Tag Archives: G. H. Hardy

Ramanujan—a math genius who left behind math formulas that took 90 years to decode

1920, the year mathematician Srinivasa Ramanujan died, is also the year he left behind mathematical formulas that may help unlock the secrets of black holes (from the Dec. 11, 2012 posting by Carol Clark for Emory University’s e-science commons blog),

“No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,” Ono [Emory University mathematician Ken Ono] says.

Expansion of modular forms is one of the fundamental tools for computing the entropy of a modular black hole. Some black holes, however, are not modular, but the new formula based on Ramanujan’s vision may allow physicists to compute their entropy as though they were.

Ramanujan was on his death bed (at the age of 32) when he devised his last formulas (from the Clark posting),

Accessed from http://esciencecommons.blogspot.ca/2012/12/math-formula-gives-new-glimpse-into.html

Accessed from http://esciencecommons.blogspot.ca/2012/12/math-formula-gives-new-glimpse-into.html

… A devout Hindu, Ramanujan said that his findings were divine, revealed to him in dreams by the goddess Namagiri.

While on his death-bed in 1920, Ramanujan wrote a letter to his mentor, English mathematician G. H. Hardy. The letter described several new functions that behaved differently from known theta functions, or modular forms, and yet closely mimicked them. Ramanujan conjectured that his mock modular forms corresponded to the ordinary modular forms earlier identified by Carl Jacobi, and that both would wind up with similar outputs for roots of 1.

No one at the time understood what Ramanujan was talking about. “It wasn’t until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,” Ono says.

This year (2012) a number of special events have been held to commemorate Ramanujan’s accomplishments (Note: I have removed links), from the Clark posting,

December 22 [2012] marks the 125th anniversary of the birth of Srinivasa Ramanujan, an Indian mathematician renowned for somehow intuiting extraordinary numerical patterns and connections without the use of proofs or modern mathematical tools. ..

“I wanted to do something special, in the spirit of Ramanujan, to mark the anniversary,” says Emory mathematician Ken Ono. “It’s fascinating to me to explore his writings and imagine how his brain may have worked. It’s like being a mathematical anthropologist.”

Ono, a number theorist whose work has previously uncovered hidden meanings in the notebooks of Ramanujan, set to work on the 125th-anniversary project with two colleagues and former students: Amanda Folsom, from Yale, and Rob Rhoades, from Stanford.

The result is a formula for mock modular forms that may prove useful to physicists who study black holes. The work, which Ono recently presented at the Ramanujan 125 conference at the University of Florida, also solves one of the greatest puzzles left behind by the enigmatic Indian genius.

Here’s a trailer for the forthcoming movie (a docu-drama) about Ramanujan, from the Clark posting,

Here’s a description of Ramanujan from Wikipedia, which gives some insight into the nature of his genius (Note: I have removed links and a footnote),

Srinivasa Ramanujan FRS (…) (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centered in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he sometimes rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G.H. Hardy, in the same league as mathematicians like Euler and Gauss.

There is a little more to Ono’s latest work concerning Ramanujan’s deathbed math functions (from the Clark posting),

After coming up with the formula for computing a mock modular form, Ono wanted to put some icing on the cake for the 125th-anniversary celebration. He and Emory graduate students Michael Griffin and Larry Rolen revisited the paragraph in Ramanujan’s last letter that gave a vague description for how he arrived at the functions. That one paragraph has inspired hundreds of papers by mathematicians, who have pondered its hidden meaning for eight decades.

“So much of what Ramanujan offers comes from mysterious words and strange formulas that seem to defy mathematical sense,” Ono says. “Although we had a definition from 2002 for Ramanujan’s functions, it was still unclear how it related to Ramanujan’s awkward and imprecise definition.”

Ono and his students finally saw the meaning behind the puzzling paragraph, and a way to link it to the modern definition. “We developed a theorem that shows that the bizarre methodology he used to construct his examples is correct,” Ono says. “For the first time, we can prove that the exotic functions that Ramanujan conjured in his death-bed letter behave exactly as he said they would, in every case.”

Ono is now on a mathematicians’ tour in India (from the Clark posting),

Ono will spend much of December in India, taking overnight trains to Mysore, Bangalore, Chennai and New Dehli, as part of a group of distinguished mathematicians giving talks about Ramanujan in the lead-up to the anniversary date.

“Ramanujan is a hero in India so it’s kind of like a math rock tour,” Ono says, adding, “I’m his biggest fan. My professional life is inescapably intertwined with Ramanujan. Many of the mathematical objects that I think about so profoundly were anticipated by him. I’m so glad that he existed.”

Between this and the series developed by Alex Bellos about mathematics in Japan (my Oct. 17, 2012 posting), it seems that attention is turning eastward where the study and development of mathematics is concerned. H/T to EurekAlert’s Dec. 17, 2012 news release and do read Clark’s article if you want more information about Ono and Ramanujan.