A mathematician who studies equations describing nature

NEW YORK: 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.”