Timeline for Is the space of nondegenerate classical paths connected?
Current License: CC BY-SA 3.0
18 events
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| Jul 9, 2013 at 8:44 | history | edited | Willie Wong | CC BY-SA 3.0 |
replaced latex.mathoverflow with mathjax http://meta.mathoverflow.net/a/385/3948
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| Dec 2, 2009 at 22:24 | vote | accept | Theo Johnson-Freyd | ||
| Nov 12, 2009 at 17:44 | answer | added | Semyon Dyatlov | timeline score: 2 | |
| Nov 9, 2009 at 7:29 | comment | added | Andrew Stacey | Okay, so we've separated the infinite dimensional stuff from the actual question. That's good: I was worried (and it seemed as though you were too) that this discussion was distracting from the main point. I'd even be happy to delete all these comments if you wanted. As for discussing infinite dimensional stuff, my vote would be to start a page on the n-lab. But I'm sufficiently intrigued by the wave that if you have a wave account, I'll have a go (on the proviso that anything interesting gets transferred to the n-lab afterwards). | |
| Nov 8, 2009 at 23:06 | history | edited | Theo Johnson-Freyd | CC BY-SA 2.5 |
moved the question to the top, and added more formulae
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| Nov 8, 2009 at 22:16 | comment | added | Theo Johnson-Freyd | But my question can be asked without it, and then it's just a differential equations question, because I can certainly exhibit a bijection between the space of classical paths and the tangent bundle, and I really want the space of nondegenerate paths to be path-connected in that topology. I believe that nondegeneracy is an open condition, but Anton challenged me on that, because my heuristic argument did live in the infinite-dimensional world. Perhaps I will edit my original question to ask it without the infinite dimensional worries. | |
| Nov 8, 2009 at 22:15 | comment | added | Theo Johnson-Freyd | Yeah, the comments are the wrong place, especially since MathOverflow only displays the top few, and my comment is slightly too long, but the Answers also are the wrong place. Spin-off blog somewhere? Or, better, Google Wave? Anyway, I'm vaguely interested in the infinite-dimensional stuff. | |
| Nov 8, 2009 at 21:18 | comment | added | Andrew Stacey | Bleugh. Comments here are really a bad place to go into these things at length. So let me ask a quick question to see whether or not it's worth continuing this discussion: does the infinite dimensional stuff actually matter? Can you prove without using anything infinite dimensional that your space is finite dimensional? If so, then I'll shut up and stop distracting you from getting an answer to your question. If, on the other hand, you are interesting in infinite dimensional structures then we can find a more suitable place to discuss it - I'm always happy to talk about such stuff! | |
| Nov 8, 2009 at 5:36 | comment | added | Theo Johnson-Freyd | Andrew: Oh, a question. Where am I taking limits/colimits? I never really took the time to learn infinite-dimensional manifolds properly --- I only really ever think about function spaces. I tend to believe the "topology" is secondary to the smooth structure: I should tell you all the smooth maps from R into my space, this collection should satisfy some axiom, and the rest of the data follows from there. Anyway, there are other trivializations of P. For example, we know the smooth structure on Hom([0,1],R^n) is, and P is isomorphic to Hom([0,1],R^n) \times R. | |
| Nov 8, 2009 at 5:22 | history | edited | Theo Johnson-Freyd | CC BY-SA 2.5 |
added parentheticals
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| Nov 8, 2009 at 5:21 | comment | added | Theo Johnson-Freyd | Anyway, though, all of Andrew's questions are beside the point, because with whatever the correct definitions are, it's true that $C$ is naturally an open subset of $TR^n \times R$, and I'm asking about a subset of it. | |
| Nov 8, 2009 at 5:19 | comment | added | Theo Johnson-Freyd | Oops! Well, for any formal power series, there is a function with those asymptotics, so I can extend to a smooth function. So I'm not too worried. | |
| Nov 7, 2009 at 21:42 | comment | added | Andrew Stacey | Your paths are smooth so you can't "extend in a straight line". There is an extension map (old result of R. Seeley) which you could use, I suppose. But I'd like to see an actual definition of open sets rather than just "I want this close to that". Infinite dimensional manifolds are tricky things, and limits/colimits like the one you are taking can really mess things up. | |
| Nov 7, 2009 at 7:50 | history | edited | Theo Johnson-Freyd |
added a tag
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| Nov 6, 2009 at 22:07 | history | edited | Theo Johnson-Freyd | CC BY-SA 2.5 |
The answer was trivially no; I modified the question.
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| Nov 6, 2009 at 21:58 | comment | added | Theo Johnson-Freyd | @Andrew: hm, that's a good question. Let t, t' be nearby lengths, with t < t', and \gamma,\gamma' paths of length t, t'. I want \gamma,\gamma' to be close if the restriction of \gamma' to [0,t] is close to \gamma. This is equivalent to extending \gamma by a straight line (we are on R^d, so this makes sense) to a path of length t', and then asking for the extension to be close to \gamma'. This does allow functions like the Action to be smooth: its derivative in a direction that changes the length is its usual functional derivative plus a boundary term. | |
| Nov 6, 2009 at 20:38 | comment | added | Andrew Stacey | Apologies for being picky, but you say that P is an infinite dimensional smooth manifold. How are you patching together the pieces with paths of different lengths? | |
| Nov 6, 2009 at 19:59 | history | asked | Theo Johnson-Freyd | CC BY-SA 2.5 |