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

115 points by mrkuchbhi 14y ago

Tycho 14y ago
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.
- rfurmani 14y ago
  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.
  - powertower 14y ago
    When you subtract any two functions that are exponential, the answer is 4?
    - Slickarango 14y ago
      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.
      - deleted 14y ago
        [deleted]
whatshisface 14y ago
I wish the article had talked more about what it had to do with black holes. Can anyone here explain that to me?
- Jun8 14y ago
  Here's a link to a paper on this: http://arxiv.org/abs/1208.4074 http://arxiv.org/abs/1208.4074. Explaining it is well beyond my capacity, though.
  - mvzink 14y ago
    The abstract, although it's also far beyond me, actually sounds even more impressive than the article lets on.
  - d0mine 14y ago
    For a layman the abstract sounds like a computer-generated hoax.
    - rfurmani 14y ago
      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.
    - jlgreco 14y ago
      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.
- frugalfirbolg 14y ago
  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 http://en.wikipedia.org/wiki/Meromorphic_function
Xcelerate 14y ago
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/0671750615/ref=sr_1_1?ie=UTF8&qid=1355776982&sr=8-1&keywords=the+man+who+knew+infinity http://www.amazon.com/The-Man-Who-Knew-Infinity/dp/067175061...)
- jlgreco 14y ago
  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.
  - nsomaru 14y ago
    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.
    - catshirt 14y ago
scotty79 14y ago
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.
- prewett 14y ago
  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...
  - scotty79 14y ago
    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.

115 points by mrkuchbhi 14y ago

Tycho 14y ago

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.

rfurmani 14y ago
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.
- powertower 14y ago
  When you subtract any two functions that are exponential, the answer is 4?
  - Slickarango 14y ago
    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.
    - deleted 14y ago
      [deleted]

whatshisface 14y ago

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

Jun8 14y ago
Here's a link to a paper on this: http://arxiv.org/abs/1208.4074 http://arxiv.org/abs/1208.4074. Explaining it is well beyond my capacity, though.
- mvzink 14y ago
  The abstract, although it's also far beyond me, actually sounds even more impressive than the article lets on.
- d0mine 14y ago
  For a layman the abstract sounds like a computer-generated hoax.
  - rfurmani 14y ago
    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.
  - jlgreco 14y ago
    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.
frugalfirbolg 14y ago
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 http://en.wikipedia.org/wiki/Meromorphic_function

Xcelerate 14y ago

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/0671750615/ref=sr_1_1?ie=UTF8&qid=1355776982&sr=8-1&keywords=the+man+who+knew+infinity http://www.amazon.com/The-Man-Who-Knew-Infinity/dp/067175061...)

jlgreco 14y ago
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.
- nsomaru 14y ago
  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.
  - catshirt 14y ago

scotty79 14y ago

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.

prewett 14y ago
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...
- scotty79 14y ago
  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.