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J Noldus might use this new Rieffel theorem

  1. Aug 13, 2006 #1

    marcus

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    Johan Noldus call home.
    I think you may have a QG paper to write.
    Marc Rieffel (I remember him slightly from a seminar years ago) just posted this. IMHO it is an intuitive or conceptual result, possibly an important one in context, and it seems "down Noldus alley". Others here might conceivably find it useful too.
    http://arxiv.org/abs/math.MG/0608266
    Vector bundles and Gromov-Hausdorff distance
    Marc A. Rieffel
    55 pages

    Subj-class: Metric Geometry; Operator Algebras

    "We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov--Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. We also develop some computational techniques, and illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus and the two-sphere. Our topic is motivated by statements concerning monopole bundles in the theoretical high-energy physics literature."
     
    Last edited: Aug 13, 2006
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  3. Aug 13, 2006 #2

    marcus

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    to paraphrase. vector bundles are "catching"
    if two base spaces are similar or close in a certain sense (gromov-hausdorff near relatives) then a vectorbundle can act like a communicable disease or bad hairstyle or a taste for piercing and tattoos-----it can spread from one manifold to the other.

    if two spaces are grom-haus cousins and one of them has a vectorbundle, then the other one could very well show up the next day with a vectorbundle just like it---or a passable imitation.
     
    Last edited: Aug 13, 2006
  4. Aug 14, 2006 #3

    Haelfix

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    That actually seems fairly intuitive in the modern Ricci flow language.
     
  5. Aug 14, 2006 #4
    Marcus, you have to distinguish physics from mathematics. It is possible for someone to do nice math, which might have something to do with our world. Usually, it turns out not to be the case. I will make a bet here : a unification between GR and QM is not going to require more than mathematical tools developped prior to 1960 (Lorentzian geometry included).

    Careful
     
  6. Aug 14, 2006 #5

    selfAdjoint

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    So what do you think will be required?
    - Mathematical tools developed after 1960 (e.g. C-Y manifolds).
    - Physical tools developed any old time.
    - Other?
    - Some combination of above?
     
  7. Aug 14, 2006 #6

    marcus

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    Hi selfAdjoint, to whom is your question addressed? C. bets that math developed after 1960 will NOT be required.
     
    Last edited: Aug 14, 2006
  8. Aug 14, 2006 #7

    Hurkyl

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    You will win that bet. All mathematical tools, in principle, reduce to elementary logic. And since we had logic before 1960...
     
  9. Aug 15, 2006 #8
    I will specify even more: everything that will be required are difference equations and some approximations to smooth geometry (so I was clearly not speaking about reductions :-) ). Certainly no category theory and masturbation alike :cool:. I repeat myself : we need new physical insight not new, or more abstract mathematics.
     
    Last edited: Aug 15, 2006
  10. Aug 15, 2006 #9

    marcus

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    I hope this is a fair analogy: anything that goes on around here could, allowing for some circumlocution, be described in Anglo-Saxon
    or, perhaps more comprehensibly, in Canterbury Tales Middle English
    or, if absolutely necessary, in the language of Rudyard Kipling's Jungle Book.
     
    Last edited: Aug 15, 2006
  11. Aug 15, 2006 #10

    Hurkyl

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    Contrary to your belief, the point of "more abstract mathematics" is (usually) not so that people can bask in their own cleverness. Things are generally much clearer when you see past all of the things that obscure what's really going on -- one use of abstraction is to put the interesting things in the limelight, while hiding, or even eliminating the uninteresting stuff.

    You're quite likely right that new physical insight is needed -- but it's much more likely to come from someone who isn't drowning in a sea of minor details.
     
  12. Aug 16, 2006 #11
    **Contrary to your belief, the point of "more abstract mathematics" is (usually) not so that people can bask in their own cleverness. Things are generally much clearer when you see past all of the things that obscure what's really going on -- one use of abstraction is to put the interesting things in the limelight, while hiding, or even eliminating the uninteresting stuff. **

    Oh dear, here we go again :rolleyes: and please do not put words in my mouth. Honestly, I do not even find abstract mathematics clever per se. And indeed, differential calculus is very uninteresting but so difficult to capture ''with the naked eye'' that nobody succeeded yet in getting it back from almost nothing. :rolleyes: But, you know, I am asking this already for two months now : give an example with some physical insight (and please not nCob). Moreover, when I was fishing for the picture people had in mind, it became very silent...

    **
    You're quite likely right that new physical insight is needed -- but it's much more likely to come from someone who isn't drowning in a sea of minor details. **

    You mean the ``mathematical cloud'' which is covering all known formulations ?! Forget it, that is the bed time story told to little children.
    It is ok to substract infinities or put in a cutoff once in a while as long as you know what you are doing physically. It seems to me that those who believe that mathematical reduction to the essentials is going to save the day are very confused and drowning into a sea of details indeed. You cannot undress without losing the physical intuition for what you are doing, that only leads to nonsense.


    Careful
     
    Last edited: Aug 16, 2006
  13. Aug 16, 2006 #12

    Hurkyl

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    Maybe because your attitude has repulsed the people in this forum who are qualified to answer it? Actually, I seem to recall them answering, and then politely refusing when you tried to drag them into a quagmire. Whether intentional or not, you certainly project the image that you believe that if it's not done your way, then it must be worthless.

    I'm not talking about trying to put physics on a rigorous footing. I'm talking about how very cumbersome it is to try and talk about advanced concepts with primitive methods. "Arrays of functions that obey certain laws of transformation" is a notorious example of what I mean.


    (ref: http://amsglossary.allenpress.com/glossary/browse?s=t&p=11)
     
  14. Aug 16, 2006 #13
    ** Maybe because your attitude has repulsed the people in this forum who are qualified to answer it? Actually, I seem to recall them answering, and then politely refusing when you tried to drag them into a quagmire. Whether intentional or not, you certainly project the image that you believe that if it's not done your way, then it must be worthless. **

    Of course, that is the easiest way to dismiss questions you do not want to answer, blame it on the attitude of the other person. I certainly do not believe that if it is not done my way, then it must be worthless. However, that does not release anybody (including myself) from the burden of having to grasp those issues which torment this problem for 80 years. Understandably, it is your DUTY to have problems with the notion of old ideas dressed in a new jacket being advertised as a fruitful and even necessary step. I have asked for an *explicit* picture in good faith and indeed I think the idea of wormholes is ridiculous.

    By the way, I have often referred to work which is not in the line of my thought but which is obviously good and stimulating to think about.

    **
    I'm not talking about trying to put physics on a rigorous footing. I'm talking about how very cumbersome it is to try and talk about advanced concepts with primitive methods. "Arrays of functions that obey certain laws of transformation" is a notorious example of what I mean. **


    But I have just told you that such advanced concepts might not be necessary at all ! Feynmann even found differential geometry ``fancy schmanzy''. If you need a new physical idea, then the first thing you do is test it on an *easy* model in which you can make some fast ad hoc computations (make sure you get the Newtonian limit and orders of magnitude right). If the idea is succesful, then you go ahead in developping or using a better language. Turning this sequence around is not a good thing to do as will be acknowledged by almost any physicist.
     
  15. Aug 16, 2006 #14

    Hurkyl

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    I call foul; you said only mathematics developed prior to 1960! :wink:


    You entirely miss my point -- one of the prime uses of abstraction is to help move "physical intuition" (or whatever analog is appropriate for the situation) into the foreground.

    Differential geometry, actually, is a beautiful example of what I mean by this. Take, for example, the notion of a connection.

    Is a connection an array of numbers that transform according to a particular law?
    Is a connection something that acts like a derivative for vector fields?
    Is a connection a matrix of one-forms satisfying a particular transformation law?

    Unfortunately, yes. Those are (roughly) the three definitions of "connection" given in Spivak, the first being the most primitive, and by far the most obtuse.

    (I should point out that I don't intend any negative connotation with "primitive" -- I just can't think of a better word at the moment)


    But what would a category theorist say? He would say that a connection is a functor that takes a path and tells you how to parallel translate along it.


    This is the sort of thing I mean. The "abstract nonsense" viewpoint zooms in directly on the key idea. The "traditional" viewpoint can talk about parallel transport, but is weighed down with a lot of additional baggage.

    The "abstract nonsense" viewpoint is more agile too -- if you suddenly wanted to talk about a new notion of space that is a bunch of points with some sort of link between them, you have to rebuild a lot of infrastructure to adapt the primitive methods... but only a tiny bit of infrastructure to adapt the abstract methods.
     
  16. Aug 17, 2006 #15
    ***
    You entirely miss my point -- one of the prime uses of abstraction is to help move "physical intuition" (or whatever analog is appropriate for the situation) into the foreground. ***

    No, I don't :wink: see my comments on your example.

    ***
    Differential geometry, actually, is a beautiful example of what I mean by this. Take, for example, the notion of a connection.

    Is a connection an array of numbers that transform according to a particular law?
    Is a connection something that acts like a derivative for vector fields?
    Is a connection a matrix of one-forms satisfying a particular transformation law?

    But what would a category theorist say? He would say that a connection is a functor that takes a path and tells you how to parallel translate along it. ***

    Sure, but that is the intuition anyone who understands connections has. But the entire difficulty is to formulate a theory which starts from such general notion. You should find a dynamics which is going to interfere out the non-smooth connections on sufficiently large scales (and that seems almost impossible). In that sense, it is much better to start from a smooth connection (or one on a regular lattice), even though it might add additional baggage or break Lorentz invariance at very high energies.
    This is not obscuring any reasoning, it is just making it possible for you to test some better ideas first.

    Btw. substracting infinities was already done at the beginning of the 20'th century when people had to dispose of the infinite energies in radiation spectra.

    Careful
     
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