It is said that mathematics is fundamental to understanding any other subject from physics to economics. Even otherwise, it remains a beautiful subject in itself and so are its practitioners – the mathematicians. As the quote from this article says “As a budding mathematician, “I had the impression that these are beautiful people,” he said. “They are honest. You need to be honest with yourself to be a mathematician. Otherwise, it doesn’t work.””
This article in Quanta is about arguably the greatest mathematician of them all – Srinivasa Ramanujan. If not the greatest, the unlikeliest for sure.
“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 article is about how Ramanujan’s century old work continues to hold mathematicians in awe even today as it shows up in a variety of related areas of mathematics. In particular, what is now known as Rogers-Ramanujan identities, which Ramanujan uncovered when he started work in Cambridge with Hardy:
“One of Ramanujan’s first tasks was to prove a general statement about his continued fractions. To do so, he needed to prove two other statements. But he couldn’t. Neither could Hardy, nor could any of the colleagues he reached out to.
It turned out that they didn’t need to. The statements had been proved 20 years earlier by a little-known English mathematician named L.J. Rogers. Rogers wrote poorly, and at the time the proofs were published no one paid any attention. (Rogers was content to do his research in relative obscurity, play piano, garden and apply his spare time to a variety of other pursuits.) Ramanujan uncovered this work in 1917, and the pair of statements later became known as the Rogers-Ramanujan identities.
Amid Ramanujan’s prodigious output, these statements stand out. They have carried through the decades and across nearly all of mathematics. They are the seeds that mathematicians continue to sow, growing brilliant new gardens seemingly wherever they fall.
…For much of the 20th century, mathematicians would delight in thinking about the strange hidden phenomena that Ramanujan had unearthed.
But it wasn’t until the late 1970s that they uncovered additional facets of the Rogers-Ramanujan identities.
…This trend of the Rogers-Ramanujan identities surfacing in various fields of mathematics continued into the 1990s and 2000s. They appeared in number theory, in the study of central functions called modular forms; in probability theory, in work on Markov chains; and in topology, in polynomials used to distinguish and classify knots. Each time, the identities could be re-proved using techniques from those fields — and each time, mathematicians could exploit those connections to produce new identities, planting more and more seeds in Ramanujan’s garden.
Every time the Rogers-Ramanujan identities pop up somewhere new, mathematicians are both surprised and unsurprised. The unexpected appearance of the identities offers a fresh connection to explore, further evidence of the mysterious unity between the many different fields of math.
“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.””
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