14,158 reputation
144106
bio website math.poly.edu/~yang
location New York, New York
age
visits member for 5 years, 1 month
seen 1 hour ago

Nov
24
reviewed Close Simplicial version of the A-infinity operad
Nov
24
reviewed Close Are compact complete geodesics closed?
Nov
24
reviewed Leave Open Reference for a strong intermediate value theorem for measures
Nov
24
reviewed Leave Open Can one properly embed a differential manifold into numerical space of double dimension?
Nov
22
answered Books on the analysis of hyperbolic partial differential equations
Nov
15
comment When is a `1-form' with continuous coefficients exact?
Doesn't the Poincare lemma hold for distributions? And if the weak derivatives of $u$ are continuous, it follows that $u$ is $C^1$.
Nov
13
comment Why does the Lefschetz Operator not Square to Zero?
Actually we can say more about the exterior product of $K$ with itself. $K^n$ is the volume form and therefore is nonzero everywhere.
Nov
9
answered Which are the recommended books for an introductory study of complex manifolds?
Nov
9
awarded  Good Answer
Nov
4
awarded  Popular Question
Nov
2
comment Is there a complex structure on the 6-sphere?
That's too bad.
Nov
2
comment Lower regularity version of Moser's theorem on volume elements
If you can somehow make the flow also satisfy either a parabolic PDE or a transverse elliptic PDE, then you would achieve the regularity you want.
Nov
2
comment Lower regularity version of Moser's theorem on volume elements
I suspect Robert is correct about the regularity. When you integrate a flow, you gain a derivative in the time direction only and not in any transverse ("spatial") directions. This can be seen from the proof of the existence and uniqueness of parameterized ODE's.
Oct
28
comment Riemann's formula for the metric in a normal neighborhood
Of the modern proofs, I like the Jacobi field approach the best. You start by using the distance from a given point as one co-ordinate function and showing that you can extend the angular variables from the tangent space at the origin. This is most easily done using Jacobi fields $J_1, \dots, J_n$,. The metric in these co-ordinates is given by $g_{ij} = J_i\cdot J_j$, so its Taylor expansion is easily calculated using the formal solution to the Jacobi equation.
Oct
27
comment Riemann's formula for the metric in a normal neighborhood
Martin, I find Christoffel symbols highly unenlightening, so that's not a simple definition to me. It seems to me that both normal co-ordinates (up to second order only) and the Riemann curvature tensor arise pretty naturally if you search for co-ordinates that simplify the 2nd order Taylor expansion of the metric as possible. But since the Hessian of the metric is a 4th order tensor, that's still a bit tricky.
Oct
27
comment Riemann's formula for the metric in a normal neighborhood
It seems to me that it's first necessary to find a "simple" definition of the Riemann curvature tensor. I'm not sure how Levi-Civita defined it.
Oct
27
comment Uniqueness of scalar curvature
Robert, oops. You're right. I once knew stuff like that.
Oct
27
comment Uniqueness of scalar curvature
I don't have a proof, but it seems to me that more is true. Namely, any scalar function defined using a Riemannian metric and its derivatives up to second order that is equivariant under diffeomorphisms has to be a function of the scalar curvature.
Oct
23
comment Why do we teach calculus students the derivative as a limit?
Depends on what you mean. If you need it for your work, you'll learn it properly through what you do. That's always the best way, because you understand why it's needed. If you just want to learn it, you succeed either by having a really good teacher or studying it on your own (and not being satisfied until you understand it inside out).
Oct
18
awarded  Yearling