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Modern math solves Ramanujan’s ‘vision’ - may clarify black holes (futurity.org)
115 points by mrkuchbhi on Dec 17, 2012 | hide | past | favorite | 22 comments

I love articles like this. Ramanujan was a very interesting man and perhaps had some of the deepest mathematical insights of anyone who existed. People are still trying to figure out what the things he wrote down meant. If you want to know more about his life, there's a great book called "The Man who Knew Infinity" (http://www.amazon.com/The-Man-Who-Knew-Infinity/dp/067175061...)

One nice example of deep insight is the Ramanujan Tau sequence, which comes from expanding out the product (1-q) (1-q^2) (1-q^3) all to the 24th power and multiplied by q, as q-24q^2+252q^3-...

Ramanujan noticed that the coefficients a_n have some remarkable arithmetic properties, namely the sequence is multiplicative: a_m a_n = a_{mn} when m and n are relatively prime. There's a more complicated formula when m and n have a common divisor, and he also conjectured that the size of a_p is upwards of 2*p^5.5 when p is prime.

This led to the beautiful study of modular forms and all of the above statements have profound explanations, the last of which was proven 58 years later by Deligne in 1974, as the Riemann Hypothesis for curves.

PS: there are plenty of random discoveries that can be made about the sequence, for example a_n is congruent mod 691 to the sum of the 11th powers of all divisors of n. This also has a good explanation that is now known. Lehmer conjectured in 1947 that a_n is never 0, which has been verified up to n=22798241520242687999, but is still an open question.

I find this very interesting since, while I am positive it has a mundane naturalistic explanation (like: the guy was really smart and had exceptional intuition), revealing the answers to questions that perhaps cannot yet even be formally asked is how I think I would go about proving that I was either from the future or another world.

Somehow get sent back in time a couple dozen thousand years? Carve a bunch of primes into the side of a cave somewhere, maybe throw in the Pythagorean theorem and a suspicious number of digits of Pi too. Messages that perhaps mean little to the contemporaries of the message.

Kind of fun to think about I think.

It gets even more interesting when you think about what 'intuition' really is, and what it means to 'know' something. Words, words m'lord.

If you're willing to leave the standpoint of a 'reality' which is based in the interactions between 'subject' and 'object' then things get really interesting and you begin to question what such an apparently great man meant when he said that what was revealed to him was done so by divinity.

To do so, of course, would mean you would have to understand the metaphysics of Indian thought and culture, which may be (rather basically) summarised to hold that the objective is merely a reflection of the absolute Subjective, i.e. divinity.

This is especially important because it is from this cultural standpoint that these visions were realised. I could go on if anyone is interested.

please do. i'd contribute more in response to encourage you, but anything i have to say on the topic is something i've drawn from my own processes and probably not worth discussing until i've at least done Wikipedia on it. and it's not really a topic i've heard discussed before, even in the context of "genius". though that is the context in which i've pieced them together. disclaimer IANAG

I think use of that acronym itself constitutes geekiness ;)

A bit of background: I've spent the last three years of my life studying Indian metaphysics in India at a traditional academy known as an 'ashram'. The environment was somewhat similar to the original Greek academies, I can imagine, but that is speculation. I also have interests in tech, programming and philosophy. Please note that I am not trying to 'prove anything' either correct or incorrect -- indeed, this would not be the forum to do so. Rather, an opportunity to be of service to another individual and present two different views of this experience of the world and thus life itself.

For the western mind, indeed, for my mind -- it is ingrained to a point of 'truth' that there is a single objective world around us which presents itself to various individuals via the senses. It literally took me 1.5 years of serious mental deconstruction, analysis and questioning before I was able to entertain the thought that my body, brain and indeed the world around me were a _product_ of the mind, and did not produce the mind itself. Consider that. It's your whole system of thinking about the world turned upside down.

In building a metaphysical system for the world and individual, the Indians defined an individual as a 'body-mind-intellect' (BMI) experiencing 'perceptions-emotions-thoughts' (PET). The former they termed the 'relative subject' and the latter the 'relative object'. It is easy to see that these cover EVERY aspect of human experience.

I am taking a shortcut and would be happy to clarify further, but there is a facet of the spiritual path in India known as bhakti yoga (the science of union with the Divine through emotion) which targets the mind in its emotional capacity. It is not uncommon, then, to use an IDOL (which is NOT the same as the goal itself) to represent the goal of absolute subjectivity which one is striving for and cannot be perceived, felt or conceived via the BMI and their PET.

This representation, symbol, takes the form of a goddess/god whose strange and crazy figures and ornamentation are intended to initially (1) evoke questions in the seeker as t to their presence and (2) inspire more devotion once their significance is intellectually understood.

Coming back to the genius mathematician. If one traces one's own moments of lucidity or inspiration, one would find that they come at times when the mind is relatively calm, composed and engaged in a subject. The extrapolation of this is the state of bhakti yoga (complete absorption in the ideas represented by the deity), and most adherents thereto ascribe the 'doing' of their thoughts, words and deeds to the deity Itself, to efface their egos and move closer to their ideal of transcending the world itself.

Now look at that simple statement: "These ideas were revealed to me by my God" [paraphrase]. The simplistic notion that an apparition appeared before him or in his mind to provide him with these incredible insights does not hold to a scientifically-inclined mind. But the notion that he had trained his mind so highly that he was able to become absorbed single-thoughtfully upon a subject to the extent that he was able to perceive subtle nuances therein, indeed subtle nuances that would seem 'magical' to a non-trained observer is not unthinkable. That he then ascribed his discoveries to a deity is something that happens daily in Indian life and as previously covered is intended as an effacement and ultimate transcendence of the ego and the world it projects.

I can highly recommend A Parthasarathy's Vedanta Treatise if you are interested further.

Though it's more historical fiction than non-fiction, I would also recommend: http://www.amazon.com/Indian-Clerk-Novel-David-Leavitt/dp/15...

This paragraph makes no sense to me:

They found that while the outputs of a mock modular form shoot off into enormous numbers, the corresponding ordinary modular form expands at close to the same rate. So when you add up the two outputs or, in some cases, subtract them from one another, the result is a relatively small number, such as four, in the simplest case.

You have two functions both of which grow to infinity, one of which is much better understood than the other. It turns out that if you subtract the two functions, they balance out perfectly so you end up with something converging to 4 instead of going to infinity.

When you subtract any two functions that are exponential, the answer is 4?

Not quite. Take two exponentially growing functions, say e^x and e^2x over the interval [0,infinity). as x-> infinity both grow without bound. So does the difference of e^(2x)-e^x because the former is just so much larger than the second. They both approach infinity, but at different rates! If the difference between two functions (in the limit) converges, that means that the two functions diverge at the same rate (i.e. both with the end behavior of e^(ax) for some constant a) This is all a little hand wavy though but I hope that clears things up.

I wish the article had talked more about what it had to do with black holes. Can anyone here explain that to me?

The authors of the paper have chosen to use N=4 topology, or supersymmetry in 4 dimensions, to simplify modeling how black holes with multiple centers decay.

In these special cases the mock modular forms, described by Ramanujan on his death bed in 1920 before anyone was talking about black holes, provide a counting function to describe the black hole's world line in string theory. In other words, they can model what's happening inside the black hole.

Using the mock modular forms was attractive because it satisfies the desire to use the holography theories about black holes to model what happens to information as matter crosses the event horizon.

The authors further justify their choice by explaining how modular forms are already used to describe characteristics of black holes in string theory, such as its Fourier coefficients (component waves) and how they change as an object crosses the wall.

The difference between a mock modular form and a modular form is that a modular form is holomorphic is differentiable at all points in Real space at infinity. A mock modular form is meromorphic, it is differentiable at almost all points in Real space at infinity. The authors account for their counting function being meromorphic by introducing a 'shadow' factor.

Edit: in particular they are complex differentiable. Wikipedia has a nice image where you can see a meromorphic function conforming to space and then a few spots where it jumps (is not continuous). A holomorphic function would conform smoothly all over. http://en.wikipedia.org/wiki/Meromorphic_function

Here's a link to a paper on this: http://arxiv.org/abs/1208.4074. Explaining it is well beyond my capacity, though.

For a layman the abstract sounds like a computer-generated hoax.

You could say that about any field that one is unfamiliar with. Heck, the most basic art concepts could seem like so to a blind man. Mathematicians have worked for ages on simplifying the concepts so every term is there because it helps a trained mathematician understand what is going on.

It seems genuine, or at least more sophisticated than computer-generated abstracts usually are. It seems to repeatedly mention the same concepts which is something usually lacking in computer-generated abstracts, though it does mix seemingly tenuously related topics.

I have no idea what any of it means though.

The abstract, although it's also far beyond me, actually sounds even more impressive than the article lets on.

Modern mathematicians find it useful for calculating properties of black holes. Ramanujan had no more to do with black holes that, say, Newton who invented calculus after all, and that's used for black-hole describing too. Its kind of a meaningless fact to toss into the article.

I hope we have handful of Ramanujan's DNA stashed somewhere so, once we accept (and achieve) human cloning, we can clone thousands Ramanujans in hopes of some of them developing such beautifully abnormal brain.

I think it would be poetic justice if in a triumph of naturalism we actually cloned him and got only boring copies because the real Ramanujan actually got his insights supernaturally, like he claimed...

There might be additional factors like mitochondrial DNA or some epigenetic factors or some accident during development of his brain. Getting 1000 boring Ramanujan clones would prove that the our skill is lacking not that we couldn't replicate a goddess.

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