“He was a problem solver unequaled,” Peter C. Sarnak, a colleague of Dr. Bourgain’s at the institute, said in an interview.

Dr. Sarnak said that Bourgain started out in an esoteric corner of mathematics with extremely difficult problems. “He just came in and started solving one problem after the other in that subject,” he said. “So he made his name there, became very famous, winning all sorts of young-up-and-coming-star prizes. But then he broadened out.”

Dr. Bourgain found that tools he had developed could also be applied to other fields of mathematics, including partial differential equations, computer science, quantum mechanics and dynamical systems, making progress on formidable problems that had stymied experts in those areas.

“There would be some big mountain in front of you,” Dr. Sarnak said, “and he would ascend halfway up, occasionally all the way up. People couldn’t understand how he got all the way up. It would take often months or years for people to understand his proofs. He liked the idea that he was way ahead and people were catching up to him all the time. He would open these doors.”

Some of Dr. Bourgain’s recent work included a “decoupling theorem” — a very abstract generalization of the Pythagorean theorem applied to oscillating waves, like light or radio waves. While Pythagoras merely showed how the length of the two shorter sides of a right triangle are related to the longer hypotenuse, the decoupling theorem proved by Dr. Bourgain and Ciprian Demeter, of Indiana University, showed similar relationships in the superposition of waves, when the individual oscillations are added together.