As a mathematician, Luis A. Caffarelli of the University of Texas at Austin tries to answer questions that sound simple, even potentially useful:

How does the shape of a piece of ice change as it melts?

Can a smooth flow of water ever spin out of control?

What is the shape of an elastic sheet stretched around an object?

These questions are not simple to answer. The behavior of these and many other phenomena in the world around us — including the gyrations of financial markets, the turbulence of river rapids and the spread of infectious diseases — can be described mathematically, using what are known as partial differential equations. The equations can often be written down simply, but finding exact solutions is devilishly difficult and indeed usually impossible.

Yet, Dr. Caffarelli, 74, was able to make major progress in the understanding of partial differential equations even when complete solutions remain elusive. For those achievements, he is this year’s winner of the Abel Prize — his field’s equivalent of the Nobel.

“Few other living mathematicians have contributed more to our understanding of partial differential equations than the Argentinian–American Luis Caffarelli,” the Abel Prize committee announced in a news release on Wednesday.

The prize is accompanied by 7.5 million Norwegian kroner, or about $700,000.

Dr. Caffarelli enjoys talking with scientists, he said in an interview. Sometimes, he suggests mathematical approaches they could try; other times, they suggest problems he could work on.

“I like to have some connection with physics, with engineering even,” Dr. Caffarelli said.

That includes what is known as the “obstacle problem.” One example is to take a balloon and squish it against a wall. “You compress it, right?” said Helge Holden, a mathematician at the Norwegian University of Science and Technology who serves as chairman of the Abel Prize committee. “What will be the interface between the wall and the balloon?”

For a flat wall, the boundary between where the balloon is touching the wall versus where it is not is pretty simple. But if there is an obstacle like a knob sticking out of the wall, the solution can become complex.

Dr. Caffarelli was able to describe specific properties of the solution.

A variation of the obstacle problem could involve determining the heating and cooling needed to keep a room within a building held at a constant temperature, even as outside temperatures warm and cool.

“These are things that really appear in real life,” Dr. Caffarelli said.

The obstacle problem is an example of what are known as free boundary problems. Another example involves melting ice.

The boundary between liquid water and ice is always 32 degrees Fahrenheit, but that surface shifts as the ice melts — hence, the boundary is free and not fixed — and that shifting surface greatly complicates the problem.

“What you’re trying to figure out is things about the shape of this free boundary,” said Carlos Kenig, a mathematician at the University of Chicago who is also an expert on partial differential equations. “He was the first person to really understand this problem in more than one dimension. And the methods that he introduced have been extremely powerful and are still being used in many other problems.”

Another important result involved the Navier-Stokes equations, which describe the dynamics of incompressible fluids. Water and air are, to a good approximation, incompressible fluids, and the Navier-Stokes equations are used to design airplane wings and to model weather patterns and ocean currents.

In reality, it would be strange if the speed of a wind gust could accelerate to infinity — and yet nothing in the equations seems to prohibit that possibility.

Dr. Caffarelli, along with two other mathematicians, Louis Nirenberg (who shared the Abel Prize in 2015) and Robert Kohn, could not prove that fluids would always flow smoothly. However, they were able to prove that such regions of infinite speed, if they existed, would have to be exceedingly small.

Some of Dr. Caffarelli’s work has also found usage in the financial world, in the pricing of certain options — contracts where someone has an opportunity but not an obligation to buy or sell something at a set price.

Dr. Holden said that Dr. Caffarelli’s papers were succinct and clear.

“He doesn’t write 200-page papers,” Dr. Holden said. “He writes short papers because there is always an ingenious idea.”

There is no Nobel Prize in mathematics, and for decades the most prestigious awards in math were the Fields Medals, awarded in small batches every four years to the most accomplished mathematicians who are 40 or younger.

The Abel, named after Niels Henrik Abel, a Norwegian mathematician, is set up more like the Nobels. Since 2003 it has been given annually to highlight important advances in mathematics. Previous laureates include Andrew J. Wiles, who proved Fermat’s last theorem and is now at the University of Oxford; John F. Nash Jr., whose life was portrayed in the movie “A Beautiful Mind”; and Karen Uhlenbeck, an emeritus professor at the University of Texas at Austin who in 2019 became the first woman to receive an Abel.

Last year, Dennis P. Sullivan, a professor of mathematics at Stony Brook University and the City University of New York Graduate Center, received the Abel for work in topology, the study of space and shapes.

While the Nobel Prizes are closely kept secrets, the Abel committee informs the winner’s institution days in advance. It then figures out how and when to share the news with the winner.

Thomas Chen, the chairman of the Texas math department, scheduled a Zoom call for Friday morning with Dr. Caffarelli and his wife, Irene Gamba, who is also a mathematician at the university. Dr. Caffarelli said he thought the call might be about someone joining the math department.

Instead, Gunn Elisabeth Birkelund, the secretary general of the Norwegian Academy of Science and Letters, which manages the Abel Prize, joined the call to tell him that he was the winner.

“It was a surprise, a total surprise,” Dr. Caffarelli said.

Dr. Caffarelli was born in Buenos Aires in 1948. After completing his Ph.D. at the University of Buenos Aires in 1972, he moved north, to the University of Minnesota where he was introduced to the obstacle problem.

In 1980, he moved to the Courant Institute of Mathematical Sciences at New York University, where he collaborated with Dr. Nirenberg and Dr. Kohn on the Navier-Stokes research. He later worked at the University of Chicago and the Institute for Advanced Study in Princeton, N.J., before returning to Courant in 1994. In 1997, he moved to the University of Texas.

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