Difference between revisions of "Scale calculus problems"
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a) find a scale-Banach space <math>\mathbb{E} = (E_\ell)_{\ell\geq 0}</math> with <math>E_0=C^1(S^1,\mathbb R^2)</math> | a) find a scale-Banach space <math>\mathbb{E} = (E_\ell)_{\ell\geq 0}</math> with <math>E_0=C^1(S^1,\mathbb R^2)</math> | ||
− | b) show that <math>\widetilde\sigma: S^1 \times \mathbb{E}\to\mathbb{E}</math> is <math>sc^1</math> and calculate <math>T\widetilde\sigma</math> | + | b) show that <math>\widetilde\sigma: S^1 \times \mathbb{E}\to\mathbb{E}</math> is <math>sc^1</math> and calculate the linearization <math>T\widetilde\sigma = (\widetilde\sigma, d\widetilde\sigma</math> |
c) iteratively show that <math>\widetilde\sigma</math> is <math>sc^k</math> for all k | c) iteratively show that <math>\widetilde\sigma</math> is <math>sc^k</math> for all k | ||
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'''YOUR SOLUTION WANTS TO BE HERE''' | '''YOUR SOLUTION WANTS TO BE HERE''' | ||
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== scale-smooth structure on <math>\mathcal{B}</math> == | == scale-smooth structure on <math>\mathcal{B}</math> == |
Revision as of 20:21, 22 May 2017
Scale calculus was developed to make reparametrization actions on function spaces smooth. All the key issues and ideas can already be seen at the example of acting on a space of nonconstant functions on , for example the shift action
The purpose of the following exercises is to give the quotient of S^1-valued functions homotopic to the identity, modulo reparametrization by shifts, the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.
Contents
differentiability of shift map
Compute the directional derivatives of the shift map , first at , then at .
Next, understand how to identify , and write down a conjectural formula for the differential .
Then recall the definition of differentiability in terms of uniform linear approximation and see whether it can be satisfied at any base point .
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local charts
Verify that has local Banach manifold charts modeled on . To set up a chart near a given point, pick a representative with and consider the map
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transition maps
For with representatives but determine the transition map .
Then find an example of a point and direction in which this transition map is not differentiable.
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alternative Banach norms
Explain why working with , H\"older, or Sobolev spaces does not resolve these differentiabliity issues. Possibly come up with other ideas for a Banach manifold structure on .
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reparametrization on Riemann surfaces
Explore the same questions for a space of nonconstant functions on a Riemann surface modulo the reparametrization action of a nontrivial automorphism group (e.g. ).
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scale smoothness of the shift map
Show that the shift action is a scale-smooth map by the following steps:
a) find a scale-Banach space with
b) show that is and calculate the linearization
c) iteratively show that is for all k
Note: The actual map above is a restriction of the map to the subset , where we view .
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scale-smooth structure on
Now use the charts and transition maps discussed above to equip with the structure of a scale-manifold (i.e.\ modeled on a scale-Banach space, with scale-smooth transition maps).
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