One morning in Katmandu, as he relaxed on a hotel porch, the structure of a proof suddenly came to him. It was a solid advance — one he didn’t forget. He discussed it with Dennis P. Sullivan, a mathematician at Stony Brook who had recently won the National Medal of Science, and the two collaborated.

In 2007, the resulting paper ran under the title “Axiomatic Characterization of Ordinary Differential Cohomology.”

What?

“It’s very hard to explain,” Dr. Simons said after a few tries. “But we solved it.”

Dr. Sullivan said that Dr. Simons, in his career, had made a series of seminal contributions and that an early one “revolutionized the consciousness of later generations.” He added that in May 2013, to celebrate Dr. Simons’s 75th birthday, four American math and science luminaries gave lectures about fields he had advanced.

Forbes magazine ranks him as the world’s 93rd richest person — ahead of Eric Schmidt of Google and Elon Musk of Tesla Motors, among others — and in 2010, he and his wife were among the first billionaires to sign the Giving Pledge, promising to devote “the great majority” of their wealth to philanthropy.

Of late, Dr. Simons said, his giving had accelerated, adding that he was particularly proud of Math for America. It awards stipends and scholarships of up to $100,000 to train high school math and science teachers and to supplement their regular salaries. The corps is expanding to 1,100 teachers, mainly in New York City, but also in Boston, Los Angeles and elsewhere.

His passion, however, is basic research — the risky, freewheeling type. He recently financed new telescopes in the Chilean Andes that will look for faint ripples of light from the Big Bang, the theorized birth of the universe.

The afternoon of the interview, he planned to speak to Stanford physicists eager to detect the axion, a ghostly particle thought to permeate the cosmos but long stuck in theoretical limbo. Their endeavor “could be very exciting,” he said, his mood palpable, like that of a kid in a candy store.