But Dr. Szemerédi’s proof was long and complicated.
“Furstenberg gave this beautiful, short proof,” said Terence Tao, a mathematician at the University of California, Los Angeles.
In 2004, Dr. Tao and Ben Green, a mathematician at the University of Oxford, cited Dr. Furstenberg and used ergodic theory arguments to prove a major result — that arbitrarily long progressions also exist among the prime numbers, the integers that have exactly two divisors: 1 and themselves.
Some notable work of Dr. Margulis, the other Abel Prize winner, addresses problems involving connected networks similar to the internet, where computers continually send messages to each other. To achieve the fastest communications, one would want to make a direct connection between every pair of computers. But that would require an impractically huge number of cables.
“These are networks that you are trying to engineer so that are very sparse on the one hand,” said Peter Sarnak, a mathematician at the Institute for Advanced Study in Princeton, N.J., “yet at the same time they have the property that if you’re trying to go from one point to another quickly with a short path, you can still do that.”
Dr. Margulis was the first to come up with a step-by-step procedure for how to create such networks, known as expander graphs.
Recasting problems, as Dr. Margulis did using ergodic theory, often does not make it easier to solve them. Dr. Sarnak said that if a student had come to him with the initial steps of what Dr. Margulis had done, he would have said: “So what? What did you do? You just reformulated it. It looks harder now.”
But the ergodic theory revealed hidden universal truth, enabling Dr. Margulis to make quick progress on some previously intractable problems. “He went from zero to solution in a couple of papers, which were stunningly original,” Dr. Sarnak said.