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Math formula may help physicists calculate entropy of a modular black hole

Emory mathematician Ken Ono has marked the 125th anniversary of the birth of Srinivasa Ramanujan, an Indian mathematician who achieved stardom for figuring out amazing numerical patterns and connections without the use of proofs or modern mathematical tools, by creating a formula for mock modular forms that may help physicists who study black holes.

Well-known English mathematician G. H. Hardy is said to have placed Ramanujan in the same league as famous mathematicians such as Euler, Gauss, Newton and Archimedes. While he had no almost no formal training in pure mathematics, Ramanujan made significant contributions to mathematical analysis, number theories and several other areas of mathematics.

“I wanted to do something special, in the spirit of Ramanujan, to mark the anniversary,” says Mr. 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.”

Mr. Ono worked with Amanda Folsom, from Yale University, and Rob Rhoades, from Stanford University.

Mr. Ono’s formula also solved one of the greatest mysteries left behind by the Indian mathematician.

Ramanujan is said to have written a letter on his death-bed in 1920 to G. H. Hardy that detailed several new functions that performed differently from known theta functions but closely mimicked them. At the time, Ramanujan theorized that his mock modular forms agreed with the ordinary modular forms earlier identified by Carl Jacobi. He believed that both would conclude with similar outputs for roots of 1.

“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,” Mr. Ono posits.

Mr. Ono and his colleagues used modern mathematical tools that were unavailable to Ramanujan to prove that a mock modular form could be calculated just as the Indian mathematician theorized.

“We proved that Ramanujan was right,” Mr. Ono says. “We found the formula explaining one of the visions that he believed came from his goddess.”

Mr. Ono explains Ramanujan’s vision using a “magic coin” analogy, in which Jacobi and Ramanujan are contemporaries that shop together. They each spend a coin in the same shop but their coins take very different paths.

“For months, the paths of the two coins look chaotic, like they aren’t doing anything in unison,” Mr. Ono says. “But eventually Ramanujan’s coin starts mocking, or trailing, Jacobi’s coin. After a year, the two coins end up very near one another: In the same town, in the same shop, in the same cash register, about four inches apart.”

Mr. Ono and his colleagues believe that Ramanujan’s work could help physicists better understand the secrets of black holes.

“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,” Mr. Ono says.

According to the Emory mathematician, the expansion of modular forms is one of the keys to computing the entropy of a modular black hole. While not all black holes are modular, the new formula may help physicists calculate their entropy as if all black holes are modular.

Mr. Ono and several graduate students looked at the paragraph in Ramanujan’s letter that offered an indistinct explanation for how the Indian mathematician arrived at the functions. The vague paragraph has been challenging mathematicians for nearly a century.

“So much of what Ramanujan offers comes from mysterious words and strange formulas that seem to defy mathematical sense,” Mr. 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.”

Mr. Ono and his graduate students successfully interpreted the paragraph.

“We developed a theorem that shows that the bizarre methodology he used to construct his examples is correct,” Mr. 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.”

Mr. Ono was able to travel to Ramanujan’s birth home in Madras to take part in a docu-drama about the famed mathematician.

“I got to hold some of Ramanujan’s original notebooks, and it felt like I was talking to him,” Mr. Ono says. “The pages were yellow and falling apart, but they are filled with formulas and class invariants, amazing visions that are hard to describe, and no indication of how he came up with them.”

Mr. Ono is glad to have presented at the Ramanujan 125 conference at the University of Florida.

“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.”

Ramanujan claimed that his mathematical discoveries were revealed to him in dreams by the goddess Namagiri.