static problems—those that do not change over time. One example involves the skin of a soap bubble stretched over a surface. The skin of the bubble is known as a minimal surface because it constantly tries to make itself as small as possible. In order to calculate the shape of such minimal surfaces, one needs differential equations. Caffarelli was interested in what minimal surfaces look like when they encounter an obstacle.
One of the most important questions when considering this problem is the size of the surface of the region where the bubble and the obstacle touch. Intuitively one would say that the bubble’s contact surface has a smooth boundary without corners or edges. But proving this mathematically is quite difficult because you have to calculate the resulting minimum area for all kinds of obstacles that requires solving an extremely large number of extremely complicated differential equations. Caffarelli set about tackling this problem in the 1970s by examining the properties of the differential equations and found that the boundary of the contact surface has no cracks or corners—if the obstacle is also smooth.
This work enabled him to focus on more complicated phenomena, such as describing the melting of an ice cube in water. Slovenian-Austrian physicist Josef Stefan had already paved the way in the late 19th century by tackling this problem and deriving two formulas. The first describes the flow of heat from the water to the ice, causing the ice to heat up and begin to melt. The second is dedicated to the vanishing contact surface between the water and ice. Both equations interact: the strength of the heat transfer depends on the surface of the ice, while the heat flow determines how quickly the surface shrinks. These so-called Stefan equations had seemed to describe the problem well.
Until the 1970s it was unclear whether they could provide abstract solutions that were divorced from the real world. The equations could predict an ice cube shape resembling a fractal, which has never been observed in nature. This was much more difficult to investigate than just looking at soap skins. The melting of ice cubes contains a temporal and spatial component. Moreover peaks, corners and edges can occur on an ice cube during the melting process, even if the cube’s original shape was smooth. You only have to imagine an ice cube in the shape of an hourglass: as soon as the connecting piece melts, two objects with a pronounced point are formed, at least for a short time.
After these achievements, Caffarelli went on to address one the most stubborn problems in physics: the well-known Navier-Stokes equations used to describe fluid flows. These are differential equations that describe the flow of liquids. The equations have raised debates among mathematicians for centuries. It is not even known whether they always give a finite and smooth solution. This means it is unclear whether the flow velocity can suddenly increase at one place or point in time to another—or whether it can even assume infinitely large values. This question is one of the seven Millennium Prize problems, for which the Clay Mathematics Institute offered awards in 2000: $1 million for the solution to each.
While walking through New York City’s Chinatown in 1980, Caffarelli and his colleagues Robert Kohn and Louis Nirenberg decided to look into the Navier-Stokes equations. Two years later they achieved a result that represents the biggest breakthrough in that field to date: If the Navier-Stokes solutions should actually contain singularities—fluid flows that exhibit jerky changes or infinitely fast speeds—that would mean that the singularities were destined to immediately disappear. This finding doesn’t solve the relevant Millennium Prize problem, but it does guarantee that, according to the equations, fluids will only behave in this strange way, if they do so at all, over a short period—a great relief to an airplane or car designer.
To this day, the 74-year-old Caffarelli continues to work tirelessly on a range of research topics and publishes several papers every year—in total, he has authored more than 320 publications over the course of his career.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
