Mathematics is the language for explaining the universe. Galileo Galilei was convinced of this already in the 16th century. But even the mundane phenomenon of melting ice cubes in a glass of water can lead to equations so complex as to overwhelm people with advanced mathematical expertise. But the Argentinian mathematician Luis Caffarelli did not stop devoting himself to just such problems in his research career. The Norwegian Academy of Sciences and the Academy of Letters awarded Caffarelli this year’s highest honor in mathematics, the Abel Prize.
Caffarelli was born in Buenos Aires in 1948. Earlier in his career, he mainly dealt with polynomial properties, i.e. algebraic formulas with more than one term, until he completed his PhD at the University of Buenos Aires. When Caffarelli received his 1973 postdoctoral fellowship at the University of Minnesota, he began to devote himself to the broad field of differential equations.
A differential equation is an equation containing derivatives that describe properties such as the rate of change of a physical system. This all sounds complicated and abstract, but it’s these kinds of equations that describe the constant physical flux of the world around us. Differential equations describe how certain variables change in time and space. It gives us a glimpse into the future because it predicts how the system will change in time and space. Suppose you throw a ball in the air. The parabolic trajectory followed by the ball can be represented as a solution of a differential equation.
Many natural phenomena (river flow speed and wind direction) depend on the time and position of the observed system. Differential equations are very difficult to solve because they contain both temporal and spatial derivatives. Caffarelli began his research by first concentrating on static problems (problems that do not change over time). As an example, the skin of a soap bubble is stretched over the surface. The foam skin is known as a minimal surface because it always tries to make itself as small as possible. Differential equations are required to compute the shape of such a minimal surface. Caffarelli was interested in what the minimum surface would look like when encountering obstacles.
One of the most important issues when considering this problem is the surface size of the area where the bubble and the obstacle meet. Intuitively, we can say that the contact surface of a bubble has a smooth boundary with no corners or edges. However, it is very difficult to prove this mathematically. This is because we need to calculate the minimum area obtainable for all kinds of obstacles, which requires solving a very large number of very complicated differential equations. Caffarelli began to address this problem in his 1970s by examining the properties of differential equations and found that if the obstacles were also smooth, there would be no cracks or corners at the boundaries of the contact surfaces.
This work allowed him to focus on more complex phenomena, such as depicting ice cubes melting in water. The Slovenian-Austrian physicist Josef Stefan had already tackled this problem in the late 19th century and paved the way by deriving two equations. The first describes the heat flow from water to ice, which heats up and begins to melt. The second is used for the disappearance of the contact surface between water and ice. Both equations interact. The strength of heat transfer depends on the surface of the ice, but heat flow determines how quickly the surface shrinks. These so-called Stefan equations seemed to explain the problem well.
Until the 1970s, it was unclear whether abstract solutions could be provided that were disconnected from the real world. This equation could predict a fractal-like ice cube shape that has never been observed in nature. This was much more difficult to investigate than looking at soap skins. Furthermore, even if the original shape of the cube is smooth, the ice cubes can develop peaks, corners and edges during the melting process. Imagine an ice cube shaped like an hourglass. As soon as the connecting part melts, two objects with prominent tips form, at least for a short time.
After these achievements, Caffarelli set out to tackle one of physics’ most stubborn problems. It is the famous Navier-Stokes equation used to describe fluid flow. These are differential equations that describe the flow of liquids. The equation has caused debate among mathematicians for centuries. It is not even known if they always give finite and smooth solutions. This means that it is unclear whether the current velocity can suddenly increase at one location or point in time at another, or whether it can take infinitely large values. This problem was one of his seven Millennium Prize problems, with the Clay Mathematics Institute providing the prize in 2000. To solve each problem he has a million dollars.
While walking through Chinatown in New York City in 1980, Caffarelli and his colleagues Robert Kohn and Louis Nirenberg decided to examine the Navier-Stokes equations. Two years later, they achieved results that represent the biggest breakthrough in the field to date. If the Navier-Stokes solution should indeed contain a singularity (fluid flow exhibiting jerky transitions or infinitely high velocities), then that means that a singularity is quickly doomed. disappear. This discovery doesn’t solve the related Millennium Prize problem, but the equations guarantee that fluids only behave in this strange way. Or a car designer.
To this day, 74-year-old Caffarelli continues to work vigorously on a variety of research topics, publishing several papers each year. In total he has authored over 320 publications during his career so far.
This article was originally science spectrum Reproduced with permission.
