SoylentNews

Arthur T Knackerbracket has processed the following story:

Ramanujan brings life to the myth of the self-taught genius. He grew up poor and uneducated and did much of his research while isolated in southern India, barely able to afford food. In 1912, when he was 24, he began to send a series of letters to prominent mathematicians. These were mostly ignored, but one recipient, the English mathematician G.H. Hardy, corresponded with Ramanujan for a year and eventually persuaded him to come to England, smoothing the way with the colonial bureaucracies.

It became apparent to Hardy and his colleagues that Ramanujan could sense mathematical truths — could access entire worlds — that others simply could not. (Hardy, a mathematical giant in his own right, is said to have quipped that his greatest contribution to mathematics was the discovery of Ramanujan.) Before Ramanujan died in 1920 at the age of 32, he came up with thousands of elegant and surprising results, often without proof. He was fond of saying that his equations had been bestowed on him by the gods.

More than 100 years later, mathematicians are still trying to catch up to Ramanujan’s divine genius, as his visions appear again and again in disparate corners of the world of mathematics.

The English mathematician G.H. Hardy, after receiving a letter from Ramanujan and recognizing his brilliance, arranged for him to study and work with him in Cambridge.

Ramanujan is perhaps most famous for coming up with partition identities, equations about the different ways you can break a whole number up into smaller parts (such as 7 = 5 + 1 + 1). In the 1980s, mathematicians began to find deep and surprising connections between these equations and other areas of mathematics: in statistical mechanics and the study of phase transitions, in knot theory and string theory, in number theory and representation theory and the study of symmetries.

Most recently, they’ve appeared in Mourtada’s work on curves and surfaces that are defined by algebraic equations, an area of study called algebraic geometry. Mourtada and his collaborators have spent more than a decade trying to better understand that link, and to exploit it to uncover rafts of brand-new identities that resemble those Ramanujan wrote down.

“It turned out that these kinds of results have basically occurred in almost every branch of mathematics. That’s an amazing thing,” said Ole Warnaar of the University of Queensland in Australia. “It’s not just a happy coincidence. I don’t want to sound religious, but the mathematical god is trying to tell us something.”

[...] In September, Ono and two collaborators — William Craig and Jan-Willem van Ittersum — published yet another application for partition identities. Rather than looking for a new source from which these identities would spring, they were able to use them for an entirely different purpose: to detect prime numbers.

They took functions that counted partitions and used them to build a special formula. When you plug any prime number into this equation, it spits out zero. When you plug in any other number, it instead spits out a positive number. In this way, the partition identities can give you a way to pick out the entire set of primes from the integers, Ono said. “Isn’t that crazy?”

“Partitions are about adding and counting,” he said. “Why would they be able to detect which numbers are prime or not, on the nose, which is a multiplication thing?”

By tapping into the rich mathematical theory of modular forms, he and his colleagues found that this formula was just a glimpse of a much larger class of prime-detecting functions — infinitely many, in fact. “That’s mind-blowing to me,” Ono said. “I hope people find it beautiful.” It indicates a deeper relation between the partitions and multiplicative number theory that mathematicians are now hoping to explore.

In some ways, it makes sense that partitions keep infiltrating every corner of mathematics. “The theory of partitions is so basic,” Andrews said. “Counting stuff and adding stuff up happens in almost every branch of mathematics.”

Still, the precise nature of those connections is hard to work out. “It’s really about getting the perspective right,” Ono said.

“This is the great thing about Ramanujan’s work,” Kanade said. “It’s not just one identity he discovered, and a dead end. It’s always the tip of an iceberg. You just have to follow it through.”

“Ramanujan is someone who can imagine things that someone like me cannot,” Mourtada said. But the development of new fields of mathematics has “given us the possibility to find new partition identities that Ramanujan could probably have found just by imagination.”

“That’s why mathematics is so important,” he added. “It allows ordinary people like me to find these miracles, too.”

Original Submission