What is Topology, and what's the big deal?

Discussion in 'Physics & Math' started by arfa brane, Nov 21, 2014.

  1. QuarkHead Remedial Math Student Valued Senior Member

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    Yes, and the most interesting and most-studied topological groups are the Lie groups of continuous symmetries. These groups are manifolds.

    They are of more than "passing interest" to mathematicians, they are relevant in many physical applications, including the favourite topics here
     
    brucep likes this.
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  3. arfa brane call me arf Valued Senior Member

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    The structure of graphs and knots seems to be related, though, at least if you construct a graph which is a knot shadow, there is a knot diagram related to it.
    About the topology of connected graphs:
    It seems to be saying that edges as connections have the same meaning as connectedness between points in open/closed sets?
     
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  5. tashja Registered Senior Member

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    Hi, Arfa. I sent your question to Prof. Tim Cochran along with a link to the thread, and here's what he said:
     
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  7. Farsight

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    Good man tashja. I thought it was good of Tim Cochran to get back to you, and with such a clear response too.

    I think there's a bit more too it than that. You start with Dirac's belt, which is essentially a Möbius strip, and you think of it as an bicycle inner tube. Then you "inflate" it into a torus. Have a look at Adrian See Adrian Rossiter's Antiprism website, and his torus animations. See the ring torus gif:

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    It's the unknot, but it's dynamical, and it's got a twist. And then you can "inflate" the torus, so it looks like the horn torus:

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    Then by allowing surfaces to pass through one another you can inflate further to the spindle-sphere torus:

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    Topologically speaking, this is still a torus, but geometrically speaking, it's a sphere. And as you know I think this is a fair depiction of the central portion of an electron, the "eye of the storm" as it were:

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    Remember, you combine the electron's radial "electric field lines" with the concentric "magnetic field lines" to depict the electromagnetic field lines, whereupon electrons and positrons move linearly and rotationally like a cyclone and anticyclone. As for why this has never come out of TQFT, I don't know. I know Michael Atiyah knows about it, Qiu-Hong Hu talked to him about it at ABB50/25. Here's his poster:
     

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    Last edited: Nov 27, 2014
  8. PhysBang Valued Senior Member

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    Wow, so Farsight's answer to a discussion of mathematics was to present the equivalent of fingerpainting.
     
  9. QuarkHead Remedial Math Student Valued Senior Member

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    Agreed.

    Farsight's understanding of topology - specifically topological spaces, topological groups and Lie groups - reduces to childish animations and a personal agenda. It's pathetic. More importantly, since it happens routinely, it gives the impression he is deliberately trying to de-rail threads where perfectly serious people are trying to be perfectly serious about physics and/or mathematics

    Surely this cannot be so?

    *wink*
     
  10. Farsight

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    My post was relevant to what rpenner was saying about the unknot and visualization, and to TQFT. It was a sincere contribution. Unlike PhysBang's.

    Methinks you're carping because you're useless at explaining things, and you can't bear it when somebody who isn't makes a contribution.
     
  11. QuarkHead Remedial Math Student Valued Senior Member

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    I agree I am useless at explaining things to people who have no understanding of mathematics, have no interest in learning any, and fail to understand that even ancient physics (let alone modern physics) gets its precision precisely through the use of mathematics

    Note mathematics - not the images and analogies like the cardboard leaves of the Ladybird books I used to read with my girls when they were little
     
  12. Farsight

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    3,492
    I understand plenty of mathematics, and its role in physics. And you're useless at explaining things full stop. For example this post was as clear as mud. And this thread was cringeworthy. You blather and babble, you throw out expressions without defining your terms, like some mage making smoke-and-mirror incantations to impress the mugs, you take a hubristic sneering pride in making everything opaque, and you clearly resent other posters who know what they're talking about. Mend your ways.

    Now can we get back on topic please? What is topology, and what's the big deal?
     
  13. PhysBang Valued Senior Member

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    That's a lot of what some would call "projection". I would just say, as others have before me, that the pot is calling the kettle black.

    You are trying to turn this discussion to your theory, wholly unproved and with no ability to produce predictions or descriptions that match the behavior of physical systems, that an electron is a single whirling photon. You use the same pictures over and over again. Unless you can truly show that your pictures add "a bit more too it than" (your words) to "a good basis in foundations of topology would seem to be necessary to make headway in or even to make sense of knot theory" (the words you quoted), you are merely derailing this thread in an attempt to glorify yourself.
     
  14. Farsight

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    I said, can we get back on topic please? What is topology, and what's the big deal?
     
  15. arfa brane call me arf Valued Senior Member

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    I'd like to see more about this "peculiar winding hole".
    I haven't seen knots described this way, including in "Knots and Physics", in which the word "topological" is found but once in the index. Then again this book is aimed at postgrads, and the author warns that it's a "rapid ascent up the side of a cliff".

    I'm not pretending that if I read the whole thing I'll have some more intuitive grasp of quantum field theories . . .
     
  16. arfa brane call me arf Valued Senior Member

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    The post in that link is plenty clear, if you understand a little set theory. Your diatribe only seems to reveal that you aren't very mathematically inclined. perhaps you think you cam "skip over" the mathematical descriptions and just "get to" the big ideas. It doesn't work that way, unfortunately.
     
  17. Farsight

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    I understand set theory. And I'm not saying one should skip over mathematical descriptions. As for diatribes, it was Quarkhead who posted a diatribe in response to my sincere on-topic post. Now can we get back on topic please? As you know it's What is topology, and what's the big deal? Perhaps you'd like to try and answer your own question, using the responses you've had so far.
     
  18. QuarkHead Remedial Math Student Valued Senior Member

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    1,740
    The question was answered exhaustively in posts #3, #8, #14 ,#15, #17 &c. What more do you expect? (kids cartoons aside}

    PS Moderators I recommend a perma-ban for Farsight for

    a) Deliberately de-railing threads he doesn't understand in order to divert them to his own agenda

    b) Insulting knowledgeable and/or thoughtful members who try to help those who ask a perfectly reasonable question, and seem willing to listen to informed responses

    c) Consistently lying about his own abilities and qualifications

    d) Rather than make a reasoned argument, instead quoting often unsafe sources, selectively quoting apparently safe sources and misunderstanding both of these

    e) Contaminating the minds of those who wish to learn physics and/or mathematics, especially those who may be just over the age of consent (according to forum rules)
     
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  19. brucep Valued Senior Member

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    4,098
    It might appear that there was a problem when you didn't offer a response to rpenner final post. He just asked you to provide a text that supported the terminology. So I got the impression you might not be able to. Otherwise who gives a crap.
     
  20. Farsight

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    3,492
    Perhaps arfa can give a digest of all those posts and explain What is topology, and what's the big deal? You know, to demonstrate that he understood them? Only he didn't respond to them.

    I'm not de-railing threads. I gave a sincere response to rpenner's post. You're exhibiting petty jealously. You're derailing this thread.

    You did that in this post.

    I have never lied about anything. Would you care to back up your assertion?

    Rubbish. I make reasoned arguments, like this, and I back them up with hard scientific evdience and references to Einstein and Minkowski and Maxwell and others. They said what they said.

    I'm not contaminating any minds. I'm not some my-theory guy. I'm quoting Einstein etc. I'm giving the physics. You've admitted that you're not a physicist. And I have had to correct you from time to time. Only you find that embarrassing and want me banned. And you aren't complaining about the ad-hominem trolls? Boy, you sure are cute. Oh, by the way, I'm John Duffield, and I live in Poole in the UK. What's your name?
     
  21. PhysBang Valued Senior Member

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    There are numerous examples of you lying about the content of your own posts on this very message boards. At this point, you have a well-earned reputation on the internet as a liar; you have done nothing to rehabilitate this image.
     
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  22. someguy1 Registered Senior Member

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    727
    The inexorable law of thread degeneration is in full display here. Maybe I'm too new around here to know about these ancient animosities, but it certainly detracts from my enjoyment of this site. Take it to Craigslist if you want to sit around and insult each other all day long.
     
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  23. arfa brane call me arf Valued Senior Member

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    I like this:
    http://en.wikipedia.org/wiki/Homeomorphism

    What it says about the meaning of this word: homeomorphism; it's what I've been thinking.

    As the quote says, topology is roughly about shape or form, i.e. "structure"; I'd say it's ok to consider it as a generalisation of geometric properties.

    You can do lots of things with a sheet of paper; some of these "morphisms" won't change some property of "the space". Mathematically a sheet of paper can be represented by a rectangular section of the real plane.
    If the "paper" can be bent or stretched (if it's plastic), then deformations to it won't preserve something (what though?), it will always be homeomorphic to a slice of the Euclidean plane (as long as no cutting or tearing is allowed).

    Draw some lines on the paper which are parallel; what kind of homeomorphisms will preserve the parallel property of these lines? etc.

    Next: some more mysterious words, such as "homotopy", and "ambient isotopy". But we already know that topos is Greek for "place", and we know about isotopes in the Periodic Table.

    And why does it all start with sets? A set is analogous to a sack or a bag with things in it, the set of things has no form as such; there is no order although you could assume there is a finite number of things in it. If you count the things in the bag, for instance, you necessarily "impose" some order on the set to do this.
    Prof Cochrane:
    "The second remark on your form" is a bit vague, but I assume refers to one of the responses on the first page of this thread.
     
    Last edited: Nov 29, 2014

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