2,002 reputation
818
bio website homepages.ulb.ac.be/~samulisi
location Nantes, France
age
visits member for 3 years, 7 months
seen Oct 23 at 15:14

Postdoc working in symplectic/contact topology. I'm currently at the Université de Nantes.


Apr
11
comment Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?
Observe that if the initial data is contained in the interior of the orthant ($\Omega$ above), then, for small time, the solution curve stays in $\Omega$, and thus you can use the standard chain rule. What's not at all clear is what behaviour you want to see at the boundary.
Apr
11
comment Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?
I don't understand your motivation. Call $\Omega$ the (strictly) positive orthant. Then the non-negative orthant is $\overline \Omega$. If your initial data $x_0 \in \Omega$, then for small time, your solution curve $x(t)$ will live in $\Omega$. However, if you have initial data at the boundary $x_0 \in \partial \overline{\Omega}$, there is no reason for your integral curve to stay in $\overline \Omega$ even infinitesimally. In the former case, one expects $V(x(t))$ to make sense for short time, whereas in the latter, you don't.
Apr
11
comment Symplectic geometry as a prequisite for Heegaard Floer homology
Do you have someone in your department who can give you some guidance? there is a lot of symplectic geometry in both of these books that is not relevant to you and there are a number of technical points about holomorphic curves that are not explained in these books. A correct answer to your question, imo, requires more information about what you know and about what you are interested in doing. Unfortunately, even with this information, I wouldn't be qualified to give you a good answer, since I don't work in Heegaard Floer homology.
Apr
8
comment Why is Cartan Formula just an avatar of Leibniz rule?
Thank you for the bounty. I am grateful to you for the video.
Apr
8
comment Vector fields as section of tangent bundle
and what is $T_xM$ for you? I think this is what jerrysciencemath wanted to know.
Apr
8
answered Structure on manifolds
Apr
8
comment Structure on manifolds
I don't understand what your second question means. Can you give some more details of what you are reading and/or looking to have clarified?
Apr
7
comment Why is Cartan Formula just an avatar of Leibniz rule?
@DamienL: I've written some details. I hope this helps explain that identity.
Apr
7
revised Why is Cartan Formula just an avatar of Leibniz rule?
added a proof of the key identity
Apr
4
comment Floquet's Theory, Hills Equation
Have you looked at the wikipedia page? en.wikipedia.org/wiki/Floquet_multiplier (Of course, to apply what they write, you will need to convert your second order equation to a first order system.)
Apr
4
answered Why is Cartan Formula just an avatar of Leibniz rule?
Apr
1
answered Algebraic interpretation of Lyapunov functions
Mar
29
comment Why is Cartan Formula just an avatar of Leibniz rule?
Thanks for the link to the video! Also, what Martin said is the answer.
Mar
29
comment What is the sense of the equality sign =?
Very nice explanation. I can go delete mine now.
Mar
27
comment Winding number of vector field on surface
It's not more general... what I described above really uses the fact that the structure group of the tangent bundle to an oriented surface can be reduced to U(1). This is not true for a general manifold. In some sense, what I describe can be generalized to a discussion of Maslov classes of Lagrangians in symplectic manifolds... but that's much fancier than what you are interested in.
Mar
26
answered Winding number of vector field on surface
Mar
26
comment Winding number of vector field on surface
Do you know anything about the zeros of these vector fields? These can make the discussion a little tricky.
Mar
22
comment Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$
My hint would be to think about what a derivation has to be, if you look at it in a coordinate chart. Another hint is to remember that you have cut-off functions and partitions of unity.
Mar
22
comment Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$
The existence of $\chi$ is key in differential geometry. See for instance en.wikipedia.org/wiki/Bump_function.
Mar
20
comment How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?
Another idea would be to take a Morse function with a unique minimum. Then, take the union of all the ascending manifolds from higher index critical points. This is of codimension 1 and thus of measure 0. I'm pretty sure the complement is the ascending manifold of the minimum and thus diffeomorphic to a ball. Oh, wait, that's the link that Jason gave. Sorry.