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Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

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Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#11

Post by jlgreco »

Xcelerate wrote: post #9

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... )

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.

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#12

Post by frugalfirbolg »

whatshisface wrote: post #2

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

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#13

Post by nsomaru »

jlgreco wrote: post #11
Xcelerate wrote: post #9

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... )

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…

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.

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#14

Post by rfurmani »

Xcelerate wrote: post #9

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.

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#15

Post by powertower »

rfurmani wrote: post #7
Tycho wrote: post #4

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?

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#16

Post by kranner »

Xcelerate wrote: post #9

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... )

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

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#17

Post by Slickarango »

powertower wrote: post #15
rfurmani wrote: post #7

Earlier quoted context omitted.

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.

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#18

Post by catshirt »

nsomaru wrote: post #13
jlgreco wrote: post #11

Earlier quoted context omitted.

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…

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…

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

Re: Modern math solves Ramanujan’s ‘vision’ - may clarify black holes

#20

Post by [deleted] »

Slickarango wrote: post #17
powertower wrote: post #15

Earlier quoted context omitted.

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.…

[deleted]
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