But Dr. Szemerédi’s proof was lengthy and sophisticated.
“Furstenberg gave this wonderful, short proof,” explained Terence Tao, a mathematician at the College of California, Los Angeles.
In 2004, Dr. Tao and Ben Environmentally friendly, a mathematician at the College of Oxford, cited Dr. Furstenberg and used ergodic theory arguments to establish a big end result — that arbitrarily lengthy progressions also exist amongst the prime figures, the integers that have just two divisors: 1 and themselves.
Some notable get the job done of Dr. Margulis, the other Abel Prize winner, addresses difficulties involving linked networks comparable to the world wide web, where computers constantly mail messages to each other. To attain the quickest communications, a person would want to make a immediate connection amongst each individual pair of personal computers. But that would call for an impractically massive quantity of cables.
“These are networks that you are trying to engineer so that are pretty sparse on the a person hand,” stated Peter Sarnak, a mathematician at the Institute for State-of-the-art Review in Princeton, N.J., “yet at the exact same time they have the assets that if you are hoping to go from just one position to a further promptly with a brief path, you can nevertheless do that.”
Dr. Margulis was the to start with to occur up with a stage-by-move technique for how to generate such networks, identified as expander graphs.
Recasting challenges, as Dr. Margulis did applying ergodic principle, normally does not make it a lot easier to clear up them. Dr. Sarnak explained that if a college student experienced arrive to him with the initial measures of what Dr. Margulis experienced completed, he would have reported: “So what? What did you do? You just reformulated it. It appears tougher now.”
But the ergodic principle disclosed hidden common fact, enabling Dr. Margulis to make speedy development on some previously intractable problems. “He went from zero to remedy in a pair of papers, which ended up stunningly original,” Dr. Sarnak said.