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A mathematician specializing in nonlinear hyperbolic PDEs, geometric analysis of pseudo-Riemannian manifolds, and general relativity.


1h
reviewed Close Convexity and curvature for plane curves
1h
comment Convexity and curvature for plane curves
I've already given analogues of Alex and Robert Bryant's comments in the comment section of my answer on MSE. See also the link that I posted to an earlier MSE discussion concerning upgrading local to global convexity. So I don't think there is a point to migrating this to MSE.
1h
comment Avoiding Ricci-flow dumbbell neck-pinch by inflating a surface
Hmm... I was being slightly imprecise in my previous comment. So lest you be misled: what I said about the inflation is more-or-less true (the width of the cylinder will be similar to, but not exactly the same as, the width of the head) if you assume the handle is sufficiently short and skinny. // Perhaps a better illustration of the problem is that the wIMCF allows changes in topology in its definition: if you start with two spheres, or a skinny torus, both will eventually become spheres under this flow.
2h
comment Avoiding Ricci-flow dumbbell neck-pinch by inflating a surface
The problem with the wIMCF is that the handle of the dumbbell will get immediately inflated to the cylinder the same width as the heads of the dumbbell. If you are happy with that then great. But as Otis noted, the definition of the wIMCF allows this kinds of jumps which is not very "flowy".
6h
comment derivative of the one parameter family of Riemannian metrics
Also, instead of typing "Dear Wong", you can make use of the comment-reply mechanism of this website to notify me of a response to my comments.
19h
comment derivative of the one parameter family of Riemannian metrics
(If you do so, feel free to leave a comment below with a link to the Math.SE question. When I have some time I'll try to give an explanation in slightly more detail.)
19h
comment derivative of the one parameter family of Riemannian metrics
It is no different from saying a one parameter family of anything. You can view a function $f:\mathbb{R}^2 \ni (x,y) \mapsto f(x,y)\in \mathbb{R}$ as a one parameter family $F(x):\mathbb{R}\ni y \mapsto f(x,y) \in \mathbb{R}$ of functions. And you can still take partial derivatives relative to $x$ without thinking about the topological vector space in which the functions on $y\in \mathbb{R}$ lives. // Anyway, if you cannot understand the pointwise definition I wrote, this is not a research level question. You should ask at Math.stackexchange.com instead.
2d
comment derivative of the one parameter family of Riemannian metrics
Why do you want to "index the set $X$"? Whether $f$ is or is not surjective is entirely inconsequential to the definition of the derivative (just think about the derivatives for real valued functions).
2d
comment derivative of the one parameter family of Riemannian metrics
Try to register your current account and then follow the instructions on this page to recover access to your other account.
2d
comment derivative of the one parameter family of Riemannian metrics
In any case, your question makes no sense: why are you fixing $f$ to be a surjective map? // If you want to understand the Ricci flow, you need to understand that the computation of the derivative of the metric **at a fixed point $p\in M^n$** requires no fancy machinery. The space of positive definite bilinear forms on $T_pM$ is a finite dimensional linear space and you can use any norm you want, since they all give the same topology and are comparable.
2d
comment derivative of the one parameter family of Riemannian metrics
Are you the same user as the one who posted mathoverflow.net/q/204344/3948 ? Instead of posting a new question, why not address the many comments that have been left there for you?
May
1
reviewed Close How to characterize a linearly-constrained subspace in a projection
Apr
30
comment Is there a relationship between Fourier transformations and cotangent spaces?
The first part of your question strongly reminds me of microlocal analysis. But unfortunately I am not able to make the connection precise. Hopefully someone who can will come along and explain.
Apr
30
comment Shot down the conjecture of Riemann?
I'm voting to close this question as off-topic because there does not appear to be an actual question in this post.
Apr
30
comment On the definition on the Ricci flow
Though I should also remark that in the case of the heat equation there is the general theory for accretive operators that apply; and the wave equation is Hamiltonian. So some of the nice features of PDEs can be extracted and abstracted to become ODE in Banach space statements. But my point is a general one about not making life more difficult than necessary for oneself.
Apr
30
comment On the definition on the Ricci flow
Just to give an example: suppose you want to think about the linear wave (or heat) equation as a second order ODE on the Sobolev space $H^1$, you have the problem that the Laplacian is not a bounded operator and so you cannot appeal to the usual theory of infinite dimensional ODEs with continuous right hand sides. The local existence theory for these equations come in many flavours (energy estimates, fundamental solutions, Fourier representations), but they all make use of details of the equation that are beyond simply "ODE in Banach space" properties.
Apr
30
comment On the definition on the Ricci flow
@DavidSpeyer: My point is that the Ricci flow is not merely an ODE in an infinite dimensional vector space. The equations the comes from a PDE are more special and some generalities one has to deal with when dealing with infinite dimensional ODEs can be avoided.
Apr
30
comment On the definition on the Ricci flow
There seems to be some fundamental misunderstanding about the Ricci flow on your part. In terms of "how can we define the derivative": it is basically no different on how you can define the Lie derivative of an arbitrary tensor field. In particular, the Ricci flow is not an ODE with values in some infinite dimensional vector space, it is a PDE with values in a finite dimensional vector bundle. If this doesn't answer your question, please edit to clarify what you actually mean.
Apr
30
comment how wiggly is a generic level set?
BTW, contrary to the question in the title, the level sets of the functions given in my first comment are close to flat.
Apr
30
comment how wiggly is a generic level set?
A simple upper bound is $\Lambda = O(\epsilon^{-1})$. Take $f(x) = \sin (2\pi x_1 \epsilon^{-1})$. It has a single Fourier component of size $\approx \epsilon^{-1}$, and its zero set is within $\epsilon/2$ of any point of $\mathbb{R}^n$. Furthermore since its derivative does not vanish on said level set, for any "sufficiently small" perturbations the level sets move only a tiny bit (implicit function theorem). For the implicit function theorem to apply, you can use a very coarse topology (that of $C^k$ functions for example). I suppose you are asking whether this is sharp?