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posted by Fnord666 on Tuesday October 29, @11:49AM   Printer-friendly
from the partitioning-things dept.

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.”


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  • (Score: 3, Insightful) by Anonymous Coward on Tuesday October 29, @02:05PM (7 children)

    by Anonymous Coward on Tuesday October 29, @02:05PM (#1379245)

    I recall reading somewhere that Ramanujan was shown this formula in a dream

    sqrt( sqrt( 2143/22 ) )

    This was before the era of computing machines and hand calculators, and it makes one wonder how exactly the mind does work.

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  • (Score: 2, Interesting) by pTamok on Tuesday October 29, @03:38PM

    by pTamok (3042) on Tuesday October 29, @03:38PM (#1379271)

    There are so many things that Ramanujan did that seem like magic:

    e.g. Stack Exchange: Mathematics: Ramanujan's approximation for π [stackexchange.com]

    ...this formula computes a further eight decimal places of π with each term in the series

    On an unrelated note, this is one of the best results given by Ramanujan which is exceedingly difficult to prove and to date (just over 10 years ago) there is no proof available within the limits of hand calculation. Also important is to note that this is an early work of Ramanujan which was done in India before he got in touch with the British mathematician G. H. Hardy.

  • (Score: 2, Interesting) by pTamok on Tuesday October 29, @03:56PM (5 children)

    by pTamok (3042) on Tuesday October 29, @03:56PM (#1379277)

    Oh, and that formula is described in the WIkipedia article on Approximations of π [wikipedia.org] underneath Miscellaneous Approximations: [wikipedia.org] Accurate to 9 digits.

    This is from Ramanujan, who claimed the Goddess of Namagiri appeared to him in a dream and told him the true value of π.

    • (Score: 1) by nostyle on Tuesday October 29, @08:29PM (3 children)

      by nostyle (11497) on Tuesday October 29, @08:29PM (#1379332) Journal

      Thanks for tracking down a citation for my recollection.

      Let me add to my comment about how mysterious the seemingly magical workings of the mind are.

      On more than one occasion, I have been halfway home, thinking about nothing in particular when into my mind's eye I will pop a line of computer code that I forgot to include among the several hundred that I had drafted that day and handed off to others to run with. Then, heading back to the office, I add that line into the mix, and things run perfectly.

      So was it my sub-conscious mind continuing to work the problem while I was consciously doing something else? Or was some benevolent angel tapping me on the shoulder and telling me to go back and fix things? Was it my ancestors? The Holy Spirit? A Hindu goddess? - I have no clue. I only know that it happens to me.

      I have often commented about the dreams and visions which befall me and borne witness to some moments when a voice in my heads came through loud and clear. You can find evidence of this here [soylentnews.org].

      Accordingly, I have no trouble believing Ramanujan's claim of having some sort of divine inspiration for his discoveries. Perhaps the "genius" part is that he was [self-] trained enough to understand the equations and their implications and arose to bring them to the attention of the world.

      --

      But I do know one and one is two

      -Sam Cooke, Wonderful World

      • (Score: 0) by Anonymous Coward on Tuesday October 29, @10:50PM

        by Anonymous Coward on Tuesday October 29, @10:50PM (#1379348)

        ...into my mind's eye I will pop a line...

        should read

        ...into my mind's eye will pop a line...

        Another edit failure. Sorry.
        -nostyle
        ..

        I've done no harm, I keep to myself
        There's nothin' wrong with my state of mental health

        -Men at Work, Who Can It Be Now?

      • (Score: 1) by pTamok on Tuesday October 29, @10:56PM

        by pTamok (3042) on Tuesday October 29, @10:56PM (#1379349)

        You are welcome.

        As for you question: it is your sub-conscious. A well known technique of coming up with solutions to things is to go and do a completely different undemanding task to encourage incubation.

        Incubation and Intuition in Creative Problem Solving [nih.gov]

        Different incubation tasks in insight problem solving: evidence for unconscious analytic thought [tandfonline.com]

        Does Incubation Enhance Problem Solving? A Meta-Analytic Review [researchgate.net]

        If you are working a problem, you first need to do some non-trivial analysis as preparation so you understand the problem: then, you go and do something different that occupies your attention that is not very demanding, but not completely undemanding, and sometimes a solution will 'just pop into your head'. The preparatory work is necessary: just goofing off at random doesn't solve problems.

      • (Score: 0) by Anonymous Coward on Thursday October 31, @06:53PM

        by Anonymous Coward on Thursday October 31, @06:53PM (#1379651)
        Yours sounds like a sub conscious.

        Ramanujan's sounds like a super conscious. 😉

        Yours is remembering stuff you forgot to do.

        His is "remembering stuff" that people decades later prove to be true and sometimes even useful.

        Ramanujan was poor, so he was likely not able to afford the paper to write and keep his proofs. Apparently he did a lot of his stuff on a slate.
    • (Score: 1) by nostyle on Friday November 01, @01:38AM

      by nostyle (11497) on Friday November 01, @01:38AM (#1379732) Journal

      Of course, nowadays, with all the computing tools we have, approximations for pi can be found much more easily. Noodling around for an hour or so using the "bc" linux utility, I was able to discover that

      cuberoot(9891/319) = 3.14159241...

      which is accurate to seven digits, and something of an improvement to cuberoot(31).

      What a tine to be alive!
      --

      What comes around goes around

      -Ratt, Round and Round