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John Nash's Mathematical Legacy
added 152 characters in body 
May 25 
awarded  Good Answer 
May 24 
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John Nash's Mathematical Legacy
Sorry. It's already in more or less a random order. I also hope everyone has the sense to take anything said by Gromov far more seriously than by me. 
May 24 
awarded  Nice Answer 
May 24 
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John Nash's Mathematical Legacy
I found the paper: Le problème de Cauchy pour les équations différentielles d'un fluide général. (French) Bull. Soc. Math. France 90 1962 487–497. The abstract seems to indicate that there were some errors in the paper? Any chance you could say more about this paper? 
May 24 
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John Nash's Mathematical Legacy
I'm not familiar with Nash's work on fluid flows. Do you have a reference for this? 
May 24 
answered  John Nash's Mathematical Legacy 
May 23 
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Vector field built from connection and metric
I have only vague thoughts. The connection defines geodesics. If you start with a frame of tangent vectors at a point, you can study how that frame changes when you parallel transport it along the connection's geodesics. And maybe if you differentiate twice, you get something useful. Your vector field appears to involve a trace of the curvature (so a Riccilike tensor), so maybe you want to look at how the determinant of the frame changes relative to the volume form of the Riemannian metric. Maybe this vector field measures the direction of maximal rate of change of this? 
May 22 
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Does the RiemannChristoffel curvature determine the connection?
eudml.org/doc/74779 
May 20 
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Specifying $L^p$ norms of derivatives
These inequalities are just 1dimensional versions of what are known as GagliardoNirenberg inequalities, which are easy to prove using integration by parts and the Holder inequality. In particular, given any $1 < p < \infty$ and $0 \le i \le j \le k$, there is such an inequality. 
May 8 
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Derivative of a conjugation of matrices
Probably the easiest way to figure this out is to write out the second order expansion of each function. 
May 3 
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Define “Mathematics Colloquium”?
My experience is that if you manage to convince the speaker to prepare a talk for an audience of nonmathematicians, then it will be a perfect talk for mathematicians who are not experts in the field. 
Apr 23 
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Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
Ok. Now try to do the global result you want in the holder case by following DeTurck's proof. Where does it break down? It has to because otherwise DeTurck would have stated and proved the global result himself. 
Apr 22 
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Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
Here's my suggestion: First, ignore the subtleties of infinitedimensional spaces and manifolds. Recall that the standard finitedimensional implicit function theorem can be used to verify whether an equation defines a submanifold or not.. Work out why your main question can be answered by an analogous infinitedimensional implicit function theorem. Now figure out what you need in order to use the Banach or tame Frechet (i.e., NashMoser) implicit function theorem to get what you want. 
Apr 22 
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Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
Perhaps you should first make sure you understand the difference between a Banach space and a Frechet space and which function spaces are which. That will then provide a guide to what kind of infinite dimensional manifold the space of metrics is. (The answer is "it depends") 
Apr 21 
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Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
The space of solutions to a nonlinear elliptic partial differential equation does not need to be a Frechet manifold. You need to know that some kind of constant rank condition holds. That's why DeTurck had to restrict to a sufficiently small ball. My guess is that a sufficient condition for the "kernel" of an underdetermined elliptic operator to be a Banach or tame Frechet manifold is for the linearized operator to be surjective. 
Apr 21 
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Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?
One has to be careful about defining what the "kernel" is of a nonlinear partial differential operator. 
Apr 20 
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Moser estimates?
Nice and simple. The more involved Moser iteration or Schauder is needed only if you need weaker regularity assumptions on the coefficients. For example, for a nonlinear PDE. 
Apr 19 
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Selfcontained book on Ricci Flow/Geometric Analysis
Since the Ricci flow is a PDE, it's not realistic to learn about it without knowing any PDE theory at all. But you don't need much. My advice is to study the books by Chow et al and consult PDE books only as needed. 
Apr 19 
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Moser estimates?
Why is right side in $C^\alpha$? 