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bio website williewong.wordpress.com
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age 31
visits member for 5 years, 3 months
seen 11 hours ago

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


1d
comment John Nash's Mathematical Legacy
In terms of keywords (since I haven't seen it mentioned on this page yet), the Nash-Kuiper theorem is one of the examples that led to the development of the h-principle. (Of course, those who would recognize this keyword probably would also know about Nash's contributions, so maybe it is not that key to include it...)
May
23
awarded  Nice Answer
May
5
answered Null geodesic congruence
May
4
comment Avoiding mean-curvature 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.
May
4
comment Avoiding mean-curvature 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".
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 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?
Apr
29
answered Derivative of a time evolution operator w.r.t. a parameter
Apr
29
comment Derivative of a time evolution operator w.r.t. a parameter
In the case that $H$ is time-independent (so independent of $x$), I think your formula is the same as Theorem 2.19 in section IX.2.6 in Kato's Perturbation theory.
Apr
29
comment Derivative of a time evolution operator w.r.t. a parameter
That formula looks awfully familiar. If I remember where I've seen it before I'll post a reference.
Apr
29
reviewed Close Suggestions on the best introductory Model Theory texts
Apr
29
reviewed Leave Open compact inclusion of domains of unbounded operators
Apr
28
comment Nonlinear Schrödinger blow-up for non radial solutions
The blow-up result by itself is not so much the emphasis of the paper, by the way. If you are willing to consider more regular initial data (which gives you (weighted) $L^2$ control) the finite-time blow-up result (not needing radial symmetry) is classical and due to Glassey in '77 (among others). In some ways Theorem 2 (the log time lower bound) of the paper is much more interesting than Theorem 1, and I am not aware of generalisations to the non-radial case.
Apr
28
comment Nonlinear Schrödinger blow-up for non radial solutions
There is very clearly a typo in your statement of the result, since $n+3 \not\leq n+2$. I am having trouble accessing Pierre's webpage right now (it is not responding), but I am assuming you are talking about the AJM paper from 2008?