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Gems of Ramanujan and their Lasting Impact on Mathematics

Ken Ono's Distinguished Lecture: Far From a Dull One

You probably know the story. Cambridge don and renowned number theoriest G.H. Hardy visited his ailing collaborator, Indian transplant and autodidact Srinivasa Ramanujan, in the hospital. Having arrived by taxi, Hardy remarked that its number, 1729, was a rather dull one. To which Ramanujan replied, "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

"Hardy probably thought, 'That isn't interesting, but it's amazing that Ramanujan happened to know that fact,'" Ken Ono (Emory University) told a packed MAA Carriage House on October 13. 

The standard version of this story, Ono explained, leads one to believe that Ramanujan recognized this extraordinary property of 1729 spontaneously. "Except it's wrong," he said.

Ono did much more in his Distinguished Lecture, "Gems of Ramanujan and their Lasting Impact on Mathematics," than shed new light on the famous tale of the first taxicab number. He recounted Ramanujan's life story and drew non-mathematical lessons from it. He described his involvement in Hollywood's adaptation of Robert Kanigel's The Man Who Knew Infinity and showed scenes from the film. Ono not only outlined mathematicians' continued efforts to fathom Ramanujan's uncanny insights, but outed Jeremy Irons as an uncoordinated tennis player.

Two-Time College Dropout

Srinivasa Ramanujan (1887-1920) was born in southern India into the Brahmin or priestly caste and remained a devout Hindu to his death. Having developed trigonometry for himself by 13, he soon turned his attention to Synopsis of Elementary Results in Pure Mathematics, fleshing out the many details the author's ellipses left to the reader.

"Ramanujan viewed the dot dot dot as a personal challenge," said Ono, who first learned of Ramanujan in the 1980s from his number theorist father.

"He told me the story about some Indian dude who died 60 years before who had visions from a goddess who inspired an entire generation of Japanese mathematicians while at the same time not knowing anything about mathematics as a two-time college dropout," Ono recalled.

Though incredulous, the teenage Ono found in the fantastical sounding tale a heartening takeaway. "What I heard was [that] my father looked up to a two-time college dropout," Ono said. "That's what I latched onto."

Those who value class rank and SAT scores above all else should not discount unconventional paths to success, Ono stressed as he sketched the outline of Ramanujan's biography. Even as Ramanujan's obsession with mathematics tanked his university career, Ono observed, he had benefactors and parents who continued to support him. Today's educational systems are too inelastic, Ono mourned, for financial and emotional backers to give could-be geniuses license to chart their own trajectories.

Math at the Movies

Everyone needs help, Ono is fond of saying, and, luckily for mathematics, Ramanujan was not afraid to seek it. Of course the unschooled Madras clerk sought help from no less than the most distinguished mathematicians of his day.

"In Ramanujan you have someone who is supremely confident of his skills but at the same time very humble," Ono explained.

Ramanujan wrote to many mathematicians, but the one who read his letter start-to-finish, saw promise in even its more outlandish formulae, and invited Ramanujan to visit him in England was G.H. Hardy - atheist, curmudgeon, and stickler for mathematical rigor.

As the mathematics consultant for The Man Who Knew Infinity, Ono helped writer and director Matt Brown figure out how to convey on screen the desperation Ramanujan (Dev Patel) felt working in isolation in India, the struggles of G.H. Hardy (Jeremy Irons) to fully grasp Ramanujan's suggestive yet perplexing scribblings, the men's differing approaches to the advancement of mathematics.

To Ono fell the tasks of explaining to non-mathy people why math might be considered beautiful, of describing "what pure mathematics is to people who really don't care," of coaching Patel and Irons on how to convincingly portray mathematicians.

"I think you would be surprised by how much they know now," Ono told listeners. And if you care to quibble with any of the math in the film, email Ono. "I stand by it completely," he said.

Legacy

Ramanujan died young, before he had a chance to fully explicate the contents of his notebooks, and mathematicians have been mining the wealth of Ramanujan's writings for decades. That 1729, for example.

"It is not true that Ramanujan saw off the top of his head that 1729 was the smallest number represented as the sum of two cubes in two different ways," Ono said. In 2013, Ono and a fellow number theoriest Andrew Granville discovered on a page of Ramanujan's notes in a Cambridge archive three rational functions expressed as power series in the coefficients of which provide near misses to Fermat's Last Theorem for exponent 3. In the bottom corner Ramanujan had written 93+103=123+1, which equals 1729.

It turns out, Ono explained, that Ramanujan had invented ingenious methods about quadratic forms long before math was ready for them. And in 1993 - 73 years after Ramanujan's death - John H. Conway and W.A. Schneeberger arrived at an almost too-good-to-be-true result known as the Fifteen Theorem by revisiting Ramanujan's notes and perfecting his methods.

Though Ono's own work owes much to Ramanujan, on October 13 he matched his eagerness to acknowledge Ramanujan's vast mathematical legacy with a keenness to derive a non-mathematical moral from the mathematician's story. "Talent can be found anywhere," Ono reminded his Carriage House audience, "even in the most unexpected and unforgiving of places. It has to be brought into the light."

Ono's lecture, part of MAA's NSA-funded Distinguished Lecture series, was live-streamed on Facebook. Visit the Videos Section on MAA's Facebook  page to watch the talk.

Katharine Merow is a freelance writer living in Washington, D.C.

 

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